Factorization of the Tenth and Eleventh Fermat Numbers...

27
TR-CS-96-02 Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent February 1996 Joint Computer Science Technical Report Series Department of Computer Science Faculty of Engineering and Information Technology Computer Sciences Laboratory Research School of Information Sciences and Engineering

Transcript of Factorization of the Tenth and Eleventh Fermat Numbers...

Page 1: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

TR-CS-96-02

Factorization of the Tenth andEleventh Fermat Numbers

Richard P. Brent

February 1996

Joint Computer Science Technical Report Series

Department of Computer ScienceFaculty of Engineering and Information Technology

Computer Sciences LaboratoryResearch School of Information Sciences and Engineering

Page 2: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

This technical report series is published jointly by the Department ofComputer Science, Faculty of Engineering and Information Technology,and the Computer Sciences Laboratory, Research School of InformationSciences and Engineering, The Australian National University.

Please direct correspondence regarding this series to:

Technical ReportsDepartment of Computer ScienceFaculty of Engineering and Information TechnologyThe Australian National UniversityCanberra ACT 0200Australia

or send email to:

[email protected]

A list of technical reports, including some abstracts and copies of some fullreports may be found at:

http://cs.anu.edu.au/techreports/

Recent reports in this series:

TR-CS-96-01 Weifa Liang and Richard P. Brent. Constructing the spannersof graphs in parallel. January 1996.

TR-CS-95-08 David Hawking. The design and implementation of a paralleldocument retrieval engine. December 1995.

TR-CS-95-07 Raymond H. Chan and Michael K. Ng. Conjugate gradientmethods for Toeplitz systems. September 1995.

TR-CS-95-06 Oscar Bosman and Heinz W. Schmidt. Object test coverageusing finite state machines. September 1995.

TR-CS-95-05 Jeffrey X. Yu, Kian-Lee Tan, and Xun Qu. On balancingworkload in a highly mobile environment. August 1995.

TR-CS-95-04 Department of Computer Science. Annual report 1994.August 1995.

Page 3: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

����������� ������ ��� ������������������� ������������� ��������!���"#���$��%&"(')��&*

+-,/.1032-+547698;:1+5<>=3?

@BA CEDGFIHGJ D 8LK�M3NOMQP/RQSUTWVGMYXUZ M5R\[^]`_OabMEXUM5cedfRQXU[gSUTWhidjXUTW[^k�[fc9XUZ M-XUM\klXUZmdfkIN!M\aWM\noM\klXUZmp MQSU]!djXqksr ]�VGMQSUP\8?qZOMtXUM\klXUZ�p MQSU]!djXukvrO]wVGMQS3TbP�d�_OSU[sNvr RQX�[fc�cx[^rvSw_vSUTW]`MwcedfRQXU[gSUP�yLTWXUZ�zs{q|i}s{>~o}�dfkIN��g�^��NOMQR\TW]!dgaNvTb�gT�XUP\81?qZOMYM\aWM\n^M\klXUZ�p MESU]!dfXqksrO]wVGMQS�TWPLd�_OSU[sNvr RQXq[gc�� noM-_OSUTW]`M5cxdgRQXU[fSUPqyLTWXUZ��v{O�v{ �l|^{I�^��dfkINm�g�f~NvM\R\TW]!dga;NOTW�gTWXUP\8!K�M`dgaWP/[�k [fXUM!d�k MEy��^�j��NvM\R\TW]!dga�NOTW�^T�XucedfRQXU[gS�[gc1XUZ MtXUZ T�S/XUM\MQksXUZ�p MQSU]!djXuksr ]wVGMESi8?qZOTbP5ksr ]wVGMESLZIdgPLcx[grOSY�vkO[iyLk�_OSUTW]`M3cedfRQXU[gSUP3dgkIN�d!�f�^�s|Q��NOM\RQTb]!dfa�NOTW�gTWXYR\[^]`_G[gP/T�XUMucxdgRQXU[fSi8525aWa�XUZOMkOMQy�cxdgRQXU[fSUP5SUM\_G[fS/XUMiN�ZOMQSUMByqMQSUMBcx[gr k N�Vl��XUZOMBM\aWaWTb_vXUTWR�R\rvSUnoMB]`MQXUZO[sN���<�.����E8�?qZ MB~^}f��NOTW�^T�X5cedgREXU[gS[fc�XUZOMLXUMQksXUZ!pOMQSU]!dfX�ksr ]wVGMES;yqdgP1c�[^rOkIN!dfc�XUMQS�dfVG[^rOX5|\~^}w�m� [^_O���lM\dfSUP1[fc�RQ[^]`_ rvX�dfXUTW[^k�89K)M-NvTbP/RQr P/PdfP/_GM\RQXUP1[fc�XUZOML_OS�dgREXUTbR\dgaGTW]`_ aWM\]`M\klX�djXUTb[gkt[gc�<�.���{sTWk R\aWrINvTWk �BXUZ MLr P/M-[gc�P/_GM\R\TbdfaW��_OrOSU_G[^P/M-ZIdjS�NvyqdfSUM^{dfkINmkO[gXUMBP/M\n^MQS�dga�[gXUZOMQS-a�djSU�^MBcedfRQXU[gSUPLcx[gr k NmSUM\R\M\klXUa���Vl��<�.���8

�I���f -¡1¢q£-¤-¥Y¦�¡1§¨£L ©�ªG«�¬®­�ª�­>­�¯l°�¬I±j²e³G¯´²�­�±i¯l°G¯l«!µB¶�±j·>¯�µq¸U±j·�¹-ºl»g¼�½G¾�¿>À9¼®Áfºo»`²�Â`Ã5Ä)ÅÇÆIÈiÉ�Ê �I�uË ±!²�ÂtÌ9­�ªvÍ�­Î±j·;¬I±�Ã-Ä�²�Â

Ï «f²�Ð�¯�ÑÒªG«!Ó�Ô7µÕÔ7Ö�¶>¬G­>×ÎØoª�Ð Ï ª�Âj²�±i¯�ÑÒªG«�Ù�Ô7µÚÔÛÆGÜ ��ÝmÞ Âjª�¶�ÑÒªG«`µÚßÛÆ�¶�±j·�¯�Ѩ¬GØo±iªG«fÂtªGÑ5Ã5Ä®¬I«j¯�ªGÑ5±j·�¯ÑÒªG«fÐ

à Æ ÄGá È Ê ��â ã\�vä

Ë ­ �vå ÜGÆ�æ�ç Þ ¯l«�è�ÜGé�¶ Ï ��� ÓIÖIê-Ñ�ª�ç;­>×�±j·>¬I±�é Ö � ÅëÙmìlÆGíYÊ � ²�Â`¬)Ѩ¬GØo±iªG«!ªGÑ3Ã5î ¶�±j·�ç>Â`×>²�Â Ï «jªv³9²�­�°)©�¯l«fЮ¬I±lïWÂð ¯ Þ ²e¯lÑ�è�ÜGñ�¶ Ï>Ï � ÆIÓ�évò�ÆIÓ�ñOêB±j·>¬I±�¬ Þ�Þ Ã5ÄάI«j¯ Ï «f²�Ð�¯ � æ�ç Þ ¯l«`¬ Ï>Ï ¬I«j¯s­�± Þ�ó ç>Âj¯s× ±i«f²�¬ Þ ×>²�³�²�Âf²eª�­ ð ó Ï «f²�Ð�¯sÂtªGѱj·�¯`ÑÒªG«fÐôé Ö à Ê � ¶�Âj¯l¯õè�Ü å ¶ Ï � ÜGÜOê �uö ª´©�¯l«fÐ�¬I± Ï «f²�Ð�¯sÂ Þ ¬I«f°G¯l«u±j·>¬G­�Ã5÷!¬I«j¯`Ì�­>ªvÍ�­�¶�¬G­>×�¬ Ï «jª ð ¬ ð ² Þ ²�Âi±j²�جI«j°�ç>Ð�¯s­�±BÐ�¬IÌG¯sÂB²e± Ï Þ ¬Gç;Âj² ð Þ ¯B±j·;¬I±Yª�­ Þeó ¬mø�­>²e±i¯w­�ç;Ð ð ¯l«YªGÑ�Ã5Ä ã Ï ¯l«g·>¬ Ï Â5ª�­ Þeó ÃYùIú âlâlâ újÃ5÷ ä ¬I«f¯ Ï «f²�Ð�¯ �û ·�¯uØoª�Ð Ï Þ ¯l±i¯BѨ¬GØo±iªG«f²eüs¬I±j²eª�­õªGÑ�±j·>¯�©�¯l«fÐ�¬I±5­�ç>Ð ð ¯l«fÂ�ÃYýIújà í ú âlâlâ ·>¬G ð ¯l¯s­�¬!Øg·>¬ Þ�Þ ¯s­�°G¯uÂf²�­>Øo¯uæ�ç Þ ¯l«sïW±j²�Ð�¯ ��þ ¯sØl¬Gç>Âi¯�±j·�¯�Ã5Ä&°G«jªOÍô«f¬ Ï ²�× Þeó ²�­ÕÂj²eül¯G¶5¬ Ð�¯l±j·�ª9×�Ím·>²�Øf·ÿѨ¬GØo±iªG«fÂõÃ5Ä�Ð�¬ ó ð ¯�²�­>¬G×>¯���ç>¬I±i¯)ÑÒªG«

à ÄIá�� � Ë ­ ±j·>²�Â)Âi¯sØo±j²�ª�­ Í�¯Î°�²e³G¯&¬ÕÂjç>ЮÐ�¬I« ó ªGÑmÍ�·>¬I±�·>¬G ð ¯l¯s­ ¬GØf·>²�¯l³G¯s× Âj²�­>Øo¯ �vå ÜGÆ � Ý ×>×;²e±j²eª�­>¬ Þ·>²�Âj±iªG«f²�Øl¬ Þ ×>¯l±j¬G² Þ Â�¬G­>×�«f¯lÑ�¯l«j¯s­;Øo¯sÂ�Øl¬G­ ð ¯õÑÒª�ç>­>×�²�­ è�ÆIÓ�¶�Ö�Ù�¶qÙGÜ êU¶q¬G­>×ÿÂiª�Ð�¯)«j¯sØo¯s­�±m«f¯sÂjç Þ ±jÂ�¬I«j¯�°�²e³G¯s­²�­ è�ÆGé9¶1Æ å ¶�Ö�ÓIê �Ë ­&±j·�¯�Ñ�ª Þ�Þ ªvÍ�²�­�°�¶��;Ä ×�¯s­�ªG±i¯sÂ�¬ Ï «f²�Ð�¯�­�ç>Ð ð ¯l«�Í�²�±j·&µ ×�¯sØl²�Ð�¬ Þ ×;²e°�²e±jÂl¶>¯ � ° � ����Å � éGÜ �� ²�Ð�² Þ ¬I« Þ�ó ¶

Ä�×�¯s­>ªG±i¯sÂ`¬�Øoª�Ð Ï ª�Âj²e±i¯´­�ç>Ð ð ¯l«tÍ�²e±j·�µÚ×�¯sØl²�Ю¬ Þ ×;²e°�²e±jÂl¶9¯ � ° � ÷mÅ �vå Æ�� ��Ë ­ Øl¬GÂi¯sÂ`Ím·�¯l«j¯�ª�­�¯�Ѩ¬GØo±iªG«Øl¬G­ ð ¯wÑ�ª�ç>­;× ð ó ×>²e³9²�Âj²eª�­�¶ Íw¯tç;Âjç>¬ Þ�Þeó Í!«f²e±i¯t²e±B¬GÂ�� Ä ªG« Ä ¬G­>× Þ ¯s¬s³G¯!²�±jÂBØoª�Ð Ï ç�±j¬I±j²eª�­®¬GÂB¬G­)¯� �¯l«fØl²�Âi¯ �©�ªG«!¯� �¬GÐ Ï Þ ¯G¶���ªG«f¬G²�­Úè�éGÜ êLÂj·>ªvÍw¯s× ±j·>¬I±!Æ �iî���� Ê � ÅëÜ!ì�����ù ý î �BË ±tÍwª�ç Þ × ð ¯m±i¯s×>²�ª�ç>Âw±iª�Í!«f²e±i¯�ª�ç�±t±j·�¯� Ó�éGÙO¸�×>²e°�²�± Ï «f²�Ð�¯�²�­ ×�¯sØl²�Ð�¬ Þuã ±j·�ª�ç>°�·&¯s¬G ó ²�­ ð ²�­>¬I« ó>äg�Ë ­ � ñGñIÓ9¶��-¬G­>×�« ó è�ÙIÓIê1ÑÒ¬GØo±iªG«f¯s×�à ý ÅëÆ å Ö �våGå ì����U÷ â �-¬G­>×>« ó ïWÂ�Ð�¯l±j·�ª9×�Ít¬GÂ�­�¯l³G¯l« Ï ç ð Þ ²�Âj·�¯s×�²�­�Ѩç Þ�Þ ¶

ð ç�±�� ² Þ�Þ ²�¬GÐ�Â�è�ñGÆIê-·>¬GÂ!¬I±i±i¯sÐ Ï ±i¯s×�±iª®«j¯sØoª�­>Âi±i«fç;Øo±�²e± �

|i�g�v|���� �"!$#&%��'�)()*,+�-/.$021�#3*,�54768�'+3+3( 9:*3�'�)();'<>=B|^|,?3}o�l{�|g|i:qzg�v{!|^|&?u�g��@BAsM\R\[^k N djS/� |g|Q��}g~v{`|^|\23�s|g{!|^|,?�|^|g{`|^|,?�|i�s{|\~g0B�g�s{G�^�C?�|i}v{I�^zEDB�^�s8

FG#&H�I:; JLK'+M� <�KON/!�JP�'+,#&+Q=-R\[^]`_OrOX�dfXUTW[gkIdgaLksr ]�VGMQStXUZOM\[gS/�l{�.1r k kOTWk �^Z dg]#_vSU[ RÒM\RQXi{-<�.���{qM\aWaWTW_OXUTWR�R\rvSUnoM�]`MQXUZO[sN�{cedfRQXU[gSUTWhidjXUTW[^k�{Gp MQSU]!dfXLksrO]wVGMQSi{TS�U�V^{$S�ULUi{$S�UXWg{$S�U�Y^{ TWklXUM\�gMQSLcedgREXU[gSUTWhidfXUTW[gk�8.1[^_l�sSUTW�^ZlX RZ |\�^�g�v{I+u8 6�8O:�SUMQksX ?;�v_GM\P/MQXLrOP/TWk �\[^]`_��badcE? <Te 8

Page 4: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

È ������������� �

Ë ­�±j·�¯ Ï ¯l«f²eª9× � ñ åGå ò � � å Ö�¶YÂi¯l³G¯l«f¬ Þ ÂjÐ�¬ Þ�Þ Ñ¨¬GØo±iªG«fÂ�ªGÑ�Ã5Ä Ñ�ªG«�³I¬I«f²eª�ç>Âmµ ß ��Íw¯l«j¯)Ñ�ª�ç;­>× ð ó ±j¬IÌ9²�­�°¬G×�³I¬G­�±j¬I°G¯�ªGÑ5±j·>¯´Â Ï ¯sØl²�¬ Þ Ñ�ªG«gÐ ã\�vä ªGÑY±j·�¯sÂi¯´Ñ¨¬GØo±iªG«f � ©>ªG«�¯� �¬GÐ Ï Þ ¯G¶>²�­ � �IÓ�Ü �Ú¯sÂi±i¯l«f­ è�Æ�� êLÑ�ª�ç;­>×αj·�¯Ñ¨¬GØo±iªG« � í ÅëÆ Ö�Æ Ö�ñGÜGÜ)ÅëÜ å ì Æ

� ý Ê � ªGÑ3Ã�� ��� ±j·�¯l«!¯� �¬GÐ Ï Þ ¯sÂ!Ю¬ ó ð ¯�ÑÒª�ç>­>× ²�­ è Ö � ¶ û ¬ ð Þ ¯õÆOê �� ²e°�­;²xø�Øl¬G­�±mѨç�«j±j·�¯l« Ï «fªG°G«j¯sÂjÂ�Ít¬GÂ�ª�­ Þeó Ï ª�ÂjÂf² ð Þ ¯�Í�²e±j·ÿ±j·�¯�×�¯l³G¯ Þ ª Ï Ð�¯s­�±�ªGÑw±j·�¯�×>²e°�²e±j¬ Þ Øoª�Ð Ï ç�±i¯l«

¬G­>×&Ð�ªG«j¯�¯���Øl²e¯s­�±�¬ Þ °GªG«f²�±j·>Ð� �BË ­ � � å Ó9¶:��ªG«j«f²�Âiª�­ ¬G­>× þ «f² Þ�Þ ·>¬I«j±�è�é ÖGê-Ѩ¬GØo±iªG«j¯s×à í ÅëÙ��Gé Ö �GÙGñ�� � Æ å Ö � å Æ �vå ì�� ÈiÈ

ð ó ±j·�¯�Øoª�­�±j²�­�ç�¯s× ÑÒ«f¬GØo±j²eª�­�Ð�¯l±j·�ª9× �uû ·�¯s­�¶>²�­ � �GñIÓ9¶ þ «j¯s­�±�¬G­>×��Yª Þ�Þ ¬I«g× è �vå ê-Ѩ¬GØo±iªG«j¯s×Ã���Å � ÆGÜGñ��GÆGéGÜGé � ÙGÙGÆGñ�� å ì�� ý È

ð ó ¬ Ð�ª9×>²xø�Øl¬I±j²�ª�­�ªGÑ��Yª Þ�Þ ¬I«f×�ïWÂ��\«f·>ª���Ð�¯l±j·�ª�×7è�Ù�¶�é å ê ��û ·>¯�Ð�ª9×>²xø1Øl¬I±j²eª�­Õ²�Ð Ï «jªO³G¯s×ÿ±j·�¯�¯���Øl²e¯s­>Ø óªGÑ��5ª Þ�Þ ¬I«f×�ïWÂtªG«f²�°�²�­>¬ Þ ¬ Þ °GªG«f²e±j·;Ð ¶�Ñ�ªG«�ÑÒ¬GØo±iªG«fÂ�ªGÑ5±j·>¯�Ñ�ªG«fÐ ã\�vä ¶ ð ó ¬�«f¬I±j²�ª�Øoª�­��\¯sØo±jç>«j¯s× ±iª ð ¯�ªGÑ5ªG«g×�¯l«Æ Ä�� È �Oµ"!�ÑÒªG« Ã���±j·�¯ÿ«f¬I±j²eª ²�Â�¬ ð ª�ç>±Îé � û ·�¯ Þ ¬I«j°G¯l«ÎѨ¬GØo±iªG« � ý È ªGÑõÃ#��Íw¬GÂ�ø;«fÂj± Ï «jªO³G¯s× Ï «f²�Ð�¯ ð ó� ² Þ�Þ ²�¬GЮÂ�è �vå ¶%$jÖIê5ç>Âj²�­�°)±j·�¯´Ð�¯l±j·�ª9× ªGÑ � ² Þ�Þ ²�¬GÐ�Ât¬G­>×�&�ç>×>× è�ñGÜ ê �û ·�¯w«f·�ª�Ð�¯l±j·�ª9×)Øoª�ç Þ ×�·>¬s³G¯`Ѩ¬GØo±iªG«j¯s×�à í ã Í�²e±j·)¬ Þ ²e±i± Þ ¯wÐ�ªG«j¯`×>²'��Ølç Þ ± ó ±j·>¬G­)Ã#�I¶�Âi¯l¯´è �vå ¶ û ¬ ð Þ ¯`ÆOê ä²eÑ�²e± ·>¬G× ð ¯l¯s­ë²�­�³G¯s­�±i¯s×ë¯s¬I« Þ ²e¯l« � � ²�Ð�² Þ ¬I« Þeó ¶`±j·�¯ Ð�ç Þ ±j² Ï Þ ¯o¸ Ï ª Þ�ó ­�ª�Ю²�¬ Þ ��ç>¬G×�«f¬I±j²�ØÕÂj²�¯l³G¯ ã �(��) �>ä

Ð�¯l±j·�ª9×Ûè å ñIêU¶YÍm·>²�Øf· ²�ÂõØlç�«f«j¯s­�± Þ�ó ±j·�¯ ð ¯sÂi±*�\°G¯s­�¯l«f¬ Þ ¸ Ï ç�« Ï ª�Âi¯+��Ð�¯l±j·�ª9× ÑÒªG«�Øoª�Ð Ï ª�Âj²e±i¯�­�ç;Ð ð ¯l«fÂ�ªGÑç Ï ±iª®¬ ð ª�ç�± � ÓGÓ�×�¯sØl²�Ð�¬ Þ ×>²e°�²e±jÂl¶�Øoª�ç Þ ×�·;¬s³G¯´Ñ¨¬GØo±iªG«j¯s× ð ªG±j· Ã í ¬G­;× Ã��I¶ ð ç>±t²�±wÍt¬GÂ!­�ªG±`¬v³ ¬G² Þ ¬ ð Þ ¯�²�­� �GñIÓ �

�LªG°�²�Øl¬ Þ�Þeó ¶�±j·�¯�­�¯� 9±�Âi±i¯ Ï ¬IÑÒ±i¯l«�±j·>¯!ÑÒ¬GØo±iªG«g²eüs¬I±j²eª�­�ªGÑqÃ#�`Ít¬GÂu±j·�¯�ÑÒ¬GØo±iªG«f²�üs¬I±j²eª�­®ªGÑ�à � �3Ë ±uÍt¬GÂuÌ9­�ªOÍ�­±j·>¬I±®Ã �ÿÅ Æ Ö�Æ Ö�ñGÜGÜ�ì �U÷,� �7û ·>¯ � Ö�ñO¸�×>²e°�²e±®Øoª�Ð Ï ª�Âf²e±i¯&­�ç;Ð ð ¯l«�«j¯sÂj²�Âi±i¯s×7¬I±i±j¬GØjÌ ð ó Ð�¯l±j·�ª9×>Â�Âjç>Øg·¬GÂ-�Yª Þ�Þ ¬I«f×Ú«f·�ª�¶#�Yª Þ�Þ ¬I«f× �/. � ¶3¬G­>×Ú±j·�¯®¯ Þ�Þ ² Ï ±j²�Ø�Ølç>«j³G¯®Ð�¯l±j·�ª9× ã æ�0\� ä ¶5Í�·>²�Øg·ÕÍ�ª�ç Þ × ·>¬v³G¯®ÑÒª�ç>­>×�iÂjÐ�¬ Þ�Þ � Ѩ¬GØo±iªG«f � Ë ±õÍt¬GÂõ±iª�ª Þ ¬I«f°G¯®±iª�Ѩ¬GØo±iªG« ð ó ±j·�¯�Øoª�­�±j²�­�ç�¯s× Ñ�«g¬GØo±j²eª�­�Ð�¯l±j·�ª�× ªG«�²e±jÂ�Âjç>ØlØo¯sÂjÂjªG«s¶�1��) �1�1û ·�¯�×>²'��Ølç Þ ± ó Íw¬GÂmø�­>¬ Þ�Þeó ªO³G¯l«fØoª�Ð�¯ ð ó ±j·�¯�²�­�³G¯s­�±j²eª�­�ªGÑu±j·�¯ ã Â Ï ¯sØl²�¬ Þ�ä ­�ç>Ð ð ¯l«�ø;¯ Þ ×�Âj²e¯l³G¯ã3�9ö © �>ä ¶ ð ¬GÂi¯s×ÿª�­ÿ¬�­�¯lÍ ²�×�¯s¬®ªGÑ2�Yª Þ�Þ ¬I«f× è�ÙGÆ ê ��Ë ­ � ���IÓ9¶��-¯s­>Âi±i«f¬9¶ �L¯s­;Âi±i«f¬9¶ ��¬G­;¬GÂjÂi¯�¬G­>×1�5ª Þ�Þ ¬I«f×�¶Í�²e±j·�±j·�¯m¬GÂjÂf²�Âi±j¬G­>Øo¯mªGÑLÐ�¬G­ ó Øoª Þ�Þ ¬ ð ªG«f¬I±iªG«fÂ�¬G­;×®¬ Ï>Ï «fª� �²�Ð�¬I±i¯ Þeó®å ÓGÓ�Í�ªG«jÌ9Âi±j¬I±j²eª�­;Â�ÂjØl¬I±i±i¯l«f¯s×�¬I«fª�ç>­>×±j·�¯´ÍwªG« Þ ×L¶9Øoª�Ð Ï Þ ¯l±i¯ Þeó ÑÒ¬GØo±iªG«f¯s×�à � ð ó �9ö © � è�ÙGÜ9¶;Ù ÖGê ��û ·�¯�ÑÒ¬GØo±iªG«g²eüs¬I±j²eª�­�²�Â

à ��ÅëÆ Ö�Æ Ö�ñGÜGÜõì å Ö�ÙGÙGéIÓ�ÆGñGÆGÙGé Ö å ñGñ Ö�ÆIÓ�ñGÜ�Ü å Ü��GÙ å Ü�éGÆGÓGÓ�Ö�ÙIÖ � � ñ å ñGÜ�ÜGé�éGÜIÖ�ÆGé�Ù å ìQ�:��� âà ��ù ·;¬G ð ¯l¯s­ÿ¬3�iÐ�ª�Âi±´Ít¬G­�±i¯s×� ­�ç>Ð ð ¯l«´²�­ÿ³ ¬I«f²�ª�ç>Â Þ ²�Âi±jÂ�ªGÑtØoª�Ð Ï ª�Âj²e±i¯�­�ç;Ð ð ¯l«fÂ�¯l³G¯l«õÂf²�­>Øo¯)±j·�¯

Ѩ¬GØo±iªG«f²eüs¬I±j²eª�­®ªGÑqà �`²�­ � ���IÓ9¶�¬G­>×�²�­)«j¯sØo¯s­�± ó ¯s¬I«fÂ�²�±u·>¬GÂuç>Âjç;¬ Þ�Þeó ð ¯l¯s­ ¾549º�Ð�ª�Âi±�Íw¬G­�±i¯s×�­�ç>Ð ð ¯l« ã Âi¯l¯G¶ÑÒªG«�¯� �¬GÐ Ï Þ ¯G¶9±j·�¯ Þ ²�Âi±w²�­76 Ï ×>¬I±i¯�Æ � �´ªGÑ�è�ÆIÓGê äg� à ��ù�Íw¬GÂ Ï «fªv³G¯s×ÎØoª�Ð Ï ª�Âj²e±i¯�²�­ � �GÙGÆ ð ó98 ª ð ²�­>Âiª�­ÿè å�� êU¶ç>Âj²�­�°9�;:¯ Ï ²�­LïWÂ`±i¯sÂi±�ª�­�±j·�¯ � � Ý 0 ��Ý ÂjÐ�¬ Þ�Þ Ñ¨¬GØo±iªG«s¶1Ö�ÙGÙ��GÆGÙ åGå ¶�Íw¬GÂmÑ�ª�ç;­>× ð ó � ¯ Þ Ñ�«g²�×�°G¯�è å Æ ê5²�­ � �GÙGÜã ¬ Þ Âiª�ª�­&±j·�¯ � � Ý 0 äg��Ý ­�ªG±j·�¯l«�ÂjÐ�¬ Þ�Þ ÑÒ¬GØo±iªG«s¶�é Ö�ñ å Ó�Ü � ñIÓ>��¶-Ít¬GÂmÑ�ª�ç>­;× ð ó þ «f² Þ�Þ ·>¬I«j±õè � � êY²�­ � �GéGÆ®ª�­¬G­ Ë\þ � å ÓIÖ ��þ «f² Þ�Þ ·>¬I«j± Þ ¬I±i¯l«õÑÒª�ç>­>× ±j·>¬I±õ±j·>¯®ØoªGÑÒ¬GØo±iªG«�Ít¬GÂ�¬�Æ�� � ¸�×>²e°�²e±�Øoª�Ð Ï ª�Âj²e±i¯ � û ·�ç>Âl¶3²e±´Ít¬GÂÌ9­�ªvÍ�­&±j·>¬I±�à ��ù�Å Ö�ÙGÙ��GÆGÙ åGå ìOé Ö�ñ å Ó�Ü � ñIÓ>��ì È �E� �û ·>²�Â Ï ¬ Ï ¯l«�×�¯sÂjØo«f² ð ¯sÂY±j·�¯�Øoª�Ð Ï Þ ¯l±i¯!Ѩ¬GØo±iªG«f²eüs¬I±j²eª�­®ªGÑ�à ��ù � 6�Âj²�­�°�æ�0�� Íw¯!ÑÒª�ç>­>×�¬�Ö�Ó ¸�×;²e°�²e±BѨ¬GØo±iªG«�;÷jù�Å Ö�éGÙ�� åGå Ù å ñGÙGÆGÆIÓGÓ � ñGÙ Ö�ÜGÆ�é Ö�ÙGé�Ó å Ö�ÜIÓ å é åGå ñ � �GÆ�ñ�� å ª�­ � Øo±iª ð ¯l«�ÆIÓ9¶ � ���GÙ ��û ·>¯�ÆGÙGÆO¸�×>²e°�²�±mØoªGѨ¬GØ^¸±iªG« È �E��� �;÷jù Ï ¬GÂjÂi¯s× ¬ Ï «jª ð ¬ ð ² Þ ²�Âi±j²�Ø Ï «f²�Ð�¬ Þ ²�± ó ±i¯sÂi±�¬G­>×�Íw¬GÂ�Âiª�ª�­ Ï «jªO³G¯s×�±iª ð ¯ Ï «f²�Ð�¯�ç>Âj²�­�°�±j·�¯Ð�¯l±j·�ª9×ÿªGÑ Ý ±iÌ9²�­Ú¬G­>× ��ªG«f¬G²�­ ã ð ¬GÂi¯s×�¶L¬ Ï;Ï «jª Ï «f²�¬I±i¯ Þeó ¶qª�­�¯ Þ�Þ ² Ï ±j²�Ø�Ølç�«j³G¯s äg� �-¬I±i¯l«s¶-¬ÎÐ�ªG«f¯�¯ Þ ¯sÐ�¯s­9¸±j¬I« ó Ï «jª�ªGÑ�Ít¬GÂ�ÑÒª�ç>­>×�¶wç>Âj²�­�° � ¯ Þ ÑÒ«f²�×>°G¯GïWÂ1�<0`ç ð ¯ 8 ª�ªG± û ·�¯lªG«j¯sÐ9� ã Âi¯l¯=$'� äg�7û ·�ç;Âl¶u±j·�¯&Øoª�Ð Ï Þ ¯l±i¯Ñ¨¬GØo±iªG«f²eüs¬I±j²eª�­�²�Â

à ��ùmÅ Ö�ÙGÙ��GÆGÙ åGå ìOé Ö�ñ å Ó�Ü � ñIÓ>��ìlÖ�éGÙ�� åGå Ù å ñGÙGÆGÆIÓGÓ � ñGÙIÖ�Ü�ÆGéIÖ�Ù�éIÓ å Ö�ÜGÓ å é å�å ñ � ��ÆGñ�� å ìQ� È î ÈÍ�·�¯l«f¯\� È î È Å � ÜIÓIÖ�ÜwìlìlìlÆ Ö�Ù åGå��Ë ­>$?$gÆO¸UÖ´Í�¯!×>¯sÂjØo«f² ð ¯tÂiª�Ð�¯`³I¬I«f²�¬G­�±jÂBªGÑ1æ�0\� ¬G­;×)±j·�¯s²e« Ï ¯l«jÑÒªG«fÐ�¬G­>Øo¯G¶�¬G­>×®²�­/$gÙ�Í�¯!×�¯sÂfØo«f² ð ¯`Âiª�Ð�¯²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­>Â�ªGÑ!æ�0\� Í�²e±j· Í�·>²�Øg·ÚÍ�¯�¬I±i±i¯sÐ Ï ±i¯s× ±iªÿÑÒ¬GØo±iªG«�à ��ù®¬G­>× ªG±j·�¯l«�©>¯l«fÐ�¬I±)­�ç>Ð ð ¯l«f �@�¯l±j¬G² Þ Â`ªGÑ3±j·�¯�Ѩ¬GØo±iªG«f²eüs¬I±j²�ª�­&ªGÑBà ��ù ¬I«f¯´°�²e³G¯s­�²�­1$gé �

Page 5: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� �

� ª Ѩ¬I«s¶L±j·>²�Â�Âjç>Ð�Ð�¬I« ó ·>¬G ð ¯l¯s­ Øg·�«jª�­�ª Þ ªG°�²�Øl¬ Þ ¶ ð ç�±�­�ªOÍôÍw¯ ð ¬GØjÌ�±i«f¬GØfÌ�¶ ð ¯sØl¬Gç>Âi¯®Ã ���´Íw¬GÂõØoª�Ð)¸Ï Þ ¯l±i¯ Þ�ó Ѩ¬GØo±iªG«j¯s×�²�­ � �GñGñ�¶3Áfº����G»iº!±j·�¯´Ñ¨¬GØo±iªG«f²eüs¬I±j²eª�­�ªGÑYà �´¬G­>× Ã ��ù �BË ­ Ѩ¬GØo±l¶

à ���wÅÇÜ � � Ö�ñ��´ìT� å Ö�ñ Ö �õì � é å �GñGñGÙGÙGéGÜ Ö �vå éIÓGÖ å Ù � Ü å ìvÜGÙGéIÓ�ñ Ö � �IÓ�é ÖGÖ�ÙGñGÜ�Ü���ÆIÓ�Ù � Ü)ì���î ý ÷Í�·�¯l«f¯M��î ý ÷�Å �vå Ü Ö�éwìlìlìoÜ Ö �våGå��Õû ·�¯�±�ÍwªÕéO¸�×;²e°�²e±õѨ¬GØo±iªG«fÂ�Íw¯l«j¯®ÑÒª�ç>­>× ð ó 0tç;­>­>²�­�°�·;¬GÐ è�ÆIÓ�¶YÜIÓIêt²�­� ñ�����¶�¬G­>×�«j¯sЮ¬G²�­>²�­�°�Ѩ¬GØo±iªG«fÂuÍw¯l«j¯!ÑÒª�ç>­>× ð ó ±j·�¯ Ï «f¯sÂi¯s­�±�¬Gç�±j·�ªG«�²�­ ��¬ ó&� �GñGñ ��û ·�¯�ÙGé ÖI¸�×>²�°�²e±uѨ¬GØo±iªG«Ï ¬GÂjÂj¯s×&¬ Ï «jª ð ¬ ð ² Þ ²�Âj±j²�Ø Ï «g²�Ð�¬ Þ ²e± ó ±i¯sÂi±l¶;¬G­;×&¬�«f²e°GªG«fª�ç>Â Ï «jª�ªGÑ5ªGÑ Ï «f²�Ð�¬ Þ ²e± ó Íw¬GÂ Ï «jªO³�²�×>¯s× ð ó ��ªG«g¬G²�­ã Âi¯l¯*$'� äg�ÎÝ Âõ×>¯l±j¬G² Þ Â�ªGÑt±j·�¯�Ѩ¬GØo±iªG«f²eüs¬I±j²�ª�­ ªGÑ`à ����·>¬s³G¯Î­�ªG± ð ¯l¯s­ Ï ç ð Þ ²�Âf·�¯s×�¶�¬ Ï ¬I«j±�Ñ�«jª�Ð�±�Íwª ð «f²�¯lѬG­>­�ª�ç;­>Øo¯sÐ�¯s­�±jÂõè�ñ9¶ � ÓIêU¶�Íw¯õ×�¯sÂfØo«f² ð ¯�±j·�¯õØoª�Ð Ï ç>±j¬I±j²eª�­&²�­1$ å��û ·�¯�«j¯s¬GÂiª�­ÿÍ�· ó à ��� Øoª�ç Þ × ð ¯�Øoª�Ð Ï Þ ¯l±i¯ Þeó Ѩ¬GØo±iªG«j¯s× ð ¯lÑÒªG«j¯�à � ¬G­;×ÿà ��ù ²�Â�±j·;¬I±m±j·�¯)×>²'��Ølç Þ ± ó ªGÑ

Øoª�Ð Ï Þ ¯l±i¯ Þeó ÑÒ¬GØo±iªG«f²�­�°�­�ç>Ð ð ¯l«g ð ó æ�0\� ²�´×>¯l±i¯l«fÐ�²�­�¯s×�Ð�¬G²�­ Þ�ó ð ó ±j·�¯�Âj²�ül¯®ªGÑ`±j·�¯��lº����I¿���� �e½I»"!�º��o¾Ï «f²�Ð�¯õѨ¬GØo±iªG«´ªGÑ�±j·>¯�­�ç>Ð ð ¯l« �´û ·�¯)Âi¯sØoª�­>×�¸ Þ ¬I«j°G¯sÂi± Ï «g²�Ð�¯õѨ¬GØo±iªG«´ªGÑwà ���´·>¬G´ÆGÆÎ×>²�°�²e±jÂ�¬G­>×Õ²�Â�Ð�ç>Øg·¯s¬GÂj²e¯l«�±iª&ø1­>× ð ó æ�0\� ±j·>¬G­ ±j·�¯®Ö�Ó ¸�×>²�°�²e±õѨ¬GØo±iªG«�ªGÑ!à ��ù�ªG«�±j·�¯®Ö �O¸�×;²e°�²e±õѨ¬GØo±iªG«�ªGÑ!Ã�� � û ·�ç>Âl¶Y±j·�¯×>²'��Ølç Þ ± ó ªGÑwØoª�Ð Ï Þ ¯l±i¯ Þeó Ѩ¬GØo±iªG«f²�­>°�Ã5Ä ð ó ±j·�¯�ÑÒ¬GÂi±i¯sÂj±�Ì9­�ªOÍ�­�Ð�¯l±j·�ª�×Õ²�Â�­�ªG±�¬�Ð�ª�­�ªG±iª�­>²�Ø�ÑÒç;­>Øo±j²eª�­ªGÑ3µ �Ý ð «g²e¯lÑ;Âjç>ЮÐ�¬I« ó ªGÑ;±j·>¯t·;²�Âi±iªG« ó ªGÑ�ÑÒ¬GØo±iªG«g²eüs¬I±j²eª�­)ªGÑ�ÃYîIú âlâlâ újà ���u²�ÂY°�²e³G¯s­)²�­ û ¬ ð Þ ¯ �I� ©�ªG«u¬�Âf²�Ð�² Þ ¬I«

·>²�Âj±iªG« ó ªGÑBà � È ú âlâlâ újà ÈiÈ ¶;Âi¯l¯�è�ÆGÙ9¶ Ï �L� Ö�ñOê �Ë ­�$gñ´Íw¯�ª�ç>± Þ ²�­�¯!¬ Ï «jª��i¯sØo±�±iª�ø�­>×�Ð�ªG«f¯!ÑÒ¬GØo±iªG«gÂ�ªGÑL©>¯l«fÐ�¬I±t­�ç>Ð ð ¯l«f ð ó æ�0\��¶�ç>Âf²�­�°´²�­�¯� Ï ¯s­>Âj²e³G¯Â Ï ¯sØl²�¬ Þ ¸ Ï ç�« Ï ª�Âi¯�·>¬I«f×�Ít¬I«j¯G¶3¬G­>× °�²�³G¯®±j·�¯�ª�­�¯�Âjç;ØlØo¯sÂjÂ�Âjª�Ѩ¬I«´òÚ¬�Æ å ¸�×>²e°�²�±õѨ¬GØo±iªG«�ªGÑ!à ���I¶3Ñ�ª�ç>­;× ²�­&�ç;­�¯ � ���GÙ �

8 ²�°GªG«jª�ç> Þeó Ï «jªO³�²�­�° Ï «f²�Ю¬ Þ ²e± ó ªGÑ�¬õ­�ç>Ð ð ¯l«�¬GÂ Þ ¬I«j°G¯�¬GÂu±j·�¯mÙGé ÖI¸�×;²e°�²e±�Ѩ¬GØo±iªG«wªGÑLà ����²�Âu¬õ­�ª�­�±i«f²e³9²�¬ Þ±j¬GÂiÌ �7Ë ­ $'�ÕÍ�¯�×>²�ÂjØlç>ÂfÂ Ï «f²�Ð�¬ Þ ²�± ó Ï «fª�ªGѨÂ�¬G­>× �iØo¯l«j±j²eø�Øl¬I±i¯sÂ,� ªGÑ Ï «g²�Ð�¬ Þ ²e± ó ÑÒªG«�±j·�¯ Ѩ¬GØo±iªG«fÂ�ªGÑ�Ã Ä ¶µÚÔ �G�I�û ·�¯�ÂjÐ�¬ Þ�Þ ¯sÂj±�©�¯l«fЮ¬I±�­�ç>Ð ð ¯l«�Í�·>²�Øg· ²�Â�­�ªG± ó ¯l±�Øoª�Ð Ï Þ ¯l±i¯ Þeó Ѩ¬GØo±iªG«j¯s× ²�Âõà � È �&Ë ±�²�ÂõÌ�­>ªvÍ�­ è�ÆIÓIê±j·>¬I±

à � È Å �G� Ö�éGñ��´ì ÆGéIÓ �våGå �GÜ´ìOéGÜ å éGéGÙGÆ���ì � �IÓ�Æ å Ö � � � ÜGé � ì � ÆGÙGé � ÜGÆ � Ü Ö � ÆGÙGÙ�é���ì ��� � í âû ·�ç>Âl¶�à � È ·;¬GÂ`¬I± Þ ¯s¬GÂi±!Âi¯l³G¯s­ Ï «f²�Ð�¯mѨ¬GØo±iªG«f ��Ë Ñ5±j·�¯�×>²�Âi±i«f² ð ç�±j²eª�­�ªGÑYÂi¯sØoª�­>×9¸ Þ ¬I«j°G¯sÂi± Ï «f²�Ð�¯mѨ¬GØo±iªG«fÂ`ªGÑÞ ¬I«j°G¯�«f¬G­>×>ª�Ð ²�­�±i¯l°G¯l«fÂ�è Ö å ¶1Ö�ñ ê3²�Â�¬G­ ó °�ç>²�×�¯G¶1²e±m²�Âmç>­ Þ ²eÌG¯ Þeó ±j·>¬I± ��� � í Øl¬G­ ð ¯�Øoª�Ð Ï Þ ¯l±i¯ Þeó ÑÒ¬GØo±iªG«f¯s×ð ó æ�0\� � ©�ªG«�Ѩç�«j±j·�¯l«!×;²�ÂjØlç>ÂjÂf²eª�­�¶9Âi¯l¯ $ � Ó �©�ªG«5ª�­�¯ Ï ª�ÂjÂj² ð Þ ¯Y¬ Ï;Ï Þ ²�Øl¬I±j²eª�­�ªGÑ9Ѩ¬GØo±iªG«f²eüs¬I±j²eª�­;ÂLªGÑ�©�¯l«fÐ�¬I±Y­�ç;Ð ð ¯l«fÂl¶Oª ð Âi¯l«j³G¯�±j·>¬I±L±j·�¯uѨ¬GØo±iªG«f²eüs¬I±j²�ª�­

ªGÑ`Ã5ùIújà �vú âlâlâ újÃ5Ä$# ��²�Âõ­>¯sØo¯sÂjÂj¬I« ó ±iª�×>¯l±i¯l«fÐ�²�­�¯�±j·>¯�Âi±i«fç;Øo±jç�«j¯�ªGÑ!¬Îø�­;²e±i¯)ø;¯ Þ ×ÚÍm²e±j· Æ È É ¯ Þ ¯sÐ�¯s­�±jÂl¶ð ¯sØl¬Gç>Âi¯)±j·�¯�Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²e³G¯´°G«jª�ç Ï ªGÑwÂjç>Øg·Ú¬®ø�¯ Þ ×�·;¬GÂ�ªG«f×�¯l«´ÆIÈ É&% � Å ÃYùlà �jÃ È ìlìlìgÃ5Ä'# � � �Ú¯�­�ªOÍÌ9­�ªvÍDZj·>¯õÂi±i«fç>Øo±jç>«j¯�ªGÑYÂfç>Øf·Îø�¯ Þ ×>ÂtÑÒªG«�Ó�Ô7µÚÔ � Æ �

�I���I���)(+*-,/.103254$687�4:9;4:, <>=@?�û ·;¬G­�Ì�Â�¬I«f¯Î×;ç�¯Î±iªBA�¯s­>×>«f²eÌ �-¯s­>Âi±i«f¬9¶�&G« � ¶uÑÒªG«�±j·�¯&æ�0\� ¬ Þ °GªG«g²e±j·>ÐÍ�·>²�Øf· Ю¬G×�¯�±j·>¯�Ѩ¬GØo±iªG«f²eüs¬I±j²eª�­ ªGÑ!à ��ù ¬G­;×�à ��� Ï ª�ÂjÂf² ð Þ ¯�!B¬G­>× ±iª �Y¯l±i¯l« ��ª�­�±i°Gª�Ð�¯l« ó ¬G­>×CAm²�«jª�Ð�²� ç ó ¬GЮ¬�ÑÒªG«`±j·�¯s²e« Ï «g¬GØo±j²�Øl¬ Þ ²�Ð Ï «fªv³G¯sÐ�¯s­�±jÂt±iª�æ�0\� � &Gª�·>­ �Yª Þ�Þ ¬I«f× Ï «jªv³9²�×�¯s×�Âiª�Ð�¯�ªGÑ5±j·�¯�ÌG¯ ó ²�×�¯s¬GÂÍ�²e±j·�·>²�Â9�b� % � ��¬G­>× �\«f·>ª��®Ð�¯l±j·�ª9×> � &Gª�·;­ þ «f² Þ�Þ ·>¬I«j±l¶ 8 ²�Øf·>¬I«f×10w«f¬G­>×>¬ Þ�Þ ¶�� ² Þ Ñ�«g²�×ED�¯ Þ�Þ ¯l«v¶@mª�­;¬ Þ ×D´­�ç�±j·�¶ &�ª�·>­ � ¯ Þ ÑÒ«f²�×�°G¯Î¬G­>×3@�¬G­>²e¯ ÞB� ·>¬G­�Ì9Â Ï «jªv³9²�×�¯s× ·>²�Âj±iªG«f²�Øl¬ Þ ²�­�ÑÒªG«fÐ�¬I±j²eª�­L¶B«j¯lÑ�¯l«f¯s­>Øo¯sÂl¶�¬G­;×�FOªG«ØoªG«j«j¯sØo±j²�ª�­>´±iªÿ×�«f¬IÑÒ±jÂ�ªGÑt±j·>²�Â Ï ¬ Ï ¯l« ��þ «gç>Øo¯/@�ª9×>Âiª�­�¶ Ý « �i¯s­ �-¯s­>Âi±i«f¬9¶��Y¯l±i¯l« ��ª�­�±i°Gª�Ð�¯l« ó ¶ 8 ª ð ¯l«j±� ² Þ ³G¯l«fÐ�¬G­ ¬G­;× � ¬GÐ � ¬I°�Âj±j¬HGY¶ &G« � Ï «fªv³9²�×�¯s× ²�­�ÑÒªG«fÐ�¬I±j²eª�­Û¬ ð ª�ç>±�ªG±j·�¯l«&¬I±i±i¯sÐ Ï ±jÂαiª ÑÒ¬GØo±iªG«�à ��ù �©�«f¬G­JIØoª�²� ��ªG«f¬G²�­ Ï «jªO³G¯s×�±j·�¯ Ï «g²�Ð�¬ Þ ²e± ó ªGÑ�±j·>¯�ÙGé ÖI¸�×>²e°�²�±�Ѩ¬GØo±iªG«tªGÑLà ��� � 0w«g¬G²e°�æ Þ ×�¯l«fÂj·;¬sÍ Ï ªG«j±i¯s×�Ð óæ�0�� Ï «jªG°G«f¬GÐ ±iª�±j·�¯ Ý � � ÓGÓGÓ � A�¬I«j³G¯ ó @�ç ð ­�¯l« Ï «fªv³9²�×�¯s× ¬(@�ç ð ­�¯l« 0w«fç>­>Øg·�¯l«�¬G­;× ¯s­>Øoª�ç�«g¬I°G¯s×Ð�¯�±iª ²�Ð Ï Þ ¯sÐ�¯s­�±�æ�0\� ª�­ ²�± � @�¯s­>­;²�Â Ý ­>×>«f²eª Þ ª�¬G­;× 8 ª ð ¯l«j± @�ç ð ­�¯l« Ï «jªv³9²�×�¯s× ¬GÂjÂj²�Âj±j¬G­>Øo¯ÎÍ�²e±j·¬GÂ Ï ¯sØo±j´ªGÑw±j·�¯>0t«fç>­>Øg·�¯l«´·>¬I«f×�Ít¬I«j¯�¬G­>× ÂiªGÑ�±EÍw¬I«f¯ � &�ª�·>­ 0t¬G­>­>ª�­Ú¬G­>×BAm¯l«fÐ�¬G­Ú±i¯ 8 ²e¯ Þ ¯ Ï «fªv³9²�×�¯sתG±j·�¯l«t¬GÂjÂj²�Âj±j¬G­>Øo¯�¬G­>×®¯s­>Øoª�ç�«f¬I°G¯sÐ�¯s­�± ��þ ª ðLK ²�­�°Gª Þ ×�°G«f¬GØl²eª�ç> Þ�ó ³Gª Þ ç>­�±i¯l¯l«f¯s×®Â Ï ¬I«f¯�Øoª�Ð Ï ç�±i¯l«tØ ó Ø Þ ¯sª�­Î¬ � Ï ¬I«fØ 0w¯s­�±i¯l«tÆIÓGÓGÓ �wû ·�¯ Ýmö 6 � ç Ï ¯l«fØoª�Ð Ï ç�±i¯l«�©;¬GØl² Þ ²e± ó Ï «jªO³�²�×>¯s×®Øoª�Ð Ï ç>±i¯l«w±j²�Ð�¯�ª�­Î¬�©>ç��j²e±jÂjçM � � ÓGÓ�¬G­>×NM �wÆGÆIÓGÓ'F � Ó9¶�¬G­>×�±j·�¯ Ý�ö 6�¸�©>ç��j²e±jÂjç90 Ý � �u«jª��i¯sØo± Ï «jªv³9²�×�¯s×�¬GØlØo¯sÂjÂ3±iª�¬�©>ç��j²e±jÂjç Ý � � ÓGÓGÓ �

Page 6: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

÷ ������������� ������������ 0wª�Ð Ï Þ ¯l±i¯�ÑÒ¬GØo±iªG«f²�üs¬I±j²eª�­&ªGÑ3Ã-Ä1¶�µ�ÅëÙ�ú âlâlâ ú �G�

µ ©>¬GØo±iªG«f²eüs¬I±j²eª�­ ��¯l±j·�ª9× @�¬I±i¯ 0wª�Ð�Ð�¯s­�±jÂÙ ���wì�� í û «f²�¬ Þ ×;²e³�²�Âj²eª�­ �vå ÜGÆ æ�ç Þ ¯l«é � ý ìQ���U÷ � ¯l¯�±i¯� 9± � ñGñIÓ �-¬G­;×�« óå ��� í ì�� ÈiÈ 0t© 8�Ý 0 � � å Ó ��ªG«j«f²�Âiª�­ ¬G­;× þ «f² Þ�Þ ·;¬I«j±ñ ��� ý ì�� ý È þ ¸ �Û«f·�ª � �GñIÓ þ «j¯s­�±�¬G­>× �Yª Þ�Þ ¬I«f× ã ��� ý úP� ý È ä

� ¯l¯�±i¯� 9± � �GñIÓ � ² Þ�Þ ²�¬GЮ ã Ï «g²�Ð�¬ Þ ²e± ó ªGÑ:��ý È ä� � í ìQ�;÷C�tìQ�:��� û «f²�¬ Þ ×;²e³�²�Âj²eª�­ � �IÓ�Ü �Ú¯sÂi±i¯l«f­ ã � í ä

�9ö © � � ���IÓ �L¯s­;Âi±i«f¬�ºl¾w½�� ã �;÷C�IúP����� ä� Ó � �wì�����ùwì���÷jù�ìC� È î È û «f²�¬ Þ ×;²e³�²�Âj²eª�­ � �GÙGÜ � ¯ Þ ÑÒ«f²�×>°G¯ ã � � ä

û «f²�¬ Þ ×;²e³�²�Âj²eª�­ � �GéGÆ þ «f² Þ�Þ ·>¬I«j± ã ����ù äæ�0�� � ���GÙ þ «j¯s­�± ã � ÷jù úP� È î È ä�G� � ý ìQ�� ý ì�� È �uìQ� ÈiÈ ìQ��î ý ÷ û «f²�¬ Þ ×;²e³�²�Âj²eª�­ � ñ���� 0tç>­;­>²�­�°�·>¬GÐ ã � ý úP�� ý äæ�0�� � �GñGñ þ «j¯s­�± ã � È �lúP� ÈiÈ úP��î ý ÷ äæ�0�� � � �GñGñ ��ªG«f¬G²�­ ã Ï «g²�Ð�¬ Þ ²e± ó ªGÑ:��î ý ÷ ä

Æ �����-¢�§��L -¡���£��������û ·�¯®º � �����q¾��"� �sÀ�» �Iºõ¼®ºl¾54:�� ã æ�0\� ä Íw¬GÂm×>²�ÂjØoªO³G¯l«j¯s× ð ó A � � � �L¯s­>Âj±i«f¬9¶ &G« � è�ÙGÙIê-²�­ � �GñGÙ � Mu¬I«f²�ª�ç>Â

Ï «f¬GØo±j²�Øl¬ Þ «j¯oø�­>¯sÐ�¯s­�±jÂuÍ�¯l«j¯�Âjç�°G°G¯sÂi±i¯s× ð ó ��ª�­�±i°Gª�Ð�¯l« ó è�ÙGñ9¶9Ù�� êU¶ � ç ó ¬GÐ�¬®è�ñIÓIêU¶�¬G­>×�ªG±j·�¯l«gÂmè � ¶ å ¶9ÆGÜOê ��Ú¯ «f¯lÑ�¯l«&±iª(è�Ù Ö>¶�éIÓ�¶ å � ê´ÑÒªG«�¬ °G¯s­>¯l«f¬ Þ ×�¯sÂjØo«g² Ï ±j²eª�­ ªGÑ�æ�0\��¶!¬G­>×ë±iª(è�Æ Ö>¶!ÖGÖ>¶ åGå ê�Ñ�ªG«&«f¯ Þ ¯l³ ¬G­�±ð ¬GØfÌ�°G«jª�ç;­>× �

�L¯s­;Âi±i«f¬9ïW´ÌG¯ ó ²�×�¯s¬ÎÍw¬GÂ�±iª�¬ Ï;Ï Þeó �Yª Þ�Þ ¬I«f×�ïWÂ��b� % � ��Ð�¯l±j·�ª9×Ûè�éGé ê ð ç�±´±iª&ÍwªG«jÌÚªv³G¯l«)¬&×>² G1¯l«j¯s­�±°G«jª�ç Ï! ��Ë Ñ`±j·�¯�Ð�¯l±j·�ª9× Ñ¨¬G² Þ Âs¶Y¬G­�ªG±j·>¯l«�°G«jª�ç Ï Øl¬G­ 𠯮±i«f²e¯s× ��û ·>²�Âõ²�Â�­>ªG± Ï ª�ÂjÂj² ð Þ ¯�Ñ�ªG«�±j·�¯5� % �Ð�¯l±j·�ª9×�¶ ð ¯sØl¬Gç>Âi¯´²�±!ç>Âi¯sÂ!¬)ø �¯s× °G«fª�ç Ï �û ª ð ¯õÂ Ï ¯sØl²xø�ØI¶�Âjç Ï>Ï ª�Âi¯�Íw¯�¬I±i±i¯sÐ Ï ±m±iª®ø�­>×&¬®ÑÒ¬GØo±iªG«�ªGÑB¬®Øoª�Ð Ï ª�Âj²�±i¯�­�ç>Ð ð ¯l«#"�¶;Ím·>²�Øf·�Í�¯�Øl¬G­

¬GÂjÂjç;Ð�¯�­�ªG±`±iª ð ¯�¬ Ï «f²�Ð�¯ Ï ªvÍw¯l«�è�ÙGÜ9¶ $gÆ � ÙOê � �-¯l± � ð ¯�±j·>¯´ÂjÐ�¬ Þ�Þ ¯sÂj± Ï «g²�Ð�¯�ÑÒ¬GØo±iªG«mªGÑ$" �BË ­ Ï «f¬GØo±j²�Øo¯²e±õ²�´×�¯sÂj²e«g¬ ð Þ ¯)±iª�«j¯sÐ�ªv³G¯�ÂfÐ�¬ Þ�Þ Ñ¨¬GØo±iªG«f ã ç Ï ±iª�Âf¬ ó7� Ó ÷ ä ð ó ±i«f²�¬ Þ ×>²�³�²�Âf²eª�­ ð ¯lÑÒªG«j¯®¬ Ï;Ï Þeó ²�­�° æ�0\��¶ð ç�±`Íw¯´ª�­ Þ�ó ­�¯l¯s×&¬GÂfÂjç>Ð�¯\�&% Ü �Ý�Þ ±j·�ª�ç�°�· �Õ²�Âmç>­>Ì�­�ªOÍ�­�¶�Í�¯�×�¯sÂjØo«g² ð ¯´±j·�¯)¬ Þ °GªG«f²e±j·>ЮÂm²�­�±i¯l«fÐ�Â�ªGÑuª Ï ¯l«f¬I±j²eª�­;Âm²�­�±j·�¯�ø�­>²e±i¯´ø;¯ Þ ×

'�Å ! à ã � ä Å � � � �`�sË ­ Ï «f¬GØo±j²�Øo¯uÍw¯uÍwªG«jÌ�Ð�ª�×;ç Þ ª(" ã ªG«YÂiª�Ð�¯l±j²�Ð�¯sÂLÐ�ª9×>ç Þ ª`¬!Ð�ç Þ ±j² Ï Þ ¯5ªGÑ)"�¶O²eÑ�±j·�¯Ð�ç Þ ±j² Ï Þ ¯�·>¬GÂ�¬�Øoª�­�³G¯s­>²�¯s­�± ð ²�­>¬I« ó «j¯ Ï «j¯sÂi¯s­�±j¬I±j²eª�­ ä ¶>¬G­;× ª�ØlØl¬GÂf²eª�­>¬ Þ�Þeó Ï ¯l«jÑ�ªG«gÐ K 0�@#Øoª�Ð Ï ç�±j¬I±j²�ª�­>ÂÍ�·>²�Øf·ÕÍ�² Þ�Þ ×�¯l±i¯sØo±�¬G­ ó ­�ª�­�±i«f²e³9²�¬ Þ ÑÒ¬GØo±iªG«�ªGÑ*" ã Ï «jª ð ¬ ð Þeó �-¶L±j·>ª�ç�°�· Ï ª�ÂjÂj² ð Þ�ó ¬&×>² Gq¯l«j¯s­�±�Ѩ¬GØo±iªG«õªGÑ" äg� �ÚªG«jÌ9²�­�°�Ð�ª�×>ç Þ ª+" ªG«�Ð�ª�×>ç Þ ª ¬ Ð�ç Þ ±j² Ï Þ ¯ ªGÑ,"�Øl¬G­ ð ¯&«f¯l°�¬I«f×�¯s× ¬GÂ�ç>Âf²�­�° ¬ «j¯s×>ç>­>×;¬G­�±«j¯ Ï «f¯sÂi¯s­�±j¬I±j²�ª�­ Ñ�ªG«m¯ Þ ¯sÐ�¯s­�±jÂ`ªGÑ-' �Ë ­7�Yª Þ�Þ ¬I«f×LïW � % � Ð�¯l±j·�ª9×�±j·>¯m°G«jª�ç Ï.! ²�Âu±j·�¯�Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²e³G¯`°G«jª�ç Ï ªGÑ�±j·�¯!ø1­>²e±i¯`ø;¯ Þ ×/' ¶�² � ¯ � ±j·�¯

Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²e³G¯�°G«jª�ç Ï ªGÑB²�­�±i¯l°G¯l«fÂmÐ�ª�×;ç Þ ª � ��Ë ­&æ�0\� ±j·�¯´°G«jª�ç Ï0! ²�Â!×>¯oø�­�¯s× ª�­&¬G­�¯ Þ�Þ ² Ï ±j²�ØmØlç>«j³G¯ �û ·�¯l«f¯õ²�Â!­�ª Þ ª�ÂjÂ!ªGÑ3°G¯s­�¯l«g¬ Þ ²e± ó ²�­�¬GÂjÂjç>Ð�²�­�°�±j·>¬I±�±j·>¯´¯ Þ�Þ ² Ï ±j²�Ø�Ølç�«j³G¯�²�Â`°�²e³G¯s­�²�­ �Ú¯s²e¯l«fÂi±i«g¬GÂjÂm­>ªG«fÐ�¬ ÞÑÒªG«fÐ

1 È Å32 � Ê5462�Ê57mú ã Æ ä

Í�·�¯l«f¯,4�¬G­>×87m¬I«f¯õØoª�­>Âi±j¬G­�±jÂmÂfç>Øf· ±j·;¬I±

Ö94 � Ê Æ å 7 È;:Å Ó ã Ü ä

²�­,' � ! Øoª�­>Âf²�Âi±jÂLªGÑ9±j·�¯wÂi¯l±LªGÑ Ï ª�²�­�±j ã 2-ú 1 ä*< '>=�' Í�·>²�Øf· Þ ²�¯Bª�­õ±j·�¯�Ølç�«j³G¯�¬G­;×´¬7� Ï ª�²�­�±-¬I±5²�­9ø�­>²e± ó �? �3Ý Øoª�Ð�Ð�ç>±j¬I±j²e³G¯w¬G­;×�¬GÂjÂjª�Øl²�¬I±j²�³G¯�°G«jª�ç Ï ª Ï ¯l«f¬I±j²eª�­�²�Â5×>¯oø�­�¯s×´²�­�¬�Âi±j¬G­;×>¬I«f×�Íw¬ ó è å ¶ ÆGÜ9¶GÙGñ�¶ åGå ê �5Ë ­

Page 7: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� î

¬GØlØoªG«f×>¬G­;Øo¯õÍ�²e±j·Î±j·>¯´ç>Âjç>¬ Þ Øoª�­�³G¯s­�±j²�ª�­�Íw¯�Í!«f²e±i¯�±j·>¯�°G«jª�ç Ï ª Ï ¯l«f¬I±j²eª�­�¬G×>×>²e±j²e³G¯ Þ�ó � �uû ·�¯ ã ¬G×>×;²e±j²e³G¯ äül¯l«jª�¯ Þ ¯sÐ�¯s­�±�ªGÑ ! ²� ? �

�L¯l± � Å�� ! � ð ¯�±j·�¯tªG«f×>¯l«YªGÑ ! �YË ­�±j·�¯ � % � Ð�¯l±j·�ª9× � Å � % � ¶ ð ç>±Y²�­)æ�0\� Íw¯t·;¬s³G¯ � Å �uÊ � %�� ¶Í�·�¯l«f¯G¶ ð ó ¬�±j·�¯lªG«j¯sÐ ªGÑ8A�¬GÂjÂi¯Îè Ö�Æ êU¶ � Å � ã ! ä Âj¬I±j²�Âiø;¯sÂ

� È�� Ö�� â ã Ö äþwó ¬Ú«j¯sÂjç Þ ±�ªGÑ @�¯sç�«g²�­�° è�Ü � êU¶t±j·�¯&«f¯sÂjç Þ ± ã Ö ä ²� ð ¯sÂi± Ï ª�ÂjÂj² ð Þ ¯G¶�²�­ ±j·>¯�Âi¯s­>Âi¯�±j·>¬I±�¬ Þ�Þ ²�­�±i¯l°G¯l« � <ã % Æ�� �5úgÊ�Æ�� � ä ¬I«f²�Âj¯�ÑÒªG«�Âiª�Ð�¯&Øg·�ª�²�Øo¯ÎªGÑ 4�¬G­>×57®²�­ ±j·�¯ �Ú¯s²e¯l«gÂi±i«f¬GÂjÂ)ÑÒªG«fÐ ã Æ äg��û ·�¯�­�ç>Ð ð ¯l«�ªGÑØlç�«j³G¯sÂ�Í�²e±j·�°�²�³G¯s­ � Øl¬G­ ð ¯`¯� Ï «j¯sÂjÂi¯s×®²�­�±i¯l«fÐ�Â�ªGÑq±j·�¯ D�«jª�­>¯sØjÌG¯l«tØ Þ ¬GÂjÂ�­�ç>Ð ð ¯l«BªGÑ � È % Ö�� ã Âi¯l¯)è�ÙGÙ ê äg�Ë ­ Ï «f¬GØo±j²�Øo¯G¶B±iªÿ¬v³Gª�²�× Øoª�Ð Ï ç�±j¬I±j²eª�­�ªGÑ!ÂC��ç>¬I«j¯�«jª�ªG±jÂs¶BÍ�¯ÎÂi¯ Þ ¯sØo±�¬ Ï Âj¯sç>×�ªI¸U«f¬G­>×>ª�Ð Ï ¬I«f¬GÐ�¯l±i¯l« 4

¬G­>×�²�­>²e±j²�¬ Þ Ï ª�²�­�± ã 2��oú 1 � ä ª�­&±j·�¯�Ølç�«j³G¯G¶q¬G­>×&±j·>¯s­�Øoª�Ð Ï ç>±i¯ 7�ÑÒ«jª�Ð ã Æ äg�!û ·�¯�Øoª�­>×;²e±j²eª�­ ã Ü ä Øl¬G­ ð ¯Øf·>¯sØjÌG¯s× ð ó ±j¬IÌ�²�­>°�¬ K 0�@ !Y²�­ ±j·�¯�ç>­ Þ ²eÌG¯ Þ�ó ¯l³G¯s­�±)±j·>¬I±�²�±´ÑÒ¬G² Þ Â´±iªÕ·�ª Þ × ±j·�¯s­ ±j·>¯ K 0�@ Ï «jª ð ¬ ð Þeó°�²e³G¯sÂ�±j·�¯�ÑÒ¬GØo±iªG« ��²�ЮÐ�¯s×>²�¬I±i¯ ÞeóG�3Ë ­ Ï «f¬GØo±j²�Øo¯�±j·�¯´±i¯sÂi±�²�Â!ç>Âjç>¬ Þ�Þeó ª�Ð�²e±i±i¯s× �Æ ���I��� < 4���9C. 6/4:25=@? �Ú¯õç>Âi¯�±j·>¯�Í�ªG«f×; �iÐ�ª�×>¯ Þ ��¬G­>×3�\ÑÒªG«fÐ9��²�­�±i¯l«fØf·;¬G­�°G¯s¬ ð ÞeóG��û ·�¯O�Ú¯s²e¯l«fÂi±i«f¬GÂfÂÑÒªG«fÐ ã Æ ä ²�Â�­�ªG±�­>¯sØo¯sÂjÂj¬I«f² Þ�ó ±j·�¯&Ð�ª�Âi±�¯���Øl²e¯s­�±®Ñ�ªG«�Øoª�Ð Ï ç�±j¬I±j²eª�­>¬ Þ Ï ç�« Ï ª�Âi¯s � � ²e±j· ã Æ ä Í�¯&·>¬v³G¯±iª Ï ¯l«jÑÒªG«fÐ�×;²e³�²�Âj²eª�­>Â�Ð�ª9×>ç Þ ª " �Îû ·�¯sÂi¯®¬I«f¯®¯� Ï ¯s­;Âj²e³G¯ ð ¯sØl¬Gç>Âi¯�±j·>¯ ó ²�­�³Gª Þ ³G¯®¬G­ ¯� 9±i¯s­>×�¯s× K 0�@Øoª�Ð Ï ç>±j¬I±j²eª�­ ��û ª&¬s³Gª�²�×Õ±j·�¯sÐ ¶LÍw¯�Øl¬G­Ú«j¯ Ï Þ ¬GØo¯ ã 25ú 1 ä ð ó ã 2%�� �ú 1 �� ä ²�­ ã Æ ä ±iª °G¯l±õ¬Î·�ª�Ð�ªG°G¯s­�¯lª�ç;Â�Ú¯s²e¯l«fÂj±i«f¬GÂjÂ!¯���ç>¬I±j²eª�­

1 È �Å32 � Ê5462� È Ê 7� � â ã Ù äû ·�¯ Ï ª�²�­�±j ã 25ú 1 ú� ä Âj¬I±j²�ÂiÑ ó ²�­�° ã Ù ä ¬I«j¯�±j·�ª�ç�°�·�±õªGÑ!¬G´«f¯ Ï «j¯sÂi¯s­�±j¬I±j²e³G¯sÂ�ªGÑt¯ Þ ¯sÐ�¯s­�±jÂõªGÑ�� È ã ' ä ¶3±j·�¯Ï «jª��i¯sØo±j²e³G¯ Ï Þ ¬G­�¯Îªv³G¯l« ' ¶w² � ¯ � ±j·�¯ Ï ª�²�­�±j ã 25ú 1 ú� ä ¬G­>× ã 2-ú 1 ú ä ¬I«j¯ «j¯l°�¬I«g×�¯s×7¬GÂ�¯���ç>²e³I¬ Þ ¯s­�±)²�Ñ :Å Ó�Ð�ª9×O� � �Ú¯�Í!«f²e±i¯ ã 2�� 1 �� ä ÑÒªG«�±j·>¯�¯���ç>²e³I¬ Þ ¯s­>Øo¯�Ø Þ ¬GÂfÂ�Øoª�­�±j¬G²�­>²�­�° ã 2-ú 1 ú� äg��û ·�¯m¬G×>×>²e±j²e³G¯�ül¯l«jª¯ Þ ¯sÐ�¯s­�± ? ²� ã Ó�� � ��Ó ä ¬G­>×&Í�¯õØl¬G­�±i¯sÂi±!ÑÒªG«�²e± ð ó Øoª�Ð Ï ç�±j²�­�° K 0�@ ã "�ú� äg�û ·�¯`¯���ç>¬I±j²eª�­ ã Ù äBÞ ¬GØfÌ�Â� ó ЮÐ�¯l±i« óG�BÝ ­®¯ Þ ¯l°�¬G­�±� ó Ð�Ð�¯l±i«f²�Øl¬ Þ ¬ Þ ±i¯l«f­>¬I±j²e³G¯�²�Â3±j·�¯ 0`¬Gç>Øf· ó Ñ�ªG«fÐ è�ÆGÜ9¶

$jÖ � ÆOê2 � Ê 1

� Ê� � Å�� 2 1 �ú ã é äÍ�·�¯l«f¯�� ²�Â�¬)Øoª�­>Âi±j¬G­�± �uË ­ Ï «f¬GØo±j²�Øo¯�Í�¯�Øg·�ª�ª�Âj¯�¬ Ï Âj¯sç>×�ªI¸U«f¬G­>×>ª�Ð Ï ª�²�­�± ã 2���� 1 ���� �� ä Í�·>²�Øg·�×>¯oø�­�¯sÂ��¶ ð ç�±�� ×�ª�¯sÂ�­>ªG±´·>¬s³G¯�±iª ð ¯�Øoª�Ð Ï ç�±i¯s× ��û ·�¯)°G«jª�ç Ï ª Ï ¯l«g¬I±j²eª�­Ú·>¬GÂ�¬& ó Ð�Ð�¯l±i«g²�Øl¬ Þ Ñ�ªG«gÐ °�²e³G¯s­²�­ è�ÆGÜ9¶ ã Ö � Æ �vä êU¶1¬G­>× ã 2�� 1 �� ä Ê ã 1 � 2 �� ä Å ? �

��ª�­�±i°Gª�Ð�¯l« ó è�ÙGñ ê-Âjç�°G°G¯sÂi±i¯s×ÿç>Âj²�­�°�±j·>¯´Ñ�ªG«gÐ7 1 È Å32 � Ê 462 È Ê 2 ãUåGä

ªG«s¶>«f¯ Ï Þ ¬GØl²�­�° ã 25ú 1 ä ð óÚã 2 �� �ú 1 �� ä ¬GÂ�¬ ð ªv³G¯G¶7 1 È �Å 2 � Ê 4�2 È �Ê 2� È â ã ñ ä

0wªG«j«f¯sÂ Ï ª�­>×>²�­�°�±iª�±j·�¯´Øoª�­;×>²e±j²eª�­ ã Ü ä Í�¯õ­>ªvÍ ·>¬v³G¯õ±j·�¯´Øoª�­;×>²e±j²eª�­ 4 È :Å Ö �ö ªG±m¯l³G¯l« ó ¯ Þ�Þ ² Ï ±j²�Ø�Ølç>«j³G¯�Øl¬G­ ð ¯´¯� Ï «j¯sÂjÂi¯s×ÿ²�­ ±j·>¯õÑÒªG«fÐ ãUåGä ªG« ã ñ ä ð ó «f¬I±j²eª�­>¬ Þ ±i«f¬G­>ÂiÑÒªG«fÐ�¬I±j²�ª�­> �

AmªvÍw¯l³G¯l«s¶ ð ó ³I¬I« ó ²�­�°.4�²�­ ãUåGä ªG« ã ñ ä ¶-Íw¯)°G¯l±´¬�Âfç���Øl²e¯s­�± Þeó�Þ ¬I«f°G¯)Ø Þ ¬GÂjÂ�ªGÑ Ï Âj¯sç>×�ªI¸U«f¬G­>×>ª�Ð Ølç>«j³G¯s �û ·�¯�¯� �¬GØo±!³I¬ Þ ç�¯�ªGÑ 7�²�­ ãUåGä ªG« ã ñ ä ²�Ât­�ªG±!²�Ð Ï ªG«j±j¬G­�±l¶ ð ç�±`²e±`²�Â`Âj²e°�­;²xø�Øl¬G­�±�Í�·>¯l±j·�¯l«#7�²�Â`¬ ��ç>¬G×>«f¬I±j²�Ø«j¯sÂj²�×>ç�¯ ã Ð�ª9× � ä ªG«�­�ªG± �uË ­Î°G¯s­�¯l«f¬ Þ Í�¯´°G¯l±�±�Íwª�×>² Gq¯l«j¯s­�±t°G«jª�ç Ï Âl¶�ªGÑYªG«f×�¯l« ��Ê � . � ¶ ð ó ³ ¬I« ó ²�­�°/7 �Ý ÂjÂjç>Ð�¯�±j·>¬I±`Íw¯´Âi±j¬I«j±�Í�²e±j·Î¬ Ï ª�²�­�±�� � Å ã 2 � � 1 � �� � ä ª�­ ã ñ äg� ©�ªG« Ï ª�Âj²e±j²e³G¯�²�­�±i¯l°G¯l«!µB¶�Íw¯�Í!«f²e±i¯

�5Ä Å(µ!� ��¬G­>×ÕÂjç Ï>Ï ª�Âi¯��-Ä&Å ã 2�Ä � 1 Ä �" sÄ äg� ��ª�­�±i°Gª�Ð�¯l« ó è�ÙGñ9¶�$ � Ó ê�Âf·�ªvÍ�Â�²�­ÿ¬Î×>²e«j¯sØo±�Ít¬ ó ±j·>¬I±l¶ÑÒªG« Ï ª�Âj²e±j²e³G¯´²�­�±i¯l°G¯l«�#Úúiµ Âjç;Øf· ±j·>¬I±$�&% :Å .��5Ä;¶�Íw¯õ·>¬v³G¯�¬G­ ½ �'� �Ò¾����I¿ � �I»^¼�À:�e½

2"%tá1Ä��� '%wá1Ä�Å� �( % #>Ä ( ã 2"% 2;Ä % )%� sÄ ä È �62*( % #>Ä ( ã 2"%$ vÄ % )% 2�Ä ä È ã � äU ,�k,+ �v{>|i}O{;|^|.-;y1Mty�SU[gXUMtXUZOMt�gSU[gr _)[g_GMQS�dfXUTW[^k�]wr a�XUTW_ aWTWRidjXUTbn^M\a���VGMQRidgrOP/M`[gcqXUZOM`dgk dgaW[^�f��yLT�XUZ�XUZOMt]wrOaWXUTW_OabTWRidjXUTWnoM

�gSU[gr _�[fc"/ St�10��1TWk�XUZOM2043�|3]`MQXUZ [sN�{ dfkIN�P/_G[^�oMB[fc"5 dgP�XUZOMBT�NvM\klXUT�X�!M\aWM\]`M\klXi8

Page 8: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

ý ������������� �

¬G­>×&¬ �IÀ ��� ���f½�¾��"�I¿ � �I»^¼)À1�e½2 È Ä �� È Ä�Å ã 2 ÈÄ % ÈÄ ä È ��Ö 2;Ä� sÄ ã 2 ÈÄ Ê 462�Ä� vÄ�Ê� ÈÄ äwâ ã\� Ó ä

Ý ­�¬ Þ ±i¯l«g­>¬I±j²e³G¯�×>¯l«f²e³I¬I±j²eª�­´ªGÑ ã � ä ò ã\� Ó ä ¶�ç>Âj²�­>°�¬G×>×>²e±j²�ª�­´¬G­>×�×>ç Ï Þ ²�Øl¬I±j²eª�­�Ñ�ªG«gÐ�ç Þ ¬GÂ-ÑÒªG«5±j·>¯�&�¬GØoª ð ²�¬G­¯ Þ�Þ ² Ï ±j²�Ø3Ѩç>­>Øo±j²eª�­ � µ È ã��qä ¶�²�Â-°�²e³G¯s­)²�­�è�ÆGÜ9¶ Ï � Ö�ÆGÆOê �uû ·>²�Â5×�¯l«f²e³I¬I±j²eª�­õЮ¬IÌG¯sÂ3±j·�¯�«f¯s¬GÂiª�­�Ñ�ªG«3¬GÂjÂiª9Øl²�¬I±j²e³9²e± óØ Þ ¯s¬I« �ö ªG±i¯�±j·>¬I± ã � ä ò ã\� Ó ä ×�ª�­�ªG±®Â Ï ¯sØl²eÑ ó ±j·>¯ 1 ¸�Øoª�ªG«f×>²�­>¬I±i¯ � ©�ªG«f±jç>­>¬I±i¯ Þeó ¶`²e±�±jç>«f­>Â)ª�ç�±�±j·;¬I±®±j·�¯ 1 ¸

Øoª�ªG«f×>²�­>¬I±i¯�²�Âw­�ªG±t«j¯���ç>²e«j¯s×�ÑÒªG«!æ�0\��¶�¬G­>×�Í�¯�Øl¬G­ Âj¬v³G¯´Í�ªG«fÌ ð ó ­�ªG±`Øoª�Ð Ï ç�±j²�­�°�²�± �BË ­�±j·>²�ÂtØl¬GÂi¯�Íw¯Í!«f²�±i¯ �5Ä�Å ã 2;Ä ��� sÄ äg�O� ²�­;Øo¯ ã 2 � 1 � ä Ê ã 2 � % 1 � ä Å ? ¶q²e°�­�ªG«f²�­>°�±j·�¯ 1 Øoª�Ð Ï ª�­>¯s­�±�¬GÐ�ª�ç>­�±j±iª®²�×>¯s­�±j²eÑ ó ²�­�° �ô¬G­>× % � �

��ª�­�±i°Gª�Ð�¯l« ó è�ÙGñ9¶ $ � Ó êtÂj·>ªvÍ�Â�·�ªvÍ ã � ä ¬G­>× ã\� Ó ä Øl¬G­ 𠯮²�Ð Ï Þ ¯sÐ�¯s­�±i¯s×Õ±iª Ï ¯l«jÑÒªG«fÐ ¬G­�¬G×;×>²e±j²eª�­¬G­>×�¬�×;ç Ï Þ ²�Øl¬I±j²�ª�­)Í�²e±j· �G� Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­> ã Ð�ª�× " äg�uû ·�¯m­�ç>Ð ð ¯l« �G� Øl¬G­ ð ¯�«f¯s×>ç>Øo¯s×�ç>­>×�¯l«wÂiª�Ð�¯Øl²e«fØlç;Ð�Âi±j¬G­>Øo¯s �

Æ � Æ ��� /4�� �H= <�� ���=4�?�û ·�¯�ø�«fÂi± Ï ·>¬GÂj¯�ªGÑ!æ�0\� Øoª�Ð Ï ç�±i¯s ���Ñ�ªG«�¬ Þ ¬I«j°G¯�²�­�±i¯l°G¯l«� � 6�Âjç>¬ Þ�Þeó ²�Âõ±j·�¯ Ï «jª9×>ç>Øo±õªGÑt¬ Þ�Þ Ï «f²�Ð�¯ Ï ªvÍw¯l«fÂ Þ ¯sÂf´±j·>¬G­ Âiª�Ð�¯ ð ª�ç;­>× ��� �Îû ·�¯l«j¯®²�Âõ­�ª�­�¯l¯s× ±iªÿØoª�Ð Ï ç�±i¯�¯� Ï Þ ²�Øl²e± Þ�óG��þwó ±j·>¯ Ï «f²�Ð�¯Î­�ç>Ð ð ¯l«�±j·�¯lªG«f¯sÐ ¶ Þ ªG°��������¬GÂ������ � �Çã Am¯l«j¯�¬G­>× ð ¯ Þ ªOÍ´¶ � Þ ªG° �Í�²e±j·>ª�ç�±`¬�Âjç ð ÂfØo«f² Ï ±t×�¯s­�ªG±i¯sÂ�±j·�¯õ­>¬I±jç�«g¬ Þ�Þ ªG°�¬I«f²e±j·>Ð ��ä©�«jª�Ð ã � ä ò ã\� Ó ä ¶�Í�¯�Øl¬G­´Øoª�Ð Ï ç�±i¯�±j·�¯-2�¬G­;×� I¸�Øoª�Ð Ï ª�­�¯s­�±jÂLªGÑ ã � �sú�� È Ä1ú�� È ÄGá�� ä ªG« ã � �sú�� È ÄGá��sú�� È ÄIá È äÑÒ«jª�Ð ±j·�¯ 27¬G­;× I¸�Øoª�Ð Ï ª�­>¯s­�±jÂ)ªGÑ ã � �sú��5Ä�ú��5ÄIá�� äg�Úû ·�ç>Âl¶3ÑÒ«jª�Ð ±j·�¯ ð ²�­>¬I« ó «j¯ Ï «j¯sÂi¯s­�±j¬I±j²eª�­ ªGÑ!±j·�¯

Ï «f²�Ð�¯ ÑÒ¬GØo±iªG«g®ªGÑ�9¶�Íw¯�Øl¬G­ Øoª�Ð Ï ç>±i¯ ã 2� ���4 � ä ²�­�� ã¨Þ ªG°� ä Å�� ã ��� ä ª Ï ¯l«f¬I±j²eª�­;Âl¶�Ím·�¯l«j¯&¯s¬GØg·ª Ï ¯l«f¬I±j²eª�­ ²�Âõ¬G­ ¬G×>×>²�±j²eª�­ ªG«)Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­ÚÐ�ª9× " �&Ë ­ Ѩ¬GØo±l¶ ã 2� � �* � ä Øl¬G­ ð ¯�Øoª�Ð Ï ç�±i¯s×�Í�²e±j·¬ ð ª�ç�± ' ����� Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²�ª�­>Â�Ð�ª�× " ¬G­;×ë¬ Øoª�Ð Ï ¬I«f¬ ð Þ ¯Õ­�ç>Ð ð ¯l«�ªGÑ�¬G×>×>²e±j²�ª�­>ÂÎÐ�ª�× "�¶mÍm·�¯l«j¯' �wÅ �G� � Þ ªG°tÆ ��Ë Ñ ��wÅ � ±j·�¯s­ ' �!Øl¬G­ ð ¯�«f¯s×>ç>Øo¯s×αiª � Ó�� Þ ªG°`Æ � ©�ªG«�×>¯l±j¬G² Þ Âl¶�Âi¯l¯M��ª�­�±i°Gª�Ð�¯l« ó è�ÙGñ9¶$ � Ó ê �Ý ±�±j·�¯�¯s­;×�ªGÑu±j·>¯õø;«gÂi± Ï ·;¬GÂi¯õªGÑ�æ�0\� Íw¯�Øg·�¯sØjÌÕ²eÑ ��´Å ? ð ó Øoª�Ð Ï ç>±j²�­�° K 0�@ ã � ú " äg��Ë ÑB±j·�¯

K 0�@ ²�Â)­�ª�­�±i«f²e³9²�¬ Þ ±j·�¯s­ ±j·�¯�ø;«fÂi± Ï ·>¬GÂi¯�ªGÑmæ�0\� ·;¬G ð ¯l¯s­ Âjç>ØlØo¯sÂfÂiÑÒç Þ ²�­�ø�­;×>²�­�°�¬ÿѨ¬GØo±iªG«�ªGÑ " �� ±j·�¯l«fÍ�²�Âi¯�Íw¯�Ю¬ ó Øoª�­�±j²�­�ç>¯®Í�²e±j· ¬�Âi¯sØoª�­>× Ï ·>¬GÂj¯ ã Âi¯l¯*$gÜ ä ð ¯lÑÒªG«j¯�±i« ó ²�­�°�¬I°�¬G²�­ Í�²�±j· ¬�×>² G1¯l«j¯s­�±Ï Âi¯sç;×�ªI¸U«f¬G­>×�ª�Ð °G«fª�ç Ï ! �

Æ � Ü ��� /4;=<�� �+<! ", 7"� .# ", < ?´Ý ­ ¬G×�³I¬G­�±j¬I°G¯®ªGÑwç>Âf²�­�° ãUåGä ªG« ã ñ ä ªO³G¯l« ã Æ ä ªG« ã Ù ä ²�Â�±j·>¬I±�±j·�¯�°G«fª�ç ϪG«f×�¯l«�²�Â�¬ Þ Ít¬ ó Â�¬�Ð�ç Þ ±j² Ï Þ ¯´ªGÑuÑÒª�ç�« ã3� ç ó ¬GÐ�¬Õè�ñIÓIê !LÂi¯l¯ è�ÙGñ9¶ Ï � ÆGéGÆOê äg�´ÝmÞ Âjª�¶1²e±�²�Â Ï ª�ÂjÂj² ð Þ ¯�±iª�¯s­;Âjç�«j¯±j·>¬I±B±j·�¯t°G«jª�ç Ï ªG«f×�¯l«u²�Â3×>²e³9²�Âj² ð Þ ¯ ð ó ñ�ú � Æ�ªG« � é � ©>ªG«u¯� 9¬GÐ Ï Þ ¯G¶�²eÑ�$+%ÛÙ�²�ÂB¬ Ï Âi¯sç>×�ªI¸U«g¬G­>×�ª�Ð#²�­�±i¯l°G¯l«s¶

� �&% Å $ È % Ù ��Ö'$qú2����� �� Å � � �'% � ú ã\�G�vä4 �94 Å ã % % �qä � ã Ü � Ê�% ä ��Ö � � %�ú

±j·�¯s­Î±j·�¯�Ølç�«f³G¯ ã ñ ä Í�²e±j·.4mÊ ÆõÅ 4� � 4� �·;¬GÂt°G«jª�ç Ï ªG«f×�¯l«!×;²e³�²�Âj² ð Þ ¯ ð ó�� Æ ��Ý Â!Âj±j¬I«j±j²�­�° Ï ª�²�­�±�Íw¯´Øl¬G­±j¬IÌG¯ ã 2���� �� �� äg�

Ü � �)( � � ��¦q£- L¤+*,( �$� ���ª�­�±i°Gª�Ð�¯l« ó ¬G­;×ÛªG±j·>¯l«f è å ¶!ÙGñ9¶�Ù��9¶!éGÆ êõ·;¬s³G¯ ×�¯sÂjØo«g² ð ¯s× Âj¯l³G¯l«f¬ Þ Íw¬ ó Â�±iª7²�Ð Ï «fªv³G¯ �L¯s­>Âj±i«f¬9ïWÂ

ªG«f²e°�²�­>¬ Þ æ�0\� ¬ Þ °GªG«f²e±j·;Ð ð ó ±j·�¯´¬G×>×;²e±j²eª�­�ªGÑB¬�Âi¯sØoª�­>× Ï ·;¬GÂi¯G¶;¬G­>¬ Þ ªG°Gª�ç;Â`±iª Ï ·>¬GÂj¯õÆ�ªGÑY±j·>¯ �5ª Þ�Þ ¬I«f×� % � Ð�¯l±j·�ª9× � �Ú¯�ª�ç�± Þ ²�­�¯�Âiª�Ð�¯´³ ¬I«g²�¬I±j²eª�­>Â`Ím·>²�Øf·&Í�¯õ·;¬s³G¯�²�Ð Ï Þ ¯sÐ�¯s­�±i¯s× � ��·>¬GÂi¯ � ²�Â`±j·�¯õÂj¬GÐ�¯õ²�­¬ Þ�Þ Øl¬GÂi¯sÂs¶�¬GÂ!×�¯sÂfØo«f² ð ¯s× ²�­1$gÆ � Æ �

�Ú¯�ç>Âjç;¬ Þ�Þeó ¬GÂjÂjç;Ð�¯�±j·;¬I±M��ª�­�±i°Gª�Ð�¯l« ó ïW´ÑÒªG«fÐ ã ñ ä ²�Â�ç>Âi¯s×ÿÍ�²�±j·ÿÂj±j¬I«j±j²�­�° Ï ª�²�­�±�� ��Å ã 2�� �!�� �� ä°�²e³G¯s­ ð óÛã\�G�väg� þ ª�ç;­>×>Â-� È ß.��� % ÓÕ¬I«j¯ Øf·�ª�Âi¯s­ ²�­ ¬G×>³ ¬G­>Øo¯ � ©�ªG«�¯� 9¬GÐ Ï Þ ¯G¶uÍw¯ Ð�²e°�·�±)Øg·�ª�ª�Âi¯� � Å � Ó ý ¬G­;×/� È Å � ÓGÓ&� � ã Âi¯l¯ $jÖ � Ö äg�3Ë ­õ±j·�¯�ÑÒª Þ�Þ ªOÍ�²�­�°�Í�¯w¬GÂjÂjç>Ð�¯�±j·>¬I±0� È21 � � ¶GÂiª3� È % � ��4 � È �

Page 9: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� í�Ú¯�×�¯oø�­>¯-�O��Å � ã � È ä % � ã ��� ä 4 � È � Þ ªG° � È ��û ·�¯)±j²�Ð�¯�«f¯���ç>²�«j¯s×�ÑÒªG« Ï ·>¬GÂj¯�Æ�²�Â�¬ Ï;Ï «jª� �²�Ð�¬I±i¯ Þ�óÏ «jª Ï ªG«j±j²�ª�­>¬ Þ ±iª �O� �

� ç Ï;Ï ª�Âj¯`±j·�¯�°G«jª�ç Ï ªG«f×�¯l« � ·>¬GÂ Ï «f²�Ð�¯`Ѩ¬GØo±iªG«f � �!ß � È ßÇìlìlì � ��·>¬GÂi¯ � Í�² Þ�Þ ç>Âjç>¬ Þ�Þeó ð ¯�Âfç>ØlØo¯sÂjÂiѨç Þ²eÑ � ��Ô ��� � �Ú¯�Âj¬ ó �iç>Âjç;¬ Þ�Þeó � ð ¯sØl¬Gç;Âi¯�²e±�²�Â Ï ª�ÂjÂf² ð Þ ¯�±j·>¬I±�¬ Ï «g²�Ð�¯�Ѩ¬GØo±iªG«�ªGÑ � ª9ØlØlç�«fÂmÍ�²e±j·&·;²e°�·�¯l«Ð�ç Þ ±j² Ï Þ ²�Øl²e± ó ²�­ � ±j·>¬G­)²�­ 9¶ ð ç�±Y±j·>²�ÂY²�Â3ç>­ Þ ²eÌG¯ Þ�ó ¬G­;×�Øl¬G­ ð ¯t­�¯l° Þ ¯sØo±i¯s×)Í�·�¯s­)Øoª�­>Âf²�×�¯l«f²�­>°�±j·�¯t¬s³G¯l«f¬I°G¯ð ¯s·>¬s³9²eª�ç�«õªGÑ!æ�0\� � ��·>¬GÂi¯�Æ�²�Âõ×>¯sÂj²e°�­�¯s×Ú±iª 𠯮Âjç>ØlØo¯sÂfÂiÑÒç Þ ²eÑ � ��Ô � È ¬G­>× � È Ô �5� �&û ª�¬�°Gª�ª9׬ Ï>Ï «fª� �²�Ð�¬I±j²eª�­�¶ Ï ·>¬GÂi¯ � ²�Â�Âfç>ØlØo¯sÂjÂiѨç Þ ²eÑ�¬ Þ�Þ Ï «f²�Ð�¯®Ñ¨¬GØo±iªG«fÂ�ªGÑ � ¬I«j¯ ¬I±�Ð�ª�Âi± � � ¶�¬G­>× Ï ·>¬GÂi¯&Æÿ²�ÂÂjç>ØlØo¯sÂfÂiÑÒç Þ ²eÑ5¬ Þ�Þ ð ç�±tª�­�¯ Ï «g²�Ð�¯�ÑÒ¬GØo±iªG«gÂ!ªGÑ � ¬I«j¯´¬I±mÐ�ª�Âi± ���s¶>¬G­>× ±j·;¬I±!ª�­�¯´Ñ¨¬GØo±iªG«m²�Â`¬I±mÐ�ª�Âi± � È �Ë ­ ±j·�¯´ÑÒª Þ�Þ ªOÍ�²�­�°�Íw¯õ×�¯sÂjØo«g² ð ¯´Âi¯l³G¯l«f¬ Þ ×;² G1¯l«f¯s­�±!³G¯l«gÂj²eª�­>Â`ªGÑ Ï ·;¬GÂi¯�Æ ã ¬ Þ Âiª®Øl¬ Þ�Þ ¯s× �iØoª�­�±j²�­�ç>¬I±j²eª�­;Â,�ð ¯sØl¬Gç>Âi¯�±j·�¯ ó Øoª�­�±j²�­�ç�¯�¬IÑÒ±i¯l« Ï ·;¬GÂi¯ �väg� � ª�Ð�¯)³G¯l«fÂj²eª�­;Â�¬I«j¯�×>²'��Ølç Þ ±m±iª&²�Ð Ï Þ ¯sÐ�¯s­�±�ç>Âj²�­�°�ª�­ Þeó ±j·�¯ÑÒªG«fÐ�ç Þ ¬I¯ ã � ä ò ã\� Ó äg� ©�ªG«�±j·>²�Â!«j¯s¬GÂiª�­�¶1Âiª�Ð�¯ Ï «jªG°G«f¬GÐ�Â!ç;Âi¯5��ª�­�±i°Gª�Ð�¯l« ó ïWÂ�ÑÒªG«fÐ ã ñ ä Ñ�ªG« Ï ·>¬GÂi¯ � ¬G­>×Øoª�­�³G¯l«j± ð ¬GØjÌ�±iª��Ú¯s²e¯l«gÂi±i«f¬GÂjÂtÑÒªG«fÐ ã Æ ä ªG« ã Ù ä Ñ�ªG« Ï ·;¬GÂi¯�Æ � ©�ªG«!×>¯l±j¬G² Þ Â�ªGÑL±j·�¯�±i«f¬G­>ÂiÑÒªG«fÐ�¬I±j²eª�­�Âi¯l¯�è�Ü9¶$jÖ � ÆOê �Ü ���I��� /4 = <���,/6��"�+6 ( . , <! ",���� < . , ?�� ç Ï>Ï ª�Âi¯ Ï ·>¬GÂi¯ � ·>¬GÂmØoª�Ð Ï ç�±i¯s×�� Å ��´Âjç;Øf· ±j·>¬I±�� :Å ? �û ·�¯��iÂj±j¬G­>×>¬I«f×&Øoª�­�±j²�­�ç>¬I±j²eª�­��Øoª�Ð Ï ç�±i¯s � � Å ã 2 �� � �� �� ä ÑÒªG«!¯s¬GØf· Ï «f²�Ð�¯ � ¶,� � � � Ô"� È ¶>¬G­;×&²�ÂÂjç>ØlØo¯sÂfÂiÑÒç Þ ²eÑ K 0�@ ã ���lú " ä ²�Â!­>ª�­�±i«f²�³�²�¬ Þ ÑÒªG«�Âiª�Ð�¯�Âjç>Øf· � � �Ú¯�Øl¬G­�¬GÐ�ªG«j±j²�ül¯´±j·�¯´Øoª�Âi±mªGÑY¬ K 0�@ ð óÑÒª Þ�Þ ªvÍm²�­�°�¬�Âjç�°G°G¯sÂj±j²eª�­ ªGÑ"�Yª Þ�Þ ¬I«f× è�é å ê � �Ú¯õØoª�Ð Ï ç�±i¯

K 0�@�� � �� Ð�ª9×/"�ú " �Í�·�¯l«f¯õ±j·�¯ Ï «jª9×>ç>Øo±�Ð�ª9× "�²�Â!±j¬IÌG¯s­ÿªv³G¯l«´¬�Âjç���Øl²�¯s­�± Þeó Þ ¬I«f°G¯�Âi¯l±mªGÑ � ¶�¬G­>× ð ¬GØjÌ�±i«f¬GØfÌ�²eÑ3±j·�¯ K 0�@²�Â`Øoª�Ð Ï ª�Âj²�±i¯ ��û ·>²�Ât«f¯s×>ç>Øo¯sÂ`±j·�¯´Øoª�Âj±�ªGÑ3¬ K 0�@#¯sÂjÂj¯s­�±j²�¬ Þ�Þeó ±iª�±j·;¬I±!ªGÑB¬�Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²�ª�­�Ð�ª�× " �û ·�¯l«f¯�²�Ât­�ª)¬G×�³I¬G­�±j¬I°G¯�²�­�ç;Âj²�­�° Ï ·>¬GÂi¯�Æ�²eÑ � �(²�ÂtØoª�Ð Ï ç�±i¯s×Îç>Âf²�­�°õ±j·>¯�Âj±j¬G­>×>¬I«f× ð ²�­;¬I« ó Ð�¯l±j·�ª9×�¶

Í�·>²�Øf·Ú±j¬IÌG¯s � ã¨Þ ªG° � ä °G«jª�ç Ï ª Ï ¯l«f¬I±j²eª�­; �®Ë ±�²�ÂõÐ�ç>Øg· Ð�ªG«f¯�¯���Øl²e¯s­�±´±iª Ï «j¯sØoª�Ð Ï ç�±i¯�¬&ÂjÐ�¬ Þ�Þ ±j¬ ð Þ ¯ªGÑ Ï ª�²�­�±jÂ�Æ�����¶!Í�·�¯l«f¯�Ó � �ÇÔ �$Âf¬ óG� û ·�¯s­�¶t°�²e³G¯s­ � ��� Ñ�ªG«ÎÂiª�Ð�¯ÿª9×>× � �l¶`Í�¯ÿØl¬G­ Øoª�Ð Ï ç�±i¯Ð�²�­ ã � ��Ê Æ ��ú � È ä �)¶!Í�·>¯l«j¯ � È ²�®±j·�¯Ú­�¯� 9± Ï «f²�Ð�¯G¶!ç;Âj²�­�° ª�­ Þeó ª�­�¯ÿ°G«jª�ç Ï ª Ï ¯l«f¬I±j²eª�­ �#û ·�ç>Âl¶`Íw¯Øl¬G­ Øoª�Ð Ï ç�±i¯ � � ÑÒªG« ¬�Âi¯���ç�¯s­>Øo¯�ªGÑ´³I¬ Þ ç�¯sÂ�ªGÑ � ²�­>Ø Þ ç>×;²�­�°Ú¬ Þ�Þ Ï «f²�Ð�¯sÂ�²�­ ã ���sú � È ê�¬G­;× Ï ª�ÂjÂj² ð Þeó²�­>Ø Þ ç>×>²�­�°�Âiª�Ð�¯�Øoª�Ð Ï ª�Âf²e±i¯s 㠲�Ñ-Æ � ²�Â�ÂjÐ�¬ Þ�Þ ¯l«w±j·>¬G­�±j·�¯�Ð�¬$ 9²�Ð�¬ Þ °�¬ Ï ð ¯l±�Íw¯l¯s­ Âjç>ØlØo¯sÂfÂj²e³G¯ Ï «f²�Ð�¯sÂ�²�­±j·�¯�²�­�±i¯l«j³I¬ Þ�ä ¶-Ím²e±j·Õª�­�¯�°G«jª�ç Ï ª Ï ¯l«g¬I±j²eª�­ Ï ¯l« Ï ª�²�­�± � ��«jªv³9²�×�¯s× � ²�Â�¬I± Þ ¯s¬GÂi±´ªGÑwªG«f×�¯l« Þ ªG° � È ¶5±j·�¯ÍwªG«jÌ�ÑÒªG« Ï ·>¬GÂi¯õÆ)²�Â`«f¯s×>ç>Øo¯s×ÎÑÒ«jª�Ð � ã � È ä °G«fª�ç Ï ª Ï ¯l«f¬I±j²eª�­>Â`±iª � ã �O� ä °G«jª�ç Ï ª Ï ¯l«f¬I±j²�ª�­> �û ·�¯�Âi±j¬G­>×>¬I«f×�Øoª�­�±j²�­�ç>¬I±j²eª�­�Øl¬G­ ð ¯õ²�Ð Ï Þ ¯sÐ�¯s­�±i¯s× ¯���Øl²�¯s­�± Þeó ²�­ � ã¨Þ ªG°*" Þ ªG°�� È ä ð ²e±jÂ!ªGÑBÂi±iªG«g¬I°G¯ �Ë ±�²�Â�­>ªG±�­>¯sØo¯sÂjÂj¬I« ó ±iªÎÂj±iªG«j¯�¬�±j¬ ð Þ ¯)ªGÑ Ï «g²�Ð�¯sÂ�ç Ï ±iª � È ¬GÂ�±j·�¯)ª�×;× Ï «f²�Ð�¯sÂ�Øl¬G­ ð ¯�°G¯s­�¯l«g¬I±i¯s× ð óÂj²e¯l³9²�­�°Î²�­ ð Þ ª9ØjÌ9Â�¬GÂ�«j¯���ç>²e«j¯s× � æu³G¯s­ Âi±iªG«g²�­�°�±j·�¯ Ï «f²�Ð�¯sÂ�±iª � � È ²�Â�ç>­>­>¯sØo¯sÂjÂj¬I« ó ¶ ð ¯sØl¬Gç>Âi¯)Íw¯�Øl¬G­Âj²e¯l³G¯�ç>Âj²�­>°&ª�×;× ²�­�±i¯l°G¯l«fÂ)Ü�úfÙ�ú å ú ��ú âlâlâG��û ·�¯�Âj²e¯l³9²�­�°&×�ª�¯sÂ�­>ªG±�­�¯l¯s×�±iª ð ¯�³G¯l« ó ¯���Øl²e¯s­�±l¶ ð ¯sØl¬Gç>Âi¯

Ð�ª�Âi±tªGÑ5±j·�¯m±j²�Ð�¯�²�Â�Â Ï ¯s­�±tª�­ÎÐ�ç Þ ±j² Ï Þ ¯o¸ Ï «f¯sØl²�Âj²eª�­�¬I«g²e±j·>Ð�¯l±j²�Øm±iª Ï ¯l«jÑ�ªG«gÐ °G«jª�ç Ï ª Ï ¯l«f¬I±j²�ª�­> � � ²e¯l³9²�­�°Øoª�ç Þ × ð ¯`«f¯ Ï Þ ¬GØo¯s× ð ó ¬�ÑÒ¬GÂj± Ï Âi¯sç>×>ªI¸ Ï «f²�Ð�¯t±i¯sÂi±l¶ ð ¯sØl¬Gç>Âi¯�²�±u×�ª�¯sÂ�­�ªG±�·�ç�«j±uæ�0\� ²eÑ�¬õÑ�¯lÍ Øoª�Ð Ï ª�Âf²e±i¯s¬I«j¯õ²�­>Ø Þ ç>×�¯s×�²�­Î±j·�¯�­�ç>Ð ð ¯l«fÂt°G¯s­�¯l«f¬I±i¯s×�¬G­>× ±i«j¯s¬I±i¯s×�¬GÂ Ï «g²�Ð�¯s �Ü � Æ ��� /4 "9 � �+.�� 4$6 = <���, 60� �H6 ('.�, <! ",���� <! 5.�, ?´û ·�¯õÂj±j¬G­>×>¬I«f×�Øoª�­�±j²�­�ç>¬I±j²eª�­�Øl¬G­ ð ¯õ²�Ð Ï «jªO³G¯s×Îò��ª�­�±i°Gª�Ð�¯l« ó ïWÂ�Ñ�ªG«fÐ ã ñ ä Øl¬G­ ð ¯tç>Âi¯s×�±j·�«jª�ç>°�·�ª�ç�±l¶�¬G­>×�Ð�ª�Âj±Y°G«fª�ç Ï ª Ï ¯l«g¬I±j²eª�­>ÂBØl¬G­ ð ¯�«f¯ Ï Þ ¬GØo¯s× ð ó ¬ÂjÐ�¬ Þ�Þ ­�ç>Ð ð ¯l«BªGÑ1Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­;Â5Ð�ª9×;" �3û ·�¯`ÌG¯ ó ²�×�¯s¬�è�ÙGñ�¶�$jÖIê1²�ÂB±j·>¬I±BÍw¯�Øl¬G­�±i¯sÂi±�²eÑ K 0�@ ã ���lú " ä²�Âm­�ª�­�±i«g²e³�²�¬ Þ Í�²e±j·�ª�ç�±mØoª�Ð Ï ç�±j²�­>° � � � �Ú¯ Ï «j¯sØoª�Ð Ï ç�±i¯�Æ����ôÑÒªG«mÓ � �ÎÔ � ¬GÂ�¬ ð ªv³G¯G¶�ç>Âf²�­�° � ã � ä°G«jª�ç Ï ª Ï ¯l«f¬I±j²eª�­; � �Ú¯`Øl¬G­)±j·�¯s­�Øoª�Ð Ï ç�±i¯ #�� Ñ�ªG« # Å � úfÆ � Ê � úiÖ�� Ê � ú âlâlâ ¶�ç>Âj²�­�°�ª�­�¯`¬ Ï>Ï Þ ²�Øl¬I±j²eª�­ªGÑ ã � ä Ñ�ªG«�¯s¬GØg· Ï ª�²�­�±�¬IÑ�±i¯l«�±j·�¯)ø;«fÂi± �)û ·�¯ Ï ª�²�­�±jÂ�#�� ¬I«j¯®ç Ï ×;¬I±i¯s×ÿ¬GÂõ­�¯sØo¯sÂjÂj¬I« ó ¶YÂiª ª�­ Þeó «j¯���ç>²e«j¯� ã¨Þ ªG°�" ä Âj±iªG«f¬I°G¯ �G� ç Ï;Ï ª�Âj¯ � Å #(ÊÕµÕ²�Ât¬ Ï «f²�Ð�¯G¶�Í�·�¯l«f¯mµÕ²�Âw¯l³G¯s­ ¬G­>×ÎÓ � µÕÔ � �uö ªOÍ � � Å ?²�Ð Ï Þ ²e¯s #��ÇÅ .�µ�� � � ²�­>Øo¯ #�� Å ã 2 % �� �� )%� ä ¬G­>× .�µ��ÇÅ ã 2�Ä!�� �� vÄ! ä ¬I«f¯�Ì9­�ªvÍm­�¶ ²�±5²�Â5Âjç���Øl²e¯s­�±±iªÎ±i¯sÂi±´²eÑ K 0 @ ã 2 % sÄ % 2;Ä � '% Iú " ä ²�Â�­�ª�­�±i«f²e³9²�¬ Þ/� �Ú¯�Øl¬G­Ú¬s³Gª�²�×ÚÐ�ª�Âi±mªGÑ�±j·�¯ K 0�@�Â�¬GÂ�²�­ $gÜ ��� ¶ð ó Øoª�Ð Ï ç�±j²�­�°�� ã 2"%�� vÄ! % 2;Ä � '% ä Ð�ª9×/" ¶5Í�·�¯l«j¯�±j·>¯ Ï «jª9×>ç>Øo±�²�Â�±j¬IÌG¯s­�ªv³G¯l«)Âi¯l³G¯l«f¬ Þ # ¬G­>×Úµ �û ·�ç>Âl¶�±j·�¯mÍwªG«jÌ�²�Â�«j¯s×>ç;Øo¯s×�±iª)¬ ð ª�ç>±�±j·�«f¯l¯mÐ�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­>ÂBÐ�ª9× " Ï ¯l« Ï «f²�Ð�¯ � �BË ±�Øl¬G­ ð ¯�«f¯s×>ç>Øo¯s×

Page 10: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

� ������������� �

±iªÿ±�ÍwªÕÐ�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­;´Ð�ª9× " ð ó ¯s­>Âjç�«g²�­�°&±j·>¬I± sÄ �Å � ¶BÍ�·>²�Øg· ²�­�³Gª Þ ³G¯sÂ)¬ Ï «j¯sØoª�Ð Ï ç>±j¬I±j²eª�­ ªGѪG«f×�¯l« � �BÝ «j¯s×;ç>Øo±j²eª�­�±iª�ª�­�¯mÐ�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­�Ð�ª�× " ²�Â Ï ª�ÂjÂj² ð Þ ¯G¶�¬I±�±j·>¯�Øoª�Âi±wªGÑ�ª�­�¯�¯� 9±i¯s­>×�¯s× K 0�@Øoª�Ð Ï ç>±j¬I±j²eª�­ Ï ¯l« Ï ª�²�­�±�#�� �`û ·;²�Â!²�Â`ÍwªG«j±j·�Í�·>² Þ ¯´²eÑ � ²�Â!Âjç���Øl²�¯s­�± ÞeóÎÞ ¬I«j°G¯ �`Ý ­�ªG±j·�¯l«�«j¯s×>ç>Øo±j²�ª�­ ð ó¬ Ѩ¬GØo±iªG«�ªGÑ`¬ Þ Ð�ª�Âi±´±�ÍwªÕØl¬G­ ð ¯�¬GØf·>²�¯l³G¯s× ð ó «f¬I±j²eª�­>¬ Þ Ï «j¯sØoª�­>×>²e±j²�ª�­>²�­�° è�éGÙ ê � ©�ªG«�Ѩç�±jç�«f¯�«j¯lÑ�¯l«f¯s­>Øo¯G¶Íw¯�¬GÂfÂjç>Ð�¯ ' È Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­>Â�Ð�ª�×." Ï ¯l«`Øoª�Ð Ï ¬I«f²�Âiª�­�ªGÑ Ï ª�²�­�±jÂl¶�Í�·�¯l«j¯ � �IÆ�Ô ' È ÔÛÜ�¶9±j·�¯�¯� �¬GØo±³I¬ Þ ç�¯�ªGÑ ' È ×�¯ Ï ¯s­>×>²�­�°�ª�­&±j·�¯õ²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­ �û ·�¯l«f¯®²�Âõ¬G­ ¬G­>¬ Þ ªG° ó Ím²e±j· � ·;¬G­�Ì�ÂsïWÂ ð ¬ ð ó ¬G­>× °�²�¬G­�±�Âi±i¯ Ï ÂÎè å Ö>¶ Ï � Ö � �Oê ��°�²�¬G­�±)Âi±i¯ Ï Âõ²�­�³Gª Þ ³G¯Î¬°G«jª�ç Ï ª Ï ¯l«f¬I±j²eª�­ ã ¬ ð ª�ç>± �G� Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­>Â ä ¬G­>× Ï ª�ÂfÂj² ð Þeó ¬G­�¯� �±i¯s­>×>¯s× K 0 @ Øoª�Ð Ï ç>±j¬I±j²eª�­�¶ ð ç�±ð ¬ ð ó Âj±i¯ Ï Â�¬v³Gª�²�× ±j·�¯´°G«fª�ç Ï ª Ï ¯l«f¬I±j²eª�­&¬G­;× ²�­�³Gª Þ ³G¯´ª�­ Þeó ' È ÔÛÜ�Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­; �û ª�×�¯sØl²�×>¯õª�­�±j·�¯õ±j¬ ð Þ ¯�Âj²eül¯���¶q­�ªG±i¯�±j·>¬I±�Âj¯l±i±j²�­�°�ç Ï ±j·�¯�±j¬ ð Þ ¯�¬G­>×�Øoª�Ð Ï ç�±j¬I±j²eª�­�ªGÑu±j·�¯ Ï ª�²�­�±jÂ#�� «j¯���ç>²e«j¯sÂ�ªGÑ`ªG«g×�¯l«�� Ê�� È � ã Æ � ä ¬ Ï>Ï Þ ²�Øl¬I±j²eª�­>Â�ªGÑ ã � äg��Ë Ñ`Âi±iªG«f¬I°G¯ ²�Âõ­�ªG±�¬�Øoª�­>Âj²�×>¯l«f¬I±j²eª�­�¶Y±j·�¯ª Ï ±j²�Ю¬ Þ � ²�Â)¬ Ï>Ï «jªT 9²�Ю¬I±i¯ Þeó � � È �IÆ � AmªvÍw¯l³G¯l«s¶ Ï «jªO³�²�×>¯s× � � È % � 1 Þ ªG°�� È ¶B±j·>¯ÎÂj¯l±i±j²�­�°Úç ÏØoª�Âi±®²�Â�� ã �O� ä ¶t¬G­>× ±j·�¯ ªO³G¯l«f¬ Þ�Þ Øoª�Âi±®ªGÑ Ï ·;¬GÂi¯�ÆÕ²�Â�¬ ð ª�ç�±/' È �O� Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­>Â�Ð�ª�×5" �7û ·�ç>Âl¶Âi±iªG«f¬I°G¯õ«j¯���ç;²e«j¯sÐ�¯s­�±jÂtÑ�ªG«!¬G­&¯���Øl²e¯s­�±!²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­�ªGÑY±j·�¯�²�Ð Ï «fªv³G¯s× Âi±j¬G­>×;¬I«f× Øoª�­�±j²�­�ç>¬I±j²�ª�­&¬I«j¯­�ªG±�Ð�ç>Øg· °G«j¯s¬I±i¯l«�±j·>¬G­ ÑÒªG«!±j·�¯´Âi±j¬G­>×;¬I«f×&Øoª�­�±j²�­�ç;¬I±j²eª�­ �

Ü � Ü ��� /4�� �+</6���� ��� � ��68.�� ('.�, <! ",���� <! 5.�, ?´û ·�¯ � ð ²e«j±j·>×;¬ ó Ï ¬I«f¬G×�ªT ��ÿØoª�­�±j²�­�ç;¬I±j²eª�­�²�Â�¬G­ ¬ Þ ¸±i¯l«f­>¬I±j²�³G¯´±iª®±j·�¯ ã ²�Ð Ï «jªO³G¯s× ä Âi±j¬G­>×;¬I«f×&Øoª�­�±j²�­�ç;¬I±j²eª�­ �uË ±!Ít¬GÂ�Âjç�°G°G¯sÂj±i¯s×�²�­ è å êY¬G­>×&·>¬G ð ¯l¯s­�²�Ð Ï Þ ¯o¸Ð�¯s­�±i¯s×�²�­ Âj¯l³G¯l«f¬ Þ ªGÑYª�ç�« Ï «fªG°G«f¬GÐ� ã Âi¯l¯9$gÙ ä ¬G­>×&²�­Î±j·�¯ Ï «jªG°G«f¬GЮÂ`ªGÑ Ýõ� �-¯s­>Âi±i«f¬�¯l±m¬ Þ è�Ü�¶�ÜGÜ9¶�Ù ÖIê �û ·�¯l«f¯w¬I«j¯`Âi¯l³G¯l«g¬ Þ ³ ¬I«g²�¬I±j²eª�­>ÂYª�­�±j·�¯ ð ²e«f±j·>×>¬ ó Ï ¬I«f¬G×�ª� )²�×�¯s¬ � �Ú¯!×�¯sÂjØo«g² ð ¯�¬�³G¯l«fÂj²eª�­)Ím·>²�Øf·�²�ÂY¯s¬G ó

±iª�²�Ð Ï Þ ¯sÐ�¯s­�±u¬G­>×�Ím·�ª�Âi¯!¯���Øl²�¯s­>Ø ó ²�ÂuØoª�Ð Ï ¬I«g¬ ð Þ ¯!±iª�±j·>¬I±�ªGÑq±j·�¯�²�Ð Ï «jªO³G¯s×®Âi±j¬G­>×;¬I«f×�Øoª�­�±j²�­�ç;¬I±j²eª�­ �©�ª Þ�Þ ªvÍ�²�­>°�¬�Âjç>°G°G¯sÂi±j²eª�­�ªGÑ � ç ó ¬GÐ�¬9¶IÍ�¯wØf·�ª�ª�Âi¯t¬ Ï ª�Âj²e±j²�³G¯ Ï ¬I«f¬GÐ�¯l±i¯l« �3û ·�¯�Øg·�ª�²�Øo¯wªGÑ��²�Â-Øoª�­>Âj²�×�¯l«j¯s×ð ¯ Þ ªvÍ � ©�ªG«�±j·�¯�Ð�ª�Ð�¯s­�±!±j·�¯�«j¯s¬G×�¯l«mØl¬G­�Âjç Ï;Ï ª�Âj¯�±j·;¬I±��Å �I�

� ç Ï;Ï ª�Âj¯G¶�¬GÂ�²�­ $gÜ ��� ¶9±j·>¬I± � ²�Âu±j·�¯mª�ç�± Ï ç�±�ªGÑ Ï ·>¬GÂi¯ �I� � ¯ Þ ¯sØo±�¬�±j¬ ð Þ ¯�Âj²eül¯�� �3Ë ÑLÂi±iªG«f¬I°G¯ Ï ¯l«gÐ�²e±jÂl¶� Âf·�ª�ç Þ × ð ¯t¬ ð ª�ç�± � �O��!�ªG±j·>¯l«jÍ�²�Âi¯`Øg·�ª�ª�Âj¯�� ¬GÂ Þ ¬I«f°G¯t¬GÂ�Âi±iªG«f¬I°G¯!Øoª�­>Âi±i«g¬G²�­�±jÂ�¬ Þ�Þ ªvÍ ã ÑÒªG«B«j¯s¬GÂiª�­;¬ ð Þ ¯¯���Øl²e¯s­;Ø ó Íw¯wª�­ Þeó ­�¯l¯s× � 1 Þ ªG°�� � äg� K ¯s­�¯l«f¬I±i¯�� Ï Âi¯sç>×�ªI¸U«f¬G­;×�ª�Ð#Ð�ç Þ ±j² Ï Þ ¯sÂ-ªGÑ �)¶�Âf¬ ó �� �Å���� �ÑÒªG«��)Å � ú âlâlâ ú�� �uû ·�¯l«f¯m²�ÂwÂiª�Ð�¯�¬G×�³I¬G­�±j¬I°G¯�²�­�Øf·�ª�ª�Âj²�­�°�±j·�¯�� ±iª ð ¯ Þ ²�­�¯s¬I«�²�­���¶9² � ¯ � � Å à ù-Ê à ���ÑÒªG«�Âiª�Ð�¯ Ï Âi¯sç>×�ªI¸U«g¬G­>×�ª�Ð à ùIú à � ã ­�ªG±�±iª�ª�ÂjÐ�¬ Þ�Þ�äg�3Ë ­�±j·>²�Â�Øl¬GÂi¯�±j·�¯ � Øl¬G­ ð ¯!Øoª�Ð Ï ç�±i¯s× ð ó ¬(�Qø�­>²e±i¯×>² Gq¯l«j¯s­>Øo¯+�®ÂjØf·�¯sÐ�¯õÍ�²e±j· � ã )� ä °G«jª�ç Ï ª Ï ¯l«f¬I±j²eª�­> ð ¯sØl¬Gç;Âi¯õ±j·�¯�o¸U±j·�×;² G1¯l«f¯s­>Øo¯sÂ!ªGÑ3±j·�¯�Ð�ç Þ ±j² Ï Þ ²e¯l«gÂ��� ¬I«j¯`Øoª�­>Âi±j¬G­�± �uÝ ­>ªG±j·�¯l« Ï ª�ÂfÂj² ð ² Þ ²e± ó ²�Â5±iª´Øf·�ª�ª�Âi¯�� ±iª ð ¯�¬ Ï «jª9×>ç>Øo±YªGÑ;ÂjÐ�¬ Þ�Þ Ï «g²�Ð�¯s � ©�ªG«B¯� 9¬GÐ Ï Þ ¯G¶²�­�ª�ç�« Ï «fªG°G«f¬GÐ�Â20�ò�æ ã Âi¯l¯ $gÙ ä Íw¯!ç>Âi¯`¬�Âi¯l±BªGÑ�Æ�� Þ ªG° È ���tª�×>× Ï «f²�Ð�¯sÂ3¬G­>×)±j¬IÌG¯�� ±iª ð ¯t¬ Ï «fª�×>ç;Øo±YªGÑ� Þ ªG° È ����ª�×>× Ï «f²�Ð�¯sÂBÑ�«fª�Ðô±j·>²�Â�Âi¯l±l¶9±j·�¯mØg·�ª�²�Øo¯�×�¯ Ï ¯s­>×>²�­>°�ª�­�±j·�¯ ð ²e±jÂ�²�­�±j·�¯ ð ²�­>¬I« ó «j¯ Ï «j¯sÂi¯s­�±j¬I±j²eª�­ªGÑ�� % �I��û ·>²�Â`ÂjØg·�¯sÐ�¯´«f¯���ç>²�«j¯s � ã '� Þ ªG° Þ ªG° � ä °G«jª�ç Ï ª Ï ¯l«g¬I±j²eª�­>Â!¬G­>× Øl¬G­ ð ¯�³G¯sØo±iªG«f²eül¯s× �Ý Ñ�±i¯l«�°G¯s­�¯l«f¬I±j²�­>°�±j·�¯,� Ï ª�²�­�±j � ¶YÍw¯�°G¯s­�¯l«f¬I±i¯ ���O� ���"!�ÑÒç>«j±j·�¯l« Ï Âi¯sç>×�ªI¸U«f¬G­;×�ª�Ð Ï ª�²�­�±jÂl¶3Âj¬ ó�$# Å � �# �)¶`Í�·>¯l«j¯&±j·�¯ �%#Ú¬I«j¯�×>²�Âi±j²�­;Øo±�Ñ�«jª�Ð ±j·�¯&� �ëû ·�¯�Øg·�ª�²�Øo¯ �'# Å Æ # ²�®Âj¬I±j²�ÂjÑÒ¬GØo±iªG« óG� ©>ªG«

¯s¬GØf·ÛÂjç>Øg· Ï ª�²�­�± � # ¶wÍ�¯&Øg·�¯sØjÌ7²eÑ � # Å . � ÑÒªG«(� Å � ú âlâlâ ú�� � û ·>²�Â�Øl¬G­ ð ¯Î×>ª�­�¯ Í�²e±j·5' È �Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­>Â`Ð�ª9×&"�¶q¬GÂm²�­�±j·�¯�×>¯sÂjØo«f² Ï ±j²�ª�­ ªGÑB±j·�¯)²�Ð Ï «jªO³G¯s×�Âi±j¬G­;×>¬I«f×�Øoª�­�±j²�­�ç>¬I±j²eª�­ÿ²�­ $gÜ � Æ �mË Ñ�ô²�ÂYÂjç���Øl²e¯s­�± Þ�óõÞ ¬I«j°G¯G¶�²e±3²�Â5Í�ªG«f±j·�Í�·;² Þ ¯�±iª�Ð�¬IÌG¯t±j·�¯� I¸�Øoª�ªG«f×>²�­>¬I±i¯sÂ3ªGÑ � ¬G­;× �)#!ç>­>²e± ó ð ó ¯� �±i¯s­;×�¯s×K 0�@#Øoª�Ð Ï ç�±j¬I±j²eª�­>Âl¶;Í�·>²�Øg·�«f¯s×>ç>Øo¯s ' È ±iª �I�mã¨û ª�«j¯s×>ç>Øo¯�±j·�¯´­�ç>Ð ð ¯l«`ªGÑ5¯� 9±i¯s­>×�¯s× K 0�@�Âl¶�Íw¯õØl¬G­°G¯s­�¯l«f¬I±i¯�±j·�¯ Ï ª�²�­�±j � # ²�­ ð ¬I±jØf·>¯s®¬G­>× «f¯s×>ç>Øo¯ ±j·�¯s²e« I¸�Øoª�ªG«g×>²�­>¬I±i¯sÂ�±iª�¬ Øoª�Ð�Ð�ª�­ ³ ¬ Þ ç>¯ ð ¯lÑÒªG«j¯Ï ¯l«jÑÒªG«fÐ�²�­�°�ª�­�¯�¯� 9±i¯s­>×�¯s× K 0�@ ��ätö ªG±i¯õ±j·>¬I±`±j·�¯ Ï ª�²�­�±j �$#�×�ª�­�ªG±�­�¯l¯s×αiª ð ¯�Âi±iªG«j¯s×L¶>¬GÂ!ª�­ Þeó ª�­�¯ã ªG«�ª�­>¯ ð ¬I±jØf· ä ²�Â`­�¯l¯s×�¯s×�¬I±�¬�±j²�Ð�¯ � �Ú¯�ª�­ Þeó ­�¯l¯s×&Âi±iªG«f¯�� ã � ä Ï ª�²�­�±j �Ë ±õ²�Â�¯s¬G ó ±iªÿÂi¯l¯�±j·>¬I±�Ð�ª�Âi±�ªGÑw±j·>¯®Øoª�Ð Ï ç�±j¬I±j²�ª�­ ÑÒªG«õ±j·�¯ ð ²e«j±j·>×;¬ ó Ï ¬I«f¬G×�ª� ³G¯l«gÂj²eª�­ÚªGÑ Ï ·;¬GÂi¯�Æ

¬GÐ�ª�ç>­�±jÂu±iªõ±j·�¯!¯l³I¬ Þ ç>¬I±j²eª�­�ªGÑ�¬ Ï ª Þeó ­�ª�Ð�²�¬ Þ ªGÑ�×�¯l°G«f¯l¯ � ¬I±����O� ���"! Ï ª�²�­�±j �3û ·�ç>Âl¶�Ѩ¬GÂi± Ï ª Þeó ­�ª�Ð�²�¬ Þ¯l³I¬ Þ ç>¬I±j²eª�­&ÂfØf·�¯sÐ�¯sÂmØl¬G­ ð ¯�ç>Âi¯s× è å ¶�Ö å ¶�éGÙ�¶�ñ ÖGê �þ ªG±j· ±j·�¯�²�Ð Ï «jªO³G¯s× Âi±j¬G­>×>¬I«g× Øoª�­�±j²�­�ç>¬I±j²�ª�­ ¬G­>×Ú±j·�¯ ð ²e«j±j·>×;¬ ó Ï ¬I«f¬G×�ª� Øoª�­�±j²�­�ç>¬I±j²eª�­ Ð�¬IÌG¯�¬ Ï ¸

Ï «jªT 9²�Ð�¬I±i¯ Þeó �O�)Øoª�Ð Ï ¬I«f²�Âiª�­>Â�ªGÑ Ï ª�²�­�±jÂ�Í�·>²�Øg·ÿ¬I«f¯)Ð�ç Þ ±j² Ï Þ ¯sÂmªGÑ �)¶LÂf¬ ó µ�� ¬G­>×Õµ ��¶L¬G­>×Õç>Âjç>¬ Þ�ÞeóÂjç>ØlØo¯l¯s×�²eÑ � È Ô"� � ¬G­;×�µ .&µ ²�Â�¬´­�ª�­�±i«f²e³9²�¬ Þ Ð�ç Þ ±j² Ï Þ ¯tªGÑ � � �uû ·�¯!²�Ð Ï «jªO³G¯s×®Âi±j¬G­>×;¬I«f×�Øoª�­�±j²�­�ç;¬I±j²eª�­

Page 11: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� �

¯s­>Âjç>«j¯s®±j·;¬I± � µ % µ� ��²�Â Ï «f²�Ð�¯G¶ ð ç>±®±j·�¯ ð ²e«j±j·;×>¬ ó Ï ¬I«f¬G×>ª� ÛØoª�­�±j²�­�ç>¬I±j²eª�­ Ð�¯l«j¯ Þeó ±j¬IÌG¯sÂ Ï Âj¯sç>×�ªI¸«f¬G­>×>ª�Ð µ ¬G­;×�µ �O� ²�­>Øo¯ � ��²�Â Ï «f²�Ð�¯G¶;²e±�Íwª�ç Þ ×�¬ Ï>Ï ¯s¬I«m±j·;¬I±m±j·�¯)²�Ð Ï «jªv³G¯s×�Âi±j¬G­>×>¬I«f×ÿØoª�­�±j²�­�ç;¬I±j²eª�­²�Â�Ð�ªG«j¯)¯���Øl²e¯s­�± � A�ªOÍ�¯l³G¯l«v¶L±j¬IÌ�²�­�°Î±j·�¯ Ï ¬I«f¬GÐ�¯l±i¯l«(;% � Ð�¬ ó Øoª�Ð Ï ¯s­>Âf¬I±i¯�ÑÒªG«�±j·>²� ��û ·�¯�­�ç;Ð ð ¯l«ªGÑBÂiª Þ ç�±j²�ª�­>ÂwªGÑ

2 È � Å � Ð�ª�× � � ã\� Æ ä²� K 0�@ ã Æ �ú � � % �väg��û ·�ç>Âs¶ ð ó Øg·�ª�ª�Âj²�­�° ´¬GÂ`¬ Ï «jª�×;ç>Øo±�ªGÑ3ÂjÐ�¬ Þ�Þ Ï «f²�Ð�¯sÂl¶�Íw¯´²�­>Øo«j¯s¬GÂi¯�±j·�¯�¯� Ï ¯sØo±i¯s×­�ç>Ð ð ¯l«´ªGÑ`Âiª Þ ç�±j²eª�­>Â�ªGÑ ã\� Æ äg� Ë ­ Ѩ¬GØo±l¶B²eÑtÆ ÎÅ Æ�� U Ü�� W Ù�� Y ìlìlìv¶Y²e±´Øl¬G­ 𠯮Âj·�ªOÍ�­Ú±j·>¬I±�±j·�¯®¯� Ï ¯sØo±i¯s׳I¬ Þ ç�¯�ªGÑ K 0�@ ã Æ �ú � % �vä Ñ�ªG«´¬�«f¬G­;×�ª�Ð Þ ¬I«j°G¯ Ï «f²�Ð�¯���²� ã ��Ê �väoã È Ê �väoã ��`Ê �vä ìlìlì � � ²�­>Øo¯ � �´²�±j·�¯ Þ ¬I«f°G¯sÂi± Ï «f²�Ð�¯�Ѩ¬GØo±iªG«õªGÑw±j·�¯�°G«jª�ç Ï ªG«g×�¯l« � ¶5²�±�²�Â�«j¯s¬GÂiª�­>¬ ð Þ ¯�±iª&¬GÂjÂjç>Ð�¯�Âf²�Ð�² Þ ¬I« ð ¯s·>¬s³9²eª�ç�«�ÑÒªG«K 0�@ ã Æ �ú � � % �väg�Bû ·�¯!­�ç>Ð ð ¯l«3ªGÑ�Âiª Þ ç�±j²�ª�­>ÂYªGÑ1¯���ç>¬I±j²eª�­ ã\� Æ ä ²�ÂB«j¯ Þ ¯l³ ¬G­�± ð ¯sØl¬Gç>Âi¯`Íw¯!¯� Ï ¯sØo± Ï ·>¬GÂi¯�Ʊiª®Âjç;ØlØo¯l¯s×�²eÑ � È Ô ����¬G­>× ã � � � # ä È �`Å � Ð�ª�× � � �û ·�¯ Ï ¬I«f¬GÐ�¯l±i¯l«��Âj·�ª�ç Þ ×�­�ªG± ð ¯�Øf·�ª�Âi¯s­�±iª�ª Þ ¬I«f°G¯G¶ ð ¯sØl¬Gç>Âi¯m±j·�¯�Øoª�Âi±�ªGÑ�°G¯s­>¯l«f¬I±j²�­�°�±j·�¯ Ï ª�²�­�±j ��

¬G­>× �)#�²�Â Ï «jª Ï ªG«j±j²eª�­>¬ Þ ±iª� ��û ª�¯s­>Âjç�«f¯�±j·>¬I±`±j·>²�Â!Øoª�Âj±�²�Â`­�¯l° Þ ²�°�² ð Þ ¯G¶9Íw¯´­�¯l¯s×& � � ��Ë ­ Ï «g¬GØo±j²�Øo¯G¶¬GÂjÂjç;Ð�²�­�° � � Ù � Æ�¶Y¬�«f¯s¬GÂiª�­>¬ ð Þ ¯�Âj±i«f¬I±i¯l° ó ²�Â�±iª&Øf·�ª�ª�Âi¯�±j·�¯ Þ ¬I«j°G¯sÂj±(�ÑÒ«jª�Ð ±j·�¯�Âi¯l±�� � úfÆ�úfÜ�úfé�ú � Æ��Âjç ð �\¯sØo±`±iª�±j·>¯õØoª�­>Âi±i«f¬G²�­�±�ÜGÆ � � �Ü � Ö ��� /4 �!�m� ( .�, < 5,���� <! 5. , ? �Yª Þ�Þ ¬I«g×�Âjç>°G°G¯sÂi±i¯s×Õ±j·�¯�ç>Âi¯�ªGÑ�±j·�¯�©3© û ±iª Â Ï ¯l¯s×ÿç Ï�Ï ·;¬GÂi¯)Æ�ÑÒªG«·>²�Â�� % � Ð�¯l±j·�ª9×�¶�¬G­;×���ª�­�±i°Gª�Ð�¯l« ó è�Ù��Iê�·>¬GÂtÂjç>ØlØo¯sÂjÂiѨç Þ�Þ�ó ²�Ð Ï Þ ¯sÐ�¯s­�±i¯s×�±j·�¯�¬G­>¬ Þ ªG°Gª�ç>Â Ï ·;¬GÂi¯�Æ�ÑÒªG«æ�0�� �Iû ·�¯t©3© û Øoª�­�±j²�­�ç>¬I±j²eª�­)Ð�¬ ó ð ¯�«j¯l°�¬I«g×�¯s×)¬GÂB¬G­�¯���Øl²e¯s­�±3°G¯s­�¯l«g¬ Þ ²�Âj¬I±j²eª�­�ªGÑ ð ªG±j·�±j·�¯`²�Ð Ï «fªv³G¯s×Âi±j¬G­>×;¬I«f×7Øoª�­�±j²�­�ç;¬I±j²eª�­ ¬G­>×7±j·�¯ ð ²e«f±j·>×>¬ ó Ï ¬I«f¬G×>ª� 7Øoª�­�±j²�­�ç>¬I±j²�ª�­ � �Ú¯Õ·>¬s³G¯ÿ­�ªG±®²�Ð Ï Þ ¯sÐ�¯s­�±i¯s× ²e±ð ¯sØl¬Gç>Âi¯�ªGÑB²e±jÂ!Øoª�Ð Ï Þ ¯� �²e± ó ¬G­>× Þ ¬I«f°G¯´Âi±iªG«f¬I°G¯õ«f¯���ç>²�«j¯sÐ�¯s­�±j �Ü � Ù ��� .�9��0� � 5=. , .�� ( .�, < 5,���� <! 5. , =@?�Ë ±3²�Â3­>¬I±jç�«f¬ Þ ±iª�¬GÂiÌ´Í�·>²�Øf·�ªGÑ;±j·�¯t¬ ð ªv³G¯`³G¯l«fÂj²�ª�­>Â-ªGÑ Ï ·>¬GÂi¯`Ʋ� ð ¯sÂi± � �Ú¯®²�­>²�±j²�¬ Þ�Þeó ²�Ð Ï Þ ¯sÐ�¯s­�±i¯s×�±j·>¯ ð ²e«j±j·>×;¬ ó Ï ¬I«g¬G×�ª� ÚØoª�­�±j²�­�ç>¬I±j²�ª�­ ð ¯sØl¬Gç>Âi¯)ªGÑw²�±jÂ�Âj²�Ð Ï Þ ²�Øl²e± óG�Ý�Þ Âiª�¶-±j·>¯®¬G ó Ð Ï ±iªG±j²�Ø�¬G­;¬ Þeó Âf²�Â�²�­7è å ¶2$ å êw²�­>×;²�Øl¬I±i¯s×Õ±j·>¬I±�²�±�Í�ª�ç Þ × ð ¯�Ѩ¬GÂi±i¯l«�±j·>¬G­Ú±j·�¯ ã ²�Ð Ï «jªO³G¯s× äÂi±j¬G­>×;¬I«f×�Øoª�­�±j²�­�ç>¬I±j²eª�­ � AmªvÍw¯l³G¯l«s¶�±j·>²�Â!Íw¬GÂmª�­�±j·>¯�¬GÂjÂjç;Ð Ï ±j²eª�­�±j·>¬I±m¬G­ÿ¬G ó Ð Ï ±iªG±j²�Øl¬ Þ�Þ�ó Ѩ¬GÂi± Ï ª Þ�ó ¸­�ª�Ð�²�¬ Þ ¯l³ ¬ Þ ç;¬I±j²eª�­�ÂjØg·�¯sÐ�¯!Íwª�ç Þ × ð ¯`ç>Âi¯s× �3Ë ­ Ï «f¬GØo±j²�Øo¯G¶�� ²�Â3«f¬I«j¯ Þeó)Þ ¬I«j°G¯`¯s­�ª�ç>°�·�Ñ�ªG«wÂjç>Øg·�¬�ÂjØf·>¯sÐ�¯±iª ð ¯�Âj²e°�­>²eø�Øl¬G­�± Þ�ó ÑÒ¬GÂj±i¯l«!±j·>¬G­&Âi±j¬G­;×>¬I«f× Ï ª Þeó ­�ª�Ð�²�¬ Þ ¯l³ ¬ Þ ç>¬I±j²eª�­ �BË ­Îª�ç�«!Ð�ª�Âi±�«j¯sØo¯s­�±m²�Ð Ï Þ ¯sÐ�¯s­�±j¬ ¸±j²eª�­�ªGÑLæ�0\� ã Ï «jªG°G«f¬GÐ K ªGÑ�$gÙ ä Í�¯�·>¬v³G¯�ç>Âj¯s×®±j·�¯�²�Ð Ï «fªv³G¯s×®Âj±j¬G­>×>¬I«f×�Øoª�­�±j²�­�ç>¬I±j²eª�­ ð ¯sØl¬Gç>Âi¯mªGÑ-²e±jÂÂ Þ ²�°�·�± Þeó�Þ ªvÍw¯l«õÂi±iªG«f¬I°G¯�«j¯���ç>²e«j¯sÐ�¯s­�±jÂ�¬G­>× ð ¯l±i±i¯l« ã Ï «f¯s×>²�Øo±i¯s× ä Ï ¯l«jÑÒªG«fÐ�¬G­>Øo¯)ÑÒªG«´ÑÒ¬GØo±iªG«g´ªGÑtç Ï ±iª&Ö�Ó×�¯sØl²�Ю¬ Þ ×;²e°�²e±j ��Ë Ñ�Âi±iªG«g¬I°G¯�«j¯���ç>²e«j¯sÐ�¯s­�±jÂ�¬G­>× Ï «jªG°G«f¬GÐ Øoª�Ð Ï Þ ¯� �²e± ó ¬I«j¯�­�ªG±´Ð�¬ �iªG«´Øoª�­>Âj²�×�¯l«g¬I±j²eª�­>Âl¶±j·�¯s­ ±j·>¯õ©3© û Øoª�­�±j²�­�ç>¬I±j²eª�­�²�Â Ï «jª ð ¬ ð Þeó ±j·>¯ ð ¯sÂj± �

Ö ��� �>¢���£-¢#�- 5¦ �Õ£��0�����Ö ���I��� � 59C4 � ��(@<+."�H= .�� � ��,/6/. 9 ", <>417�4��+= ? �L¯l±wµ � ã " ä ß7µ È ã " ä ß âlâlâ ð ¯m±j·�¯ Ï «f²�Ð�¯!Ѩ¬GØo±iªG«fÂ�ªGÑ-¬Ï ª�Âj²e±j²e³G¯�²�­�±i¯l°G¯l« " �uû ·�¯�µ ã " ä ¬I«f¯õ­�ªG±!­�¯sØo¯sÂjÂf¬I«f² Þeó ×>²�Âi±j²�­>Øo± � ©�ªG«mØoª�­�³G¯s­>²e¯s­>Øo¯õÍ�¯´±j¬IÌG¯�µ ã " ä Å �²eÑ$" ·;¬GÂ Þ ¯sÂjÂt±j·>¬G­ � Ï «g²�Ð�¯�ÑÒ¬GØo±iªG«g �©�ªG« à ß � ¶uÂjç Ï>Ï ª�Âi¯®±j·>¬I± � ß�� � ß âlâlâ ß�� # ß Ó � ©>ª Þ�Þ ªvÍm²�­�°�MB¯l«gÂj·>²eÌ è�ñ � êU¶BÍw¯Î×�¯oø�­�¯�� # Å� # ã � �sú âlâlâ ú�� # ä ð ó

� # Å Þ ²�Ð����� � � " � � Ô " Ô�� úBµ ã " ä Ô "����`ÑÒªG« ��Å � ú âlâlâ ú à �� â

Ë ­�Ñ�ªG«gÐ�¬ Þ�Þeó ¶�� # ã � �sú âlâlâ ú�� # ä ²�Ât±j·�¯ Ï «jª ð ¬ ð ² Þ ²e± ó ±j·;¬I±�¬ Þ ¬I«j°G¯´«g¬G­>×�ª�Ð ²�­�±i¯l°G¯l« " ·>¬GÂ!²e±j �I¸U±j· Þ ¬I«j°G¯sÂi±Ï «f²�Ð�¯�Ѩ¬GØo±iªG«�¬I±�Ð�ª�Âi±,"����G¶LÑÒªG«��&Å � ú âlâlâ ú à ��û ·�¯�Øl¬GÂj¯s à Å � ¬G­>× à Å ÆάI«j¯�«j¯ Þ ¯l³I¬G­�±´±iª&æ�0\�ã Âi¯l¯ $?$jÖ � ÆO¸UÖ � Ö äg� Ë ±�²�Â)Øoª�­�³G¯s­>²�¯s­�±�±iªÚ×�¯oø1­�¯�� � ã � � ä Å � ²eÑ � � % � ¶�¬G­;×!� � ã � � ä Å ÓÚ²eÑ � � � Ó �MB¯l«fÂj·;²eÌÕè�ñ � ¶ û ·>Ð �L� ê5Âf·�ªvÍ�Â!±j·>¬I±

� # Å#" �%$ù " �%$�& U' $ ìlìlì(" � U' W � �*) + #� % + �J% ìlìlì % + #�,� + � âlâlâ � + # # � � + #+ � âlâlâ + # # � + # â ã\� Ü ä

Page 12: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��ù ������������� �

D´­�ç�±j· ¬G­>× û «f¬ ð>ð �3¬I«f×>ª�è Ö�ñ êLª ð Âi¯l«f³G¯õ¬G­�²�­�±i¯l«j¯sÂi±j²�­�°�Øoª�­>­�¯sØo±j²eª�­�Í�²e±j·Î±j·�¯´×>²�Âi±i«f² ð ç>±j²eª�­®ªGÑBØ ó Ø Þ ¯Þ ¯s­�°G±j·;ÂB²�­)¬m«g¬G­>×�ª�Ð Ï ¯l«fÐ�ç�±j¬I±j²eª�­ �YË ­�Ѩ¬GØo±l¶�±j·�¯`×>²�Âi±i«f² ð ç>±j²eª�­´ªGÑ �I¸U±j· Þ ¬I«f°G¯sÂi± Ï «f²�Ð�¯�Ѩ¬GØo±iªG«fÂBªGÑ�µq¸�×>²e°�²e±«f¬G­>×>ª�Ð ²�­�±i¯l°G¯l«gÂ`²�Â`¬G ó Ð Ï ±iªG±j²�Øl¬ Þ�Þ�ó ±j·�¯�Âj¬GÐ�¯´¬GÂ!±j·�¯´×>²�Âi±i«g² ð ç�±j²eª�­�ªGÑ �G¸U±j· Þ ª�­�°G¯sÂi±!Ø ó Ø Þ ¯sÂ!²�­�«g¬G­>×�ª�ÐÏ ¯l«fÐ�ç�±j¬I±j²eª�­>ÂtªGÑ3µÚª ð �i¯sØo±j ��û ·�ç>Âl¶�±j·>¯ Þ ²e±i¯l«f¬I±jç�«f¯�ª�­ «f¬G­>×�ª�Ð Ï ¯l«gÐ�ç�±j¬I±j²�ª�­>´è�Ü��9¶ å Ù ê-²�Â`«j¯ Þ ¯l³I¬G­�± �

�Ú¯õ×�¯oø1­�¯

� ã � ä Å � � ã\� � � ä ÑÒªG« � % Ó�ú� È ã � ä Å � È ã\� ú � � � ä ÑÒªG« � ß � ú ã\� Ö ä� ã ��ú�� ä Å � È ã �#� ��ú � � � ä ÑÒªG« � Ô�� Ô � â

Ë ­�Ñ�ªG«gÐ�¬ Þ�Þeó ¶ � ã � ä ²�Â�±j·�¯ Ï «fª ð ¬ ð ² Þ ²�± ó ±j·>¬I±)µ� � Ô " ¶ � È ã � ä ²�Â�±j·>¯ Ï «jª ð ¬ ð ² Þ ²e± ó ±j·>¬I±�µ�È Ô " ¶w¬G­>×� ã ��ú�� ä ²�Â�±j·�¯ Ï «jª ð ¬ ð ² Þ ²e± ó ±j·>¬I± ð ªG±j· µ�È Ô " ¬G­>× µ� � Ô "��;¶!ÑÒªG« ¬ Þ ¬I«j°G¯�«g¬G­>×�ª�Ð ²�­�±i¯l°G¯l«."Í�²e±j· Þ ¬I«j°G¯sÂi± Ï «f²�Ð�¯�ÑÒ¬GØo±iªG«�µ � ¬G­>×�Âi¯sØoª�­;×9¸ Þ ¬I«j°G¯sÂi± Ï «f²�Ð�¯´Ñ¨¬GØo±iªG«�µ È ��ö ªG±i¯)±j·>¬I± � ã � ä Å � ã ��ú �vä ¬G­>×� È ã � ä Å � ã ��ú�� äg�û ·�¯mÑÒç;­>Øo±j²eª�­ � ²�Â�ç;Âjç>¬ Þ�Þeó Øl¬ Þ�Þ ¯s×/@�²�ØfÌ�Ð�¬G­LïWÂtÑÒç>­;Øo±j²eª�­�¬IÑÒ±i¯l« @�²�ØjÌ9Ð�¬G­ è�ÜGÆ êU¶9±j·�ª�ç�°�·ÎÂiª�Ð�¯�¬Gç�±j·>ªG«f«j¯lÑÒ¯l«�±iª � �άGÂ/@�²�ØfÌ�Ð�¬G­LïWÂ�Ѩç>­>Øo±j²�ª�­�¶`¬G­>× MB¯l«gÂj·>²eÌÇè�ñ � ê�Øl¬ Þ�Þ Â�� �ÿÅ �* � ±j·�¯ @�²�ØjÌ9Ð�¬G­9¸ K ª�­>Øf·>¬I«fªv³Ñ¨ç>­>Øo±j²eª�­ �Ë ±!²�Â`Ì9­�ªvÍ�­ ã Âj¯l¯�è å ê ä ±j·>¬I± � Âj¬I±j²�Â\ø;¯sÂ`¬�×>² G1¯l«j¯s­�±j²�¬ Þ ¸�×;² G1¯l«f¯s­>Øo¯�¯���ç>¬I±j²�ª�­� � ã � ä Ê � ã � % �vä Å Ó ã\� Ù ä

ÑÒªG« � ß �I��û ·�ç>Âl¶ � ã � ä Ю¬ ó ð ¯´Øoª�Ð Ï ç�±i¯s× ð ó ­�ç>Ð�¯l«f²�Øl¬ Þ ²�­�±i¯l°G«f¬I±j²eª�­ ÑÒ«jª�Ð� ã � ä Å �� " �� # � �

ã � ä � � ã\� é ä

ÑÒªG« � % �I��û ·�¯�Ѩç>­>Øo±j²�ª�­ � Ð�¬ ó ð ¯´Øoª�Ð Ï ç�±i¯s×&Ñ�«jª�Ð è å ¶>¯���­ �-ã Ü � Ü ä ê

� ã ��ú�� ä Å � ã � ä Ê " � # �

� # � � ã � ä� % � � � â ã\�våGä

ÑÒªG« � Ô� Ô � �uû ·�¯�ÑÒªG«fÐ�ç Þ ¬õÑ�ªG« � °�²e³G¯s­�²�­ÿè å �9¶ Ï � ÖGÖ å êL²�Â�²�­>ØoªG«j«j¯sØo±l¶9¬GÂtØl¬G­ ð ¯�Âj¯l¯s­ ð ó Øoª�­>Âj²�×�¯l«f²�­�°±j·�¯ Þ ²�Ð�²e±t¬GÂ��� � Ê �û ·�¯w«j¯sÂjç Þ ±j ã\� Ù ä ò ã\�våGä Ñ�ª Þ�Þ ªvÍ ÑÒ«jª�Ð MB¯l«fÂj·>²eÌ1ïWÂ5°G¯s­�¯l«f¬ Þ «j¯sÂjç Þ ± ã\� Ü ä ¶�¬ Þ ±j·�ª�ç�°�·)²e±3²�Â Ï ª�ÂfÂj² ð Þ ¯�±iª�×>¯l«f²e³G¯

±j·�¯sÐ Ð�ªG«j¯õ×>²e«f¯sØo± Þeó ¶>¬GÂ!²�­ è å ¶;Ö å ¶>Ö�ñ ê �� ·>¬I« Ï ¬G ó Ð Ï ±iªG±j²�Ø�«j¯sÂfç Þ ±jÂ�Ñ�ªG« � ¬I«j¯�°�²e³G¯s­ ð ó ×�¯ þ «fç;² �i­ÿè�Æ � ¶>Ù å êU¶�¬G­>×άG­�¬G ó Ð Ï ±iªG±j²�Ø�«f¯sÂjç Þ ±�Ñ�ªG« �

²�Â`Âi±j¬I±i¯s×ÿ²�­ è å ê ��û ª Ï «f¯s×>²�Øo±t±j·�¯ Ï ¯l«jÑÒªG«fÐ�¬G­>Øo¯�ªGÑ Ï ·>¬GÂj¯ � ªGÑBæ�0\� ²e±!²�Ât¯s­�ª�ç>°�· ±iª�Ì9­�ªvÍ ±j·>¬I±� ã � % �vä� ã � ä � % Þ ªG° � ã � ä � � Þ ªG° � ã\� ñ ä

¬GÂ � � � �

Ö � Æ ����4��&� 5= <! 5( ��,���2 �-=� 5= .�� � ���=4���? �Ú¯ÿø;«gÂi±�°�²�³G¯ÿ¬ Âj²�Ð Ï Þ ¯G¶!·�¯sç�«g²�Âi±j²�Ø ¯� Ï Þ ¬G­;¬I±j²eª�­ ªGÑõÍ�· óÏ ·>¬GÂj¯ � ªGÑ�æ�0\� Í�ªG«fÌ� � Ý ÂfÂjç>Ð�¯Î±j·>¬I±l¶wÂiªÕÑÒ¬I«�¬GÂ�²�±jÂ Þ ¬I«j°G¯sÂi± Ï «f²�Ð�¯�Ѩ¬GØo±iªG« � ��²�Â)Øoª�­>Øo¯l«f­�¯s×�¶�±j·�¯°G«jª�ç Ï ªG«f×�¯l« � ð ¯s·>¬v³G¯sÂ Þ ²eÌG¯�¬õ«f¬G­>×�ª�Ð ²�­�±i¯l°G¯l«`­�¯s¬I«�� �Bû ·>²�Âu²�Â�­>ªG± ��ç>²e±i¯�ØoªG«j«f¯sØo±l¶ ð ç�±�²�Â�¬G­®¬GØlØlç�«g¬I±i¯¯s­�ª�ç�°�·�¬ Ï>Ï «jªT 9²�Ð�¬I±j²eª�­Ú±iª&ª ð ±j¬G²�­ ¬G ó Ð Ï ±iªG±j²�Ø�«j¯sÂjç Þ ±j ��Ë ­ $jÖ � Ö�Íw¯®±j¬IÌG¯�Ì�­�ªOÍ�­ ×;²e³�²�ÂiªG«fÂ�ªGÑ � ²�­�±iª¬GØlØoª�ç>­�± �

�L¯l± �7Å Þ ªG°�� � Þ ªG° � � ¶LÂjª � � Å � � � � �õû ·�¯ Ï «jª ð ¬ ð ² Þ ²e± ó ±j·>¬I±�ª�­�¯�Ølç�«j³G¯�Âjç>ØlØo¯l¯s×;Â�²�­&ø1­>×>²�­�°�±j·�¯Ñ¨¬GØo±iªG« �Ú²�Â�Ø Þ ª�Âi¯�±iª�±j·>¯ Ï «jª ð ¬ ð ² Þ ²e± ó ±j·;¬I± � ��Ô ���l¶q¬G­>×�Øl¬G­ ð ¯õ¬ Ï>Ï «jª� �²�Ð�¬I±i¯s× ð ó � ã � äg��û ·�ç>Âl¶�±j·�¯¯� Ï ¯sØo±i¯s× ­�ç>Ð ð ¯l«�ªGÑwØlç�«f³G¯sÂ�Ñ�ªG«´Âfç>ØlØo¯sÂj´²�Â� B�´Å � � � ã � äg��Ý Â�¯s¬GØg· Ølç�«j³G¯�«j¯���ç>²e«f¯sÂ�¬ ð ª�ç>±�' ������Å' � � � � � Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²�ª�­> ã Ð�ª�× " ä ¶>±j·�¯�¯� Ï ¯sØo±i¯s×�­�ç>Ð ð ¯l«`ªGÑBÐ�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­> ã Ð�ª9× " ä ²�Â

� ã � ä Å ' �L� � � � � � ã � äwâ ã\� � ä

Page 13: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� ���

@�² Gq¯l«j¯s­�±j²�¬I±j²�­�°�Í�²e±j·Î«f¯sÂ Ï ¯sØo±`±iª ��¶>Í�¯´Âj¯l¯´±j·>¬I±!±j·�¯�Ð�²�­>²�Ð�ç>Ð � ã � ä ª�ØlØlç�«gÂ`Í�·�¯s­Þ ªG°��ÎÅ % � È � ã � ä � � ã � ä!â

6�Âj²�­�° ã\� Ù ä ¬G­;× ã\� ñ ä ¶;Íw¯´ª ð ±j¬G²�­&±j·�¯´ÑÒª Þ�Þ ªvÍm²�­�°)¬G ó Ð Ï ±iªG±j²�Ø�«j¯sÂfç Þ ±jÂtÑÒªG«!±j·�¯´ª Ï ±j²�Ð�¬ Þ Ï ¬I«f¬GÐ�¯l±i¯l«g �

Þ ªG°�� Å � � ã � % �vä� ã � ä � � È Þ ªG° � ú ã ÆIÓ ä

Þ ªG°���� Å � ã � % �vä� ã � ä � � Þ ªG° � ú ã Æ �vä

Þ ªG° � Å % Þ ªG° � ã � ä � � Þ ªG° ��ú ã ÆGÆ äÞ ªG° � Å Þ ªG°�' � % ���� ã � Þ ªG° � ã � äiä �ëÆ � Þ ªG° ��ú ã ÆGÜ ä

Þ ªG° ã � � � � ä Å % � È �� � ) Þ ªG° � ã � ä� , � � â ã Æ Ö ä

û ·�¯�«j¯sÂjç Þ ± ã Æ Ö ä ²�ÂYÐ�ªG«j¯t×�¯ Þ ²�Øl¬I±i¯w±j·>¬G­�±j·�¯wªG±j·�¯l«B¬G ó Ð Ï ±iªG±j²�Ø�«j¯sÂjç Þ ±j ð ¯sØl¬Gç>Âi¯t²e±5²�­�³Gª Þ ³G¯sÂ3Øl¬G­>Øo¯ Þ�Þ ¬I±j²eª�­ªGÑ3±j·�¯�±i¯l«fÐ�Â!ªGÑYªG«g×�¯l« � Þ ªG° � ²�­ ã Æ �vä ¬G­>× ã ÆGÆ äg�©�«jª�Ð ã ÆIÓ ä ¬G­>× ã ÆGÜ ä Í�¯õª ð ±j¬G²�­

Þ ªG° � � � Æ Þ ªG°�� Þ ªG° Þ ªG° � ã ÆGÙ ä¬G � � � �uû ·�ç>Âl¶ � Å � ã ��� ä ÑÒªG«�¬G­ ó�� % Ó �Ö � Ü ��� 4:,/= <'� ��� = ��,���2 �-=! = .�� � ���=4 ��? ��ª�×>ç Þ ª´¬G­�ç>­ Ï «jªv³G¯s× ð ç�± Ï Þ ¬Gç;Âj² ð Þ ¯w¬GÂjÂjç>Ð Ï ±j²eª�­�«j¯l°�¬I«g×>²�­�°±j·�¯õ×;²�Âi±i«f² ð ç�±j²eª�­�ªGÑ Ï «g²�Ð�¯�ÑÒ¬GØo±iªG«gÂ!ªGÑ3«f¬G­>×�ª�Ð ²�­�±i¯l°G¯l«fÂm²�­3�iÂj·�ªG«f±,��²�­�±i¯l«j³ ¬ Þ Âl¶7�L¯s­>Âj±i«f¬&è�ÙGÙ ê5·>¬GÂ�Ð�¬G×�¯±j·�¯ÿ¬I«j°�ç>Ð�¯s­�± Þ ¯s¬G×;²�­�° ±iª ã ÆGÙ ä «g²e°GªG«jª�ç> � Am¯�Âf·�ªvÍ�Â�±j·>¬I± Ï ·>¬GÂi¯ � ªGÑ´æ�0�� ¶�Í�·�¯s­ ¬ Ï;Ï Þ ²e¯s× ±iª ¬Øoª�Ð Ï ª�Âj²e±i¯´²�­�±i¯l°G¯l«�" Í�²�±j· ÂjÐ�¬ Þ�Þ ¯sÂj± Ï «f²�Ð�¯�Ѩ¬GØo±iªG« �-¶�Ím² Þ�Þ ø�­>× ��²�­ ¬G­ ¯� Ï ¯sØo±i¯s×�­�ç>Ð ð ¯l«

� � ã � ä Å ¯� Ï )�� ã Æ�Ê�� ã\�väiä�Þ ªG°�� Þ ªG° Þ ªG° � , ú ã ÆGé ä

ªGÑ�Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²�ª�­> ã Ð�ª�×5" ä ¶�Ím·�¯l«j¯ ±j·�¯ � � ã\�vä � ±i¯l«gÐ�±i¯s­>×>Â�±iªÚül¯l«fª ¬GÂO� � � � û ·�¯ ¯� Ï ¯sØo±i¯s׫fç>­;­>²�­�°�±j²�Ð�¯´²� �

� ã �-ú " ä Å � ã " ä � � ã � ä ú ã Æ åGäÍ�·�¯l«f¯�� ã " ä ²�Ât±j·�¯õ±j²�Ð�¯�«j¯���ç>²e«j¯s×αiª�Ð�ç Þ ±j² Ï Þeó ­�ç>Ð ð ¯l«gÂ`Ð�ª�×;ç Þ ª;" �û ·�¯wѨ¬GØo±iªG« � ã " ä ²�Â5²�Ð Ï ªG«j±j¬G­�± ð ¯sØl¬Gç>Âi¯wÍ�¯`¬I«j¯`²�­�±i¯l«j¯sÂi±i¯s×)²�­)©�¯l«gÐ�¬I±B­�ç>Ð ð ¯l«fÂ$" Å Ã5Ä�ÅëÆIÈ É Ê �

Í�·>²�Øf· Ð�¬ ó ð ¯�³G¯l« ó Þ ¬I«f°G¯ �Îþwó Ít¬ ó ªGÑ!Øoª�Ð Ï ¬I«f²�Âiª�­L¶-±j·�¯�ª ð ³9²eª�ç>Â�Ð�¯l±j·�ª9× ªGÑ!ÂC��ç>¬I«f²�­>° ã Ð�ª�× � ä µ±j²�Ð�¯sÂs¶�Ñ�ªG«!¯s¬GØg·&��Ô ��ªGÑY±j·>¯�Ñ�ªG«fÐ ã\�vä ¶;±j¬IÌG¯sÂm±j²�Ð�¯��

ã �5ú " ä Å ��^Æ #>Ä µ�� ã¨Þ ªG°�� ä È � â ã ÆGñ ä0wª�Ð Ï ¬I«f²�­�°�±j·�¯ ð ª�ç;­>×> ã Æ åGä ¬G­;× ã ÆGñ ä ¶-Í�¯�Âi¯l¯)±j·>¬I±�±j·�¯)Âi¯sØoª�­>×ÚÐ�¯l±j·>ª�×Õ²�Â Ï «j¯lÑÒ¯l«f¬ ð Þ ¯)²eÑ ã¨Þ ªG°�� ä �Oµ²�Â�Âjç���Øl²e¯s­�± Þ�ó ÂjÐ�¬ Þ�Þtã ±j·�ª�ç�°�·Ú²e±�Øl¬G­Ú­�ªG± ð ¯�±iª�ª ÂjÐ�¬ Þ�Þ ¶1²�­�³9²e¯lÍ#ªGÑ ã\�väiäg�õË ­ Ï «f¬GØo±j²�Øo¯G¶Læ�0\� ²�Â�ª�­ Þ�óÑÒ¯s¬GÂj² ð Þ ¯õª�­�Ã5ÄÎÑ�ªG«�Ð�ª9×�¯l«f¬I±i¯�µ ��±j·�¯ Þ ²�Ð�²�±!²�Â�¬ ð ª�ç�±m±j·�¯�Âj¬GÐ�¯�¬GÂm±j·�¯ Þ ²�Ð�²e±`ªGÑBÑÒ¯s¬GÂj² ð ² Þ ²e± ó ªGÑ �;:¯ Ï ²�­�ïW±i¯sÂi± ã Ølç>«j«j¯s­�± Þeó µÚÔ ÆGÆ�¶;Âi¯l¯�è�ÆGé9¶1Æ å ê äg�Ö � Ö ����4��&� 5= <! 5( ��,���2 �-=� 5= .���� ���=4�� ? �L¯s­;Âi±i«f¬9ïWÂB«j¯sÂfç Þ ± ã ÆGé ä ¬ Ï>Ï Þ ²e¯sÂYª�­ Þeó ±iª Ï ·>¬GÂj¯ � ¬G­>×�¬GÂjÂjç;Ð�¯s±j·>¬I±´±j·>¯ �Ú¯s²e¯l«fÂi±i«g¬GÂjÂ�ÑÒªG«fÐ�²�´ç;Âi¯s× ��û ª Ï «j¯s×>²�Øo±�±j·>¯®²�Ð Ï «fªv³G¯sÐ�¯s­�±õ×�¯l«f²�³G¯s×ÕÑ�«jª�Ð Ï ·>¬GÂi¯®Æ ¬G­;×Ú±j·�¯ç>Âi¯�ªGÑO��ª�­�±i°Gª�Ð�¯l« ó ïWÂ�ÑÒªG«fÐ ¶tÍw¯�·>¬v³G¯�±iª�ç;Âi¯�·�¯sç�«g²�Âi±j²�ØάI«j°�ç>Ð�¯s­�±j � �Ú¯Õ¬GÂjÂjç>Ð�¯�±j·>¬I±�±j·�¯&°G«fª�ç ϪG«f×�¯l« � ð ¯s·>¬s³G¯s ã Âiª�Ѩ¬I«m¬GÂ`±j·>¯õ×>²�Âi±i«g² ð ç�±j²eª�­�ªGÑB²e±jÂ Þ ¬I«f°G¯sÂi±!±�Íwª Ï «g²�Ð�¯�ÑÒ¬GØo±iªG«gÂm¬I«j¯�Øoª�­>Øo¯l«f­�¯s× ä`Þ ²eÌG¯´¬«f¬G­>×>ª�Ð ²�­�±i¯l°G¯l«m­�¯s¬I« � � �>¶�Í�·�¯l«j¯ ��±j¬IÌG¯sÂm¬GØlØoª�ç;­�±�ªGÑ3Ì�­�ªOÍ�­&ÂjЮ¬ Þ�Þ ×>²e³9²�ÂiªG«fÂtªGÑ � �wË Ñ ��ª�­�±i°Gª�Ð�¯l« ó ïWÂÑÒªG«fÐ ²�Âmç>Âj¯s×�Í�²�±j·�±j·�¯)Ølç�«j³G¯)Øf·>ª�Âi¯s­Õ¬GÂ�²�­=$gÆ � Ü�¶�±j·�¯s­���Å � Æ ��ÝôÞ ²�Ð�²e±i¯s×�¬GÐ�ª�ç>­�±mªGÑu¯� Ï ¯l«f²�Ð�¯s­�±j¬ Þ

Page 14: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

� È ������������� �

×>¬I±j¬)è å ¶ Ù Ö;¶ Ù�� ê�Øoª�­9ø;«fЮÂL±j·>¬I±Y±j·>²�Â5¬GÂjÂjç>Ð Ï ±j²eª�­´²�Â5«j¯s¬GÂiª�­>¬ ð Þ ¯G¶I±j·�ª�ç�°�· Ï ¯l«g·>¬ Ï ÂLÂ Þ ²e°�·�± Þ�ó Øoª�­;Âi¯l«j³I¬I±j²e³G¯�!æ�0�� Ð�¬ ó Ï ¯l«fÑ�ªG«fÐ Â Þ ²e°�·�± Þeó ð ¯l±i±i¯l«!±j·>¬G­ Ï «j¯s×>²�Øo±i¯s× �Ë Ñ�æ�0\� ²�Â�ç>Âi¯s× Í�²�±j· Ï ¬I«f¬GÐ�¯l±i¯l«f ����¬G­>× � È ¶�¬G­>× ±j·�¯ ã ²�Ð Ï «jªv³G¯s× ä Âi±j¬G­;×>¬I«f× Øoª�­�±j²�­�ç;¬I±j²eª�­ ²�¬ Ï>Ï Þ ²e¯s×�¶1±j·�¯s­ÕÍ�¯�¯� Ï ¯sØo±´¬�ÑÒ¬GØo±iªG«�±iª ð ¯�ÑÒª�ç>­>×Õ²eÑ � È Ô �5�´¬G­>× � �õÔ � È ��Ë Ñ � Å Þ ªG° ã � � � ä � Þ ªG° ���

¬G­>× ��Å Þ ªG° � È � Þ ªG° ���l¶�±j·>¯s­ ã ð ó ª�ç�«m·>¯sç�«f²�Âi±j²�Ø�¬GÂjÂjç>Ð Ï ±j²eª�­ ä ±j·�¯ Ï «jª ð ¬ ð ² Þ ²e± ó ±j·>¬I±�±j·>²�Â!ª�ØlØlç�«gÂ�ÑÒªG«¬G­ ó ª�­�¯�Ølç�«j³G¯)²� � ã ��ú�� äg�´û ·�ç>Âs¶;±j·�¯)¯� Ï ¯sØo±i¯s×Õ­�ç>Ð ð ¯l«�ªGÑ�Ølç�«f³G¯sÂm«f¯���ç>²�«j¯s× ð ó æ�0���²� ã ��ú�� ä Å� � � ã ��ú�� ä ¶1¬G­;× ã ¬GÂjÂjç>Ю²�­�°)±j·>¬I± � ²�Â�ÂjÐ�¬ Þ�Þ�ä ±j·�¯ Ï «jª ð ¬ ð ² Þ ²e± ó ªGÑBÂjç;ØlØo¯sÂjÂ!¬IÑÒ±i¯l«m¬I±�Ð�ª�Âj± � Ølç>«j³G¯sÂ�²�Â

� % ã\� % � ã ��ú�� äiä � 4 � % ¯� Ï ã %�� � ä�â ã Æ�� äË Ñ`±j·�¯ ð ²�«j±j·>×>¬ ó Ï ¬I«f¬G×�ªT Øoª�­�±j²�­�ç>¬I±j²eª�­ ²�Â�ç>Âi¯s×L¶5±j·�¯ Ï ¯l«jÑÒªG«fÐ�¬G­>Øo¯�×�¯ Ï ¯s­>×>´ª�­�±j·�¯�¯� Ï ª�­�¯s­�±��¶

¬G­>× ²e±`²�Â Ï ª�ÂjÂf² ð Þ ¯mÑ�ªG« Ï ·;¬GÂi¯´Æ�±iª�Âjç>ØlØo¯l¯s× Í�²�±j· � � % � È � ©�«jª�Ð ã\� Ü ä ¶;±j·�¯ Ï «fª ð ¬ ð ² Þ ²�± ó ªGÑ5Âfç>ØlØo¯sÂjÂ`ÑÒªG«ª�­�¯õØlç>«j³G¯´²�Â!¬ Ï>Ï «jª� �²�Ð�¬I±i¯ Þeó�� ã Æ ä�� " � � �

ù " � #��� � % ¯� Ï %�� � � ã � � � ä ' � � � � ) �� % + % � ,

� + � �+� ú

Í�·�¯l«f¯�±j·�¯)Âjç>Ð ²�Âmªv³G¯l«�±j·>¯ � ã Æ ä Ï ª�ÂfÂj² ð Þ ¯õ³ ¬ Þ ç>¯sÂ�ªGÑ � ��Ð�ª9×�Æ G¶ � «g¬G­�°G¯sÂ�ªv³G¯l«´±j·>¯�ØoªG«j«f¯sÂ Ï ª�­>×>²�­�°³I¬ Þ ç�¯sÂ`ªGÑ K 0�@ ã Æ �ú � � % �vä ¶�¬G­;×�� Å Þ ªG° ã � � � ä � Þ ªG° ���#% Æ �wÝ ÂjÂjç>Ю²�­�°�Ø Þ ª�Âi¯´±iª�±j·�¯´ª Ï ±j²�Ð�¬ Þ Øf·>ª�²�Øo¯ªGÑ Ï ¬I«f¬GÐ�¯l±i¯l«fÂ�ÑÒªG«�Ѩ¬GØo±iªG«fÂ�ªGÑ�ÜIÓÕ±iªÚÖ�ÓÚ×>¯sØl²�Ð�¬ Þ ×>²e°�²e±jÂs¶u­�ç>Ð�¯l«f²�Øl¬ Þ ²�­�±i¯l°G«f¬I±j²eª�­ ²�­>×;²�Øl¬I±i¯sÂ�±j·>¬I±�±j·�¯¯� Ï ¯sØo±i¯s×ÎØoª�Âi±wªGÑ�±j·�¯ ð ²e«j±j·;×>¬ ó Ï ¬I«f¬G×>ª� ®Øoª�­�±j²�­�ç;¬I±j²eª�­ ã Ím²e±j·�ª�ç�±�ÑÒ¬GÂi± Ï ª Þeó ­>ª�Ð�²�¬ Þ ¯l³I¬ Þ ç>¬I±j²eª�­ ä ²� � Ù�±iª�ÆIÓ� Ð�ªG«j¯ ±j·>¬G­ ÑÒªG«�±j·�¯ ã ²�Ð Ï «jªO³G¯s× ä Âi±j¬G­>×;¬I«f× Øoª�­�±j²�­�ç>¬I±j²�ª�­7²eÑ ÚÅ é�¶w¬G­;× �� ±iª � Ö Ð�ªG«j¯²eÑ ®Å � Æ �®þ ¯sØl¬Gç>Âi¯�±j·�¯�¬G­>¬ Þeó Âj²�Â�²�ÂõÂj²�Ð Ï Þ ¯l«s¶�²�­Õ±j·�¯�ÑÒª Þ�Þ ªOÍ�²�­�°ÎÍw¯�¬GÂjÂjç>Ð�¯�±j·>¯®²�Ð Ï «fªv³G¯s× Âi±j¬G­;×>¬I«f×Øoª�­�±j²�­�ç;¬I±j²eª�­ �û ª&Øf·�ª�ª�Âi¯�ª Ï ±j²�Ð�¬ Þ Ï ¬I«f¬GÐ�¯l±i¯l«gÂl¶LÍw¯®­�ªG±i¯�±j·>¬I±´±j·>¯�±j²�Ð�¯)±iª Ï ¯l«jÑÒªG«fÐ ð ªG±j· Ï ·>¬GÂi¯sÂ�ª�­Õª�­�¯�Ølç�«j³G¯

²�Â Ï «jª Ï ªG«j±j²eª�­>¬ Þ ±iª ' � ���3Ê' È �O� ¶ Ï «jªv³9²�×�¯s×�ªv³G¯l«f·�¯s¬G×;Â�Âfç>Øf·�¬GÂm±j¬ ð Þ ¯�²�­;²e±j²�¬ Þ ²eüs¬I±j²�ª�­ ¬I«j¯�­�¯l° Þ ²e°�² ð Þ ¯ �û ·�¯´Øoª�­;Âi±j¬G­�±j ' ��¬G­>×&' È ×�¯ Ï ¯s­>×Ϊ�­&×>¯l±j¬G² Þ ÂtªGÑ3±j·�¯´²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­ ã Âi¯l¯9$gÜ äg�Ë ­ Ï «f²�­>Øl² Ï Þ ¯G¶>²�ÑBÍ�¯�Ì�­>¯lÍ �Ú²�­�¬G×>³ ¬G­>Øo¯G¶qÍw¯�Øoª�ç Þ ×�Øg·�ª�ª�Âi¯ �5��¬G­>× � È ±iªÎЮ²�­>²�Ð�²�ül¯�±j·�¯�¯� Ï ¯sØo±i¯s׫fç>­Î±j²�Ð�¯G¶�Í�·>²�Øg· ²�Â Ï «fª Ï ªG«f±j²eª�­>¬ Þ ±iª�±j·�¯´¯� Ï ¯sØo±i¯s×&­�ç>Ð ð ¯l«!ªGÑ3Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­>ÂwÐ�ª9× " �

� Å ã ' � � � Ê ' È � � ä � � ã ��ú�� ä!â8 ¯sØl¬ Þ�Þ ±j·>¬I±��ë¬G­>× � ¬I«j¯®Ñ¨ç>­>Øo±j²�ª�­>Â�ªGÑ2����¬G­>×�� È ¶BÂiª&±j·;²�Âõ²�Â�¬&Âj²�Ð Ï Þ ¯ Ï «jª ð Þ ¯sÐ ªGÑ!Ð�²�­;²�Ð�²eüs¬I±j²eª�­²�­ÿ±�Íwª�³I¬I«f²�¬ ð Þ ¯s � � ç Ï>Ï ª�Âi¯�±j·>¬I±�±j·�¯�Ð�²�­;²�Ð�ç;Ð ²�Â������ � ��û ¬ ð Þ ¯sÂ�ªGÑwª Ï ±j²�Ð�¬ Þ Ï ¬I«f¬GÐ�¯l±i¯l«f´¬I«f¯�°�²e³G¯s­²�­Õè�Ü�¶ å ¶9Ù Ö>¶�Ù��9¶ å � êU¶9Í�²e±j·�¯s¬GØf· Ï ¬ Ï ¯l«tÐ�¬IÌ9²�­�°�Â Þ ²e°�·�± Þeó ×>² Gq¯l«j¯s­�±�¬GÂjÂjç;Ð Ï ±j²eª�­> �5Ë ­ û ¬ ð Þ ¯�Æ�Í�¯�°�²e³G¯�¬ÂjÐ�¬ Þ�Þ ±j¬ ð Þ ¯`ªGÑ Þ ªG° ��ù ����� � Ñ�ªG«BѨ¬GØo±iªG«fÂ�ªGÑ�� ×�¯sØl²�Ð�¬ Þ ×>²�°�²e±j � �Ú¯m¬GÂjÂjç>Ð�¯`±j·;¬I± ' �wÅ �G� � Þ ªG°`Æ�¶6' È Å � ¶¬G­>× Þ ªG° ��ù � 4 � % Ó â Ù �\� ª�Ð�¯´Øoª�Ð Ï ç�±i¯s×&³ ¬ Þ ç>¯sÂ`ªGÑ�� ã � ä ¬I«j¯õ¬ Þ Âjª�Âj·�ªvÍm­&²�­ û ¬ ð Þ ¯õÆ�¶>Í�·>¯l«j¯

� ã � ä Å ã¨Þ ªG°����� � ä ÈÞ ªG°�� Þ ªG° Þ ªG°��

úÂiª

����� � Å ¯� Ï ) � � ã � ä�Þ ªG°�� Þ ªG° Þ ªG° � , â� ²�­;Øo¯´±j·�¯�¯� Ï ¯sØo±i¯s×ÿ«fç>­�±j²�Ð�¯)²�Â�²�­;Âi¯s­>Âj²e±j²�³G¯´±iª�Øg·>¬G­�°G¯sÂ�²�­ ����¬G­>× � È ­�¯s¬I«�±j·�¯�ª Ï ±j²�Ð�¬ Þ ³ ¬ Þ ç�¯sÂl¶²e±�²�Â�­�ªG±�²�Ð Ï ªG«j±j¬G­�±�±iª&Øg·�ª�ª�Âi¯�±j·�¯sÐ�¬GØlØlç>«f¬I±i¯ ÞeóG��Ë ­ Ï «f¬GØo±j²�Øo¯G¶-±j·�¯�Âj²e°�­>²eø�Øl¬G­�± Ï ª�²�­�±�²�Â�±j·>¬I±´Íw¯�×�ª

­�ªG±!Ì9­�ªOÍ ��²�­ ¬G×�³I¬G­>Øo¯ � Mu¬I«f²eª�ç>Â!Âj±i«f¬I±i¯l°�²e¯sÂ�·;¬s³G¯ ð ¯l¯s­ Âjç>°G°G¯sÂi±i¯s×&±iª�ªv³G¯l«fØoª�Ð�¯�±j·>²�Â!×;²'��Ølç Þ ± óG�2� ç�«Âi±i«f¬I±i¯l° ó ·>¬G ð ¯l¯s­�±iª�²�­>Øo«f¯s¬GÂi¯)���t¬GÂ!¬�Ѩç>­>Øo±j²eª�­�ªGÑY±j·�¯�­�ç;Ð ð ¯l«tªGÑYØlç>«j³G¯s � Í�·;²�Øf· ·>¬v³G¯ ð ¯l¯s­Î±i«f²e¯s×�¶ç>Âj²�­�°�±j·>¯�ÑÒ¬GØo±�±j·>¬I±�ÑÒªG«´±j·�¯�ª Ï ±j²�Ð�¬ Þ Øf·�ª�²�Øo¯�Í�¯�¯� Ï ¯sØo±)� � � � ±iª ð ¯�¬ ð ª�ç>±�ÜGÜIÓ&Ñ�ªG«�ÜIÓ ¸�×;²e°�²e±�ÑÒ¬GØo±iªG«g¬G­>× ±iª ð ¯�ÑÒ¬G²�« Þeó ²�­>Âi¯s­>Âj²�±j²e³G¯)±iª�±j·�¯�Âj²eül¯�ªGÑt±j·�¯�Ѩ¬GØo±iªG« � K ²e³G¯s­����s¶-Íw¯�Øf·>ª�ª�Âi¯�� È Âiª&±j·�¯®±j²�Ð�¯�ÑÒªG«Ï ·>¬GÂj¯�Æ ²�´¬ ð ª�ç�±�·>¬ Þ Ñw±j·>¬I±õÑÒªG« Ï ·;¬GÂi¯ ��ã ±j·>²�´Øf·�ª�²�Øo¯�²�´­>ªG±õѨ¬I«õÑÒ«jª�Ð ª Ï ±j²�Ю¬ Þ�äg�®Ë Ñ �5� 4 � Ó ý ±j·>²�°�²e³G¯sÂ3� È 4 � ÓGÓ&��� �

� ­>Øo¯ ¬ÕѨ¬GØo±iªG«M� ·>¬G ð ¯l¯s­ ÑÒª�ç>­>×�¶�Í�¯&Øl¬G­7Øoª�Ð Ï ç�±i¯ ±j·�¯ ¯���Øl²�¯s­>Ø ó�� ¶�×�¯oø1­�¯s× ¬GÂ�±j·�¯ «f¬I±j²eªÚªGѱj·�¯´¯� Ï ¯sØo±i¯s× ±j²�Ð�¯�±iª®ø�­>×���Í�²�±j·Îª Ï ±j²�Ð�¬ Þ Ï ¬I«f¬GÐ�¯l±i¯l«fÂ`±iª�±j·�¯�¯� Ï ¯sØo±i¯s×�±j²�Ð�¯�Í�²�±j·Î±j·�¯ Ï ¬I«f¬GÐ�¯l±i¯l«fÂ

Page 15: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� ���

� � ������� � æ Ï ¯sØo±i¯s×&Í�ªG«jÌÎÑÒªG«�æ�0��

×>²e°�²e±jÂ�� Þ ªG° ��ù ����� � �ÆIÓ å�� ÜGÙ �I� é åGåÜIÓ � � Ù å �I� é��GÙÖ�Ó �G�I� Ö � �I��å Ó åÙIÓ � Ü � ÆGÆ �I��å�� ééIÓ � Ö � ñIÓ �I��å ÆGÜ

¬GØo±jç>¬ Þ�Þ�ó ç>Âi¯s× � ©�ªG«m¬G­�¯� �¬GÐ Ï Þ ¯�²�­�Í�·>²�Øf·�Í�¯�Âi±j¬I«j±i¯s×ÎÍm²e±j· � � ±iª�ª�ÂjЮ¬ Þ�Þ ð ç�±�°G«g¬G×>ç>¬ Þ�Þeó ²�­>Øo«j¯s¬GÂi¯s×�²e±±iª®¬�³I¬ Þ ç�¯´Ø Þ ª�Âi¯�±iª®ª Ï ±j²�Ð�¬ Þ ¶�Âi¯l¯ û ¬ ð Þ ¯õÜ �©�«jª�Ð ±j·�¯�¬G ó Ð Ï ±iªG±j²�Ø ð ¯s·>¬v³�²eª�ç>«mªGÑ�±j·�¯�Ѩç>­>Øo±j²�ª�­> � ã � ä ¬G­>× � ã ��ú�� ä ¶L²e±�Øl¬G­ ð ¯�Âj·�ªOÍ�­ÿ±j·>¬I±�±j·�¯

¯� Ï ¯sØo±i¯s×�Â Ï ¯l¯s×>ç Ï�� ð ¯sØl¬Gç;Âi¯uªGÑ9±j·�¯�ç>Âj¯BªGÑ Ï ·>¬GÂi¯�Æ ã Âj±j¬G­>×>¬I«f×´Øoª�­�±j²�­�ç>¬I±j²eª�­ ä ¶GØoª�Ð Ï ¬I«j¯s×´±iª��jç>Âi±Lç;Âj²�­�°Ï ·>¬GÂj¯ � ¶�²�Â�ªGÑ-ªG«f×>¯l« Þ ªG° Þ ªG°�� �YË ±`²�Âw¬I«j°�ç�¯s×β�­Õè å ¶ $ å êq±j·;¬I±w±j·>¯ ð ²�«j±j·>×>¬ ó Ï ¬I«f¬G×�ªT �Øoª�­�±j²�­�ç>¬I±j²�ª�­Î°�²e³G¯s¬ÎÂ Ï ¯l¯s×>ç Ï ªGÑ�ªG«f×>¯l« Þ ªG° � ã ±j·�ª�ç�°�·Õª�­ Þeó ²eÑu¬G ó Ð Ï ±iªG±j²�Øl¬ Þ�Þeó Ѩ¬GÂi± Ï ª Þeó ­�ª�Ð�²�¬ Þ ¯l³ ¬ Þ ç;¬I±j²eª�­Õ²�Â�ç>Âi¯s×%!�ÑÒªG«ª�ç�«�²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²�ª�­>Â�±j·>¯�Â Ï ¯l¯s×>ç Ï ²�Â�ªGÑuªG«f×�¯l« Þ ªG° Þ ªG°�� äg��û ·�¯�Â Ï ¯l¯s×>ç Ï ÑÒªG«�±j·�¯)©Y© û Øoª�­�±j²�­�ç;¬I±j²eª�­²�Â Ï «jª ð ¬ ð Þeó ªGÑ�ªG«g×�¯l« Þ ªG° � ¬I±�Ð�ª�Âi± � Ý�Þ ±j·�ª�ç>°�·�±j·>¯sÂi¯ Â Ï ¯l¯s×>ç Ï Â�¬I«j¯ ²�Ð Ï ªG«j±j¬G­�±�²�­ Ï «g¬GØo±j²�Øo¯G¶�±j·�¯ ó¬I«j¯&±j·>¯lªG«j¯l±j²�Øl¬ Þ�Þeó ²�­>Âf²e°�­>²xø�Øl¬G­�±l¶ ð ¯sØl¬Gç>Âj¯&±j·�¯ ó Øl¬G­ ð ¯�¬ ð ÂiªG« ð ¯s× ²�­�±iª ±j·>¯ � ã\�vä ±i¯l«fÐ ²�­ �L¯s­>Âj±i«f¬9ïW«j¯sÂjç Þ ± ã ÆGé äg� û ·�ç>Âl¶BÍw¯ ¯� Ï ¯sØo± � ã � ä � Æÿ¬G � � � ¶�²�­>×�¯ Ï ¯s­>×�¯s­�±õªGÑmÍ�·�¯l±j·�¯l« Ï ·;¬GÂi¯&Æ�²�Â�ç;Âi¯s× �û ¬ ð Þ ¯�Æ®Âj·�ªvÍmÂ!±j·>¬I±m±j·�¯�Øoª�­�³G¯l«j°G¯s­>Øo¯�²�Â`³G¯l« ó Â Þ ªvÍ(¬G­>×&±j·;¬I± � ã � ä 4 �Iâ�å Ñ�ªG« �Õ²�­ ±j·�¯�ÆGÙ®±iª®Ö�Ù®×>²e°�²e±«f¬G­�°G¯ �

�Ú¯ ­�ªG±i¯&¬ Þ ¬GØjÌ�ªGÑm ó Ð�Ð�¯l±i« ó ²�­ ã Æ�� ä Í�·>²�Øg·�Ð�¬ ó ð ¯�ªGÑm²�­�±i¯l«j¯sÂi±�±iªÚØo« ó Ï ±iªG°G«f¬ Ï ·�¯l«fÂ&è å ÓGê �ÚË Ñ �²�Â�Ð�ç>Øg· Þ ¬I«j°G¯l«�±j·;¬G­ )¶3Âj¬ ó � Å � ÓGÓ )¶3±j·�¯s­�±j·�¯ Ï «jª ð ¬ ð ² Þ ²e± ó ªGÑ`Ѩ¬G² Þ ç�«j¯�²�´¯� Ï ã % � ÓGÓ ä ¶uÂiª�Í�¯Î¬I«j¯¬ Þ Ð�ª�Âj±!Øo¯l«j±j¬G²�­&±iª�ø�­>× ¬)Ѩ¬GØo±iªG« ��� ­&±j·�¯�ªG±j·�¯l«m·;¬G­>×�¶�²eÑ � ²�Â`Ð�ç>Øg·�ÂjÐ�¬ Þ�Þ ¯l«`±j·>¬G­� �¶>Âj¬ ó � Å � � ÓGÓ9¶±j·�¯s­ � % ¯� Ï ã %�� � ä 4�� � 4 Ó â Ó � ²�Â�ÂjÐ�¬ Þ�Þ ¶ ð ç�±�­�ªG±�¯� Ï ª�­�¯s­�±j²�¬ Þ�Þ�ó Âiª ��û ·�ç>Âl¶Læ�0\� ·>¬GÂõ¬ ­�ª�­9¸­�¯l° Þ ²�°�² ð Þ ¯)Øf·>¬G­;Øo¯®ªGÑwø�­>×>²�­�°�Ѩ¬GØo±iªG«fÂõÍ�·>²�Øg· ¬I«j¯®Ð�ç>Øf· Þ ¬I«j°G¯l«õ±j·;¬G­Ú¯� Ï ¯sØo±i¯s× � ©�ªG«õ¯� �¬GÐ Ï Þ ¯G¶5²eÑw±j·�¯ÍwªG«jÌ Ï ¯l«jÑÒªG«fÐ�¯s×�²�Â)Âjç���Øl²e¯s­�±�±iª�ø1­>× ÜIÓ ¸�×;²e°�²e±�ÑÒ¬GØo±iªG«fÂ�Í�²e±j· Ï «jª ð ¬ ð ² Þ ²e± ó Ó � Ù�¶�±j·�¯s­ Í�²e±j·�±j·�¯ Âj¬GÐ�¯ÍwªG«jÌ&±j·�¯l«j¯)²�Â�¬ ð ª�ç�±mª�­>¯�Øg·>¬G­>Øo¯�²�­ � ÓGÓ�ªGÑuø�­>×>²�­>°�¬�Ѩ¬GØo±iªG«�ªGÑuÖ�ÓÎ×>²e°�²�±jÂ�¬G­>×ÿ¬ ð ª�ç�±mª�­>¯�Øg·>¬G­>Øo¯�²�­� ÓGÓGÓGÓ�ªGÑ-ø�­>×>²�­�°�¬)Ѩ¬GØo±iªG«�ªGÑ3ÙIÓ�×>²e°�²e±j ã Íw¯´×�ª�­�ªG±!¬I±i±i¯sÐ Ï ±`±iª ð ¯ Ï «j¯sØl²�Âi¯ ð ¯sØl¬Gç;Âi¯�±j·�¯ Ï «jª ð ¬ ð ² Þ ²e±j²e¯sÂ×�¯ Ï ¯s­>×αiª®Âjª�Ð�¯´¯� 9±i¯s­�±!ª�­�·�ªvÍ ±j·�¯ Ï ¬I«f¬GÐ�¯l±i¯l«fÂ2����¬G­>× � È ¬I«j¯õØg·�ª�Âi¯s­ äg�

Ù ���q£ #�+����� § 3* �6��#�> 5¡6�>¡1§Ò£-  �©�ªG«mÑÒç�±jç>«j¯�«j¯lÑÒ¯l«j¯s­>Øo¯G¶>Íw¯´×�¯sÂjØo«f² ð ¯�Âi¯l³G¯l«g¬ Þ ªGÑ3ª�ç�«!²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­>ÂtªGÑBæ�0\� �� ? � ç�«�ø;«fÂi±�²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­®Íw¬GÂ�Í!«f²�±i±i¯s­®²�­ � �GñGÙ�¶9Ю¬G²�­ Þeó ²�­7�3¬GÂjØl¬ Þ ¶ ð ç>±�Í�²e±j·®Âiª�Ð�¯m¬GÂfÂi¯sÐ ð Þ ¯l«

±iªÎÂ Ï ¯l¯s×�ç Ï Þ ¬I«j°G¯o¸�²�­�±i¯l°G¯l«�Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²�ª�­> ��Ë ±mç;Âi¯s× ��ª�­�±i°Gª�Ð�¯l« ó ïWÂ�ÑÒªG«fÐ� ãUåGä ¬G­;× ã ñ ä Ñ�ªG« Ï ·>¬GÂj¯ � ¶¬G­>×�Øoª�­�³G¯l«j±i¯s× ð ¬GØjÌ�±iª�Ú¯s²e¯l«fÂi±i«f¬GÂfÂu­�ªG«fÐ�¬ Þ ÑÒªG«fÐ ã Æ ä ÑÒªG« Ï ·>¬GÂi¯!Æ�¶�Ím·>²�Øf·)ç>Âj¯s×)±j·�¯ ð ²e«j±j·;×>¬ ó Ï ¬I«f¬G×�ª� ²�×�¯s¬ �"8 ¬I±j²eª�­>¬ Þ Ï «j¯sØoª�­>×>²�±j²eª�­>²�­�°�è�éGÙ ê1Íw¬GÂwç>Âi¯s×®±iª�Â Ï ¯l¯s×�ç Ï®Ï ª Þeó ­�ª�Ð�²�¬ Þ ¯l³ ¬ Þ ç;¬I±j²eª�­®²�­ Ï ·>¬GÂi¯�Æ �uû ·�¯²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­ ¬GØf·>²�¯l³G¯s× ' �wÅ � Ó�� Þ ªG°tÆ�¬G­;× ' È Å � �IÆ ã «f¯sØl¬ Þ�Þ ±j·>¬I±l¶;¬GÂ�²�­1$gÆ � Ævò$gÜ � Ü�¶q±j·�¯´­�ç;Ð ð ¯l«ªGÑ;Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­>Â�Ð�ª�× " Ï ¯l«5Ølç�«j³G¯t²�ÂY¬ ð ª�ç�± ' ���5�vÊ ' È �O� äg� �u«jªG°G«g¬GÐ Ý «g¬G­õª�­�³I¬I«f²eª�ç>Â5Ð�¬GØf·>²�­�¯sÂl¶²�­>Ø Þ ç>×>²�­�° � ç>­&Ü�¬G­;× M Ý�� ¶�¬G­>× ÑÒª�ç>­>× Ð�¬G­ ó ÑÒ¬GØo±iªG«fÂ�ªGÑ3ÆGÙ®×�¯sØl²�Ð�¬ Þ ×;²e°�²e±jÂtªG« Þ ¯sÂfÂõè � ñ ê �')?�Ý Âj²�Ð Ï Þ ¯ û ç�« ð ª �3¬GÂjØl¬ Þ ²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­õÍt¬GÂYÍ!«f²e±i±i¯s­�²�­ � �GñGé�Ñ�ªG«3¬G­ ËQþ � ��0Ûèd� ê � ��«jªG°G«f¬GÐ þ

ç>Âi¯s´±j·�¯/0t¬Gç>Øg· ó ÑÒªG«fÐ ã é ä ¬G­>× ª�­ Þeó Ï ·>¬GÂi¯ � ²�Â�²�Ð Ï Þ ¯sÐ�¯s­�±i¯s× ��û ·�¯�²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­ÕªGÑtÐ�ç Þ ±j² Ï Þ ¯o¸Ï «j¯sØl²�Âj²eª�­ ¬I«f²e±j·>Ð�¯l±j²�Ø®ç>Âi¯sÂ�Âj±i«f²�­�°�Â�ªGÑm×�¯sØl²�Ð�¬ Þ ×;²e°�²e±jÂl¶BÂiªÚ²�Â)Âj²�Ð Ï Þ ¯ ð ç>±�²�­>¯���Øl²e¯s­�± � ��«jªG°G«f¬GÐ þ ²�ÂÐ�¬G²�­ Þeó ç>Âi¯s×αiª�°G¯s­�¯l«g¬I±i¯´±j¬ ð Þ ¯s´è � ñ êU¶�±j¬IÌ9²�­�°�²�­�±iª�¬GØlØoª�ç>­�±�¬ Þ °G¯ ð «g¬G²�Ø�¬G­>× Ý ç>«f²eÑÒ¯sç>² Þ�Þ ²�¬G­)ÑÒ¬GØo±iªG«fÂ�è � ÖGêU¶¬G­>×�¬GØlØo¯sÂjÂj²�­>°m¬�×>¬I±j¬ ð ¬GÂi¯wªGÑ�ªv³G¯l« � �IÓ9újÓGÓGÓ�Ì9­�ªvÍm­´Ñ¨¬GØo±iªG«f �BÝ ÂY¬ ð ó Ï «jª9×>ç>Øo±l¶ Ï «jªG°G«f¬GÐ þ Øl¬G­ Ï «jª9×>ç>Øo¯Þ ²�Âj±jÂtªGÑYØoª�Ð Ï ª�Âj²�±i¯sÂ!Í�·>²�Øf· ¬I«j¯´ç>Âj¯s×&¬GÂ�²�­ Ï ç�±w±iª�ªG±j·�¯l« Ï «jªG°G«f¬GÐ�Âl¶�¯ � ° � Ï «jªG°G«f¬GЮ 0�ò�æ ð ¯ Þ ªOÍ �

Page 16: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

�U÷ ������������� �

��? � ·�¯s­�¬�³G¯sØo±iªG« Ï «jª9Øo¯sÂjÂiªG«fÈ ð ¯sØl¬GÐ�¯!¬v³ ¬G² Þ ¬ ð Þ ¯t¯s¬I« Þeó ²�­ � �GñGñ�¶�¬�©�ªG«f±i«f¬G­ Ï «jªG°G«g¬GÐ ��M�© Ý 0 Ít¬GÂÍ!«f²�±i±i¯s­ ã ð ¬GÂi¯s× ª�­ Ï «jªG°G«f¬GÐ Ý ¶�Í�²�±j· Âiª�Ð�¯õ²�Ð Ï «jªv³G¯sÐ�¯s­�±jÂm¬G­>×&Âj²�Ð Ï Þ ²xø�Øl¬I±j²�ª�­>Â�è �G� ê äg� MB¯sØo±iªG«f²eüs¬I±j²eª�­²�Â�¬GØlØoª�Ð Ï Þ ²�Âj·�¯s× ð ó Í�ªG«jÌ9²�­�°�ª�­Ú¬Î­�ç;Ð ð ¯l«�ªGÑ�¯ Þ�Þ ² Ï ±j²�Ø´Ølç�«f³G¯sÂ�²�­ Ï ¬I«f¬ Þ�Þ ¯ Þ ×>ç�«f²�­�° Ï ·>¬GÂj¯ �I� ��·>¬GÂi¯�Ʋ�Ð Ï Þ ¯sÐ�¯s­�±jÂ5±j·>¯ ð ²e«j±j·>×;¬ ó Ï ¬I«f¬G×�ª� �²�×�¯s¬�¬GÂu²�­7$gÜ � Ü � @�ç>«f²�­�° Ï ·>¬GÂi¯�Æ�±j·>¯ Ï «jªG°G«f¬GÐ#ÍwªG«jÌ9ÂBª�­�ª�­ Þeó ª�­�¯Ølç�«j³G¯)¬I±�¬�±j²�Ð�¯G¶ ð ç�±�±j¬IÌG¯sÂ�¬G×�³ ¬G­�±j¬I°G¯�ªGÑu±j·�¯�³G¯sØo±iªG«õç>­;²e±jÂm×;ç�«f²�­�° Ï ª Þeó ­�ª�Ð�²�¬ Þ ¯l³I¬ Þ ç>¬I±j²eª�­ � 6�­ Þ ²eÌG¯Ï «jªG°G«g¬GÐ Ý ¶ ð ªG±j· Ï ·>¬GÂi¯sÂYç>Âi¯G��ª�­�±i°Gª�Ð�¯l« ó ïWÂ3Ñ�ªG«gÐ ã ñ ä ¶�Í�²�±j·�' ��Å �G� � Þ ªG°`Æ�¬G­>× ' È Å �I�3û ·�¯w²�­>²e±j²�¬ ÞÏ ª�²�­�±�²�Â�ç>Âjç>¬ Þ�Þ�ó Øf·�ª�Âj¯s­ ±iª&¯s­>Âfç�«j¯)±j·>¬I±´±j·�¯�°G«fª�ç Ï ªG«f×�¯l«�²�Â�×>²e³9²�Âj² ð Þ ¯ ð ó � Æ�¶3¬G´×�¯sÂjØo«g² ð ¯s×ÿ²�­ $gÆ � Ü ��ÿç Þ ±j² Ï Þ ¯o¸ Ï «j¯sØl²�Âj²�ª�­ ¬I«f²e±j·;Ð�¯l±j²�Ø ã Í�²e±j· ð ¬GÂi¯ ÆIÈ ý ä ²�­ ±j·�¯ ²�­>­�¯l« Þ ª�ª Ï Â ²�Â Ï ¯l«jÑÒªG«fÐ�¯s×ëç>Âj²�­�° ×>ª�ç ð Þ ¯o¸Ï «j¯sØl²�Âj²eª�­ � ª�¬I±j²�­>°I¸ Ï ª�²�­�±`ª Ï ¯l«f¬I±j²eª�­; ������� ¬G­>×����� �� � ª Ï ¯l«f¬I±j²�ª�­>Â!¬I«j¯õç;Âi¯s× ±iª®Â Ï Þ ²e±`¬ Ï «jª9×>ç>Øo±`²�­�±iª·>²e°�·�¬G­>× Þ ªOÍ`¸UªG«f×�¯l« Ï ¬I«j±j � � Ï ¯l«g¬I±j²eª�­>Â�Í�·>²�Øg·�¬I«j¯�­�ªG±w±j²�Ð�¯o¸�Øo«g²e±j²�Øl¬ Þ ¶9Âjç>Øg·�¬GÂ`²�­ Ï ç�±�¬G­;×�ª�ç>± Ï ç�±l¶�¬I«j¯Ï ¯l«jÑÒªG«fÐ�¯s× ç;Âj²�­�° ±j·�¯ �1� Ï ¬GØjÌI¬I°G¯ è ÖIê � �u«fªG°G«f¬GÐ 0ôÑÒª�ç>­>× ±j·>¯®Ñ¨¬GØo±iªG«f²eüs¬I±j²eª�­ ªGÑ�à ��� ã Âi¯l¯($ åGä ¬G­>×Ð�¬G­ ó ÑÒ¬GØo±iªG«gÂl¶9ªGÑ-Âj²eül¯�ç Ï ±iª�Ö�Ó�×>¯sØl²�Ð�¬ Þ ×;²e°�²e±jÂl¶�­>¯l¯s×�¯s×®ÑÒªG«�è � Ù9¶ � é�¶ � ñ ê � D�¯ Þ�Þ ¯l«mè Ö�éIêqç>Âi¯s× Ï «jªG°G«f¬GÐ 0±iª�ø�­;×ÎÑÒ¬GØo±iªG«fÂmç Ï ±iª�Ü���×>²e°�²e±jÂtªGÑ 0tç Þ�Þ ¯s­�­�ç>Ð ð ¯l«f �� ? ��M�© Ý 0 ¬ Þ Âiª&«fç>­;´Í�²e±j· Ð�²�­�ªG«�Øf·>¬G­�°G¯sÂ�ª�­�ªG±j·�¯l«�Ю¬GØf·>²�­>¯sÂõÍ�²�±j· ©�ªG«j±i«f¬G­ Øoª�Ð Ï ² Þ ¯l«fÂl¶5¯ � ° �

� ç>­´Ö�ÍwªG«jÌ�Âj±j¬I±j²eª�­> � ©�ªG«3Ð�¬GØf·>²�­�¯sÂ5ç;Âj²�­�° Ë æ�æ�æ � ª�¬I±j²�­�°I¸ Ï ª�²�­�±5¬I«f²e±j·>Ð�¯l±j²�ØB±j·�¯ ð ¬GÂj¯�Ð�ç>Âi± ð ¯u«f¯s×>ç>Øo¯s×±iª�Æ È ÷ �uÝ�Þ ±j·�ª�ç�°�·Î±j·>¯�Í�ªG«jÌ9Âi±j¬I±j²eª�­;Â!×�ª�­�ªG±!·>¬v³G¯´³G¯sØo±iªG«mç;­>²e±jÂl¶9±j·�¯�³G¯sØo±iªG«f²�ül¯s×�Øoª9×�¯�«fç>­>Âw¯���Øl²�¯s­�± Þeóð ¯sØl¬Gç>Âi¯�ªGÑ-±j·>¯�¯�Gq¯sØo±!ªGÑY¬GÐ�ªG«j±j²eüs²�­>° Þ ª�ª Ï Âi±j¬I«j±jç Ï ªO³G¯l«f·�¯s¬G×>Â!ªv³G¯l«�Âi¯l³G¯l«g¬ Þ Ølç�«j³G¯sÂ`¬G­>×ÎÌG¯l¯ Ï ²�­�°�Ð�ª�Âi±Ð�¯sÐ�ªG« ó ¬GØlØo¯sÂjÂi¯sÂ�²�­ ¬�ÂfÐ�¬ Þ�Þ ÍwªG«jÌ9²�­�°�Âj¯l± ã ¬G­>× ·�¯s­;Øo¯�²�­ ±j·�¯�Øl¬GØg·�¯ äg� �u«jªG°G«g¬GÐ @ Ñ�ª�ç;­>× ±j·�¯��;÷jùѨ¬GØo±iªG«�ªGÑBà ��ù ã Âj¯l¯9$gé äg��N? �5¬I±i¯w²�­ � ��� Ö�¶�ª�ç�« � ç>­õÖ�Øoª9×�¯wÍw¬GÂ Ï ªG«j±i¯s×´±iª�±j·�¯w©>ç��j²e±jÂjç Ý � � ÓGÓGÓ9¶�Í�·>²�Øf·õ²�Â5¬ Ï ¬I«f¬ Þ�Þ ¯ Þ Ð�¬GØf·;²�­�¯

ç>Âj²�­�°�ÆGÙ ��·�ü � Ï ¬I«gØ Ï «jª9Øo¯sÂjÂiªG«fÂ)è�ÜGÙ9¶1Ö�Ü ê �mû ·�¯�Ð�¬GØg·>²�­�¯�¬s³I¬G² Þ ¬ ð Þ ¯õ±iªÎç>Â�·>¬G � ÆGñ Ï «jª9Øo¯sÂjÂiªG«f � æ�¬GØg·Ï «jª9Øo¯sÂjÂiªG«�ÍwªG«jÌ9Â�ª�­�ª�­>¯�ªG«´Ð�ªG«j¯�Ølç>«j³G¯sÂ�²�­>×�¯ Ï ¯s­>×>¯s­�± Þeó ¶1¬G­>×ÕØoª�Ð�Ð�ç>­>²�Øl¬I±i¯sÂ�«j¯sÂfç Þ ±j 㠲eÑ�¬G­ ó�ä ¬IÑÒ±i¯l«¯s¬GØf· Ï ·>¬GÂi¯ � �u«jªG°G«f¬GÐ#æÕÑÒª�ç>­>×�±j·�¯��;÷��5Ѩ¬GØo±iªG«uÐ�¯s­�±j²eª�­�¯s×)²�­7$gÙ � Æ�¶�¬G­>×�ªG±j·�¯l«BÑÒ¬GØo±iªG«gÂ3Ñ�ªG«3¬G­�¯� �±i¯s­>Âf²eª�­ªGÑ3±j·�¯ 0tç>­;­>²�­�°�·>¬GÐ Ï «fª��\¯sØo±õè � ñGê �� ?wË ­ @�¯sØo¯sÐ ð ¯l« � ��� Ö�¶`¬ ­>¯lÍ ²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­ Ít¬GÂ�Í�«f²e±i±i¯s­ ²�­ 0 ÑÒªG«�±j·�¯�@�ç ð ­>¯l«/0w«fç>­>Øg·�¯l«

ã Âi¯l¯�$gñ äg��Ë ­>²�±j²�¬ Þ�Þeó ±j·�¯70 Øoª�×>¯�Ít¬GÂ�Ð�ª9×�¯ Þ�Þ ¯s×�ª�­ Ï «jªG°G«g¬GÐ þ ¶�Âiª ç>Âj¯s×�±j·>¯90`¬Gç>Øf· ó ÑÒªG«fÐ � ��·>¬GÂi¯�ÆÍt¬GÂ-²�Ð Ï Þ ¯sÐ�¯s­�±i¯s×�ç>Âj²�­�°`¬!Âj±iªG«f¬I°G¯o¸U¯���Øl²e¯s­�±5³G¯l«gÂj²eª�­�ªGÑ9±j·�¯ ð ²�«j±j·>×>¬ ó Ï ¬I«f¬G×>ª� ´²�×�¯s¬9¶IÐ�ç;Øf·´¬GÂ5×�¯sÂjØo«f² ð ¯sײ�­1$gÜ � Ü � �u«jªG°G«f¬GÐ © Ñ�ª�ç>­;× ¬O� Èjí Ѩ¬GØo±iªG«�ªGÑBà ��� ã Âj¯l¯9$gñ äg�

� ?�Ë ­ &�ç>­�¯ � ���GÙ ±j·�¯ 0w«gç>­>Øf·>¯l« Ï «fªG°G«f¬GÐ Ít¬GÂ�«j¯lÍ!«f²�±i±i¯s­7±iª ç>Âi¯�±j·�¯ ��ª�­�±i°Gª�Ð�¯l« ó ÑÒªG«fÐ ã ñ ä±j·�«jª�ç>°�·�ª�ç�±l¶5¬G­;×Ú±iª�ç>Âj¯�¬&Âi±iªG«f¬I°G¯o¸U¯���Øl²�¯s­�±�²�Ð Ï «jªv³G¯s× Âj±j¬G­>×>¬I«f× Øoª�­�±j²�­�ç>¬I±j²eª�­ÚÑÒªG« Ï ·>¬GÂi¯®Æ ã ¬GÂ�×�¯o¸ÂjØo«f² ð ¯s×Õ²�­ $gÜ � Æ äg� ��«jªG°G«f¬GÐ K ²�Â�Âf²e°�­>²xø�Øl¬G­�± Þeó Ѩ¬GÂi±i¯l«´±j·>¬G­ Ï «jªG°G«f¬GÐ © ã ¬ ð ª�ç�±´ÜGÙ� Ѩ¬GÂi±i¯l«´ª�­Ú¯s¬GØg·Ølç�«j³G¯G¶�¬G­>× ±j·�¯Î¯� Ï ¯sØo±i¯s× ­�ç>Ð ð ¯l«�ªGÑmØlç�«f³G¯sÂ�²�Â�«f¯s×>ç>Øo¯s× ð ¯sØl¬Gç>Âi¯�±j·�¯Î°G«jª�ç Ï ªG«f×�¯l«�²�Â�×>²e³9²�Âj² ð Þ ¯ ð ó� Æ äg��û ·>¯ 0w«fç>­;Øf·�¯l« Ï «jªG°G«f¬GÐ�Â�© ¬G­>× K ×�ªõ­>ªG±�ç>Âi¯�¯� �±i¯s­>×>¯s× K 0 @ëØoª�Ð Ï ç�±j¬I±j²eª�­; � ©�ªG«w±j·>²�ÂB«j¯s¬GÂjª�­�¶±j·�¯®Øoª�­;Âi±j¬G­�±j ' �õÅ � Æ�� Þ ªG°!Æ ã Ñ�ªG« Ï «jªG°G«f¬GÐ K ä ¬G­>×+' È Å Ü ¬I«j¯ Þ ¬I«j°G¯l«´±j·>¬G­ Ñ�ªG« Ï «jªG°G«f¬GÐ� 0�ò�æ �û ·�¯m¯�G1¯sØo±tª�­�«j¯ Þ ¬I±j²e³G¯ Ï ¯l«jÑÒªG«fÐ�¬G­>Øo¯�²�Â Þ ¯sÂfÂ�±j·>¬G­�Ð�²e°�·�±�¬ Ï;Ï ¯s¬I«v¶ ð ¯sØl¬Gç>Âi¯�±j·�¯mªv³G¯l«f¬ Þ�Þ Ï ¯l«jÑÒªG«fÐ�¬G­>Øo¯�²�ÂÐ�ªG«j¯�Âi¯s­>Âj²e±j²�³G¯�±iª ' � ±j·>¬G­ ±iª ' È �Ù ���I��� /4&9 �/2 <! �� 2� 5('��<! 5.�, ��257�. � <89 ? ��ª�Âi±YªGÑ>±j·�¯tØoª�Âi±5ªGÑ�æ�0\� ²�Â-²�­ Ï ¯l«jÑ�ªG«gÐ�²�­�°!Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­>ÂÐ�ª9× " �1� ç�« Ï «jªG°G«f¬GÐ�Â Ý ¸ K ¬ Þ�Þ ç>Âi¯®±j·�¯�Ø Þ ¬GÂjÂj²�Øl¬ Þ � ã µ È ä ¬ Þ °GªG«g²e±j·>Ð�±iªÕÐ�ç Þ ±j² Ï Þ�ó µq¸ ð ²e±�­�ç>Ð ð ¯l«f �D´¬I«f¬I±jÂfç ð ¬9ïWÂ)¬ Þ °GªG«f²e±j·;Ð$è Ö å ¶�$jÖ � Ü � ÜOêmªG«�ªG±j·>¯l« �\Ѩ¬GÂi±,�Ú¬ Þ °GªG«g²e±j·>Ð�Â�è�ÆGé�¶BÆGñ êwÍwª�ç Þ × ð ¯ Ï «f¯lÑ�¯l«f¬ ð Þ ¯®ÑÒªG«Þ ¬I«j°G¯`µ �3û ·�¯!Øo«fª�ÂjÂiªv³G¯l« Ï ª�²�­�±B×�¯ Ï ¯s­>×;Â5ª�­®×�¯l±j¬G² Þ Â3ªGÑ1±j·�¯!²�Ð Ï Þ ¯sÐ�¯s­�±j¬I±j²eª�­ � ��ªG«f¬G²�­�è�éGÆ9¶ 0t· � ÙOê�Âj±j¬I±i¯s±j·>¬I± D´¬I«g¬I±jÂjç ð ¬9ïWÂuÐ�¯l±j·�ª�×�²�Â3Í�ªG«j±j·�Í�·>² Þ ¯�ÑÒªG«uµÚßÛñIÓGÓ´ª�­�¬´ÜGÆO¸ ð ²e±BÍwªG«jÌ�Âj±j¬I±j²eª�­ �Bû ·�¯!Øo«jª�ÂfÂiªv³G¯l« Ï ª�²�­�±ª�­ ¬�é ÖI¸ ð ²e±´³G¯sØo±iªG« Ï «jª9Øo¯sÂjÂiªG«õ²�Â Ï «jª ð ¬ ð Þ�ó Â Þ ²e°�·�± ÞeóÿÞ ¬I«j°G¯l« �>� ­ ¬(0w«fç;­>Øf·�¯l«�±j·>¯�Øo«jª�ÂjÂiªO³G¯l«�²�ÂõÐ�ç>Øg·Þ ¬I«j°G¯l« ð ¯sØl¬Gç;Âi¯�±j·�¯õÐ�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­�±j²�Ð�¯´²�Â!¯sÂjÂi¯s­�±j²�¬ Þ�Þeó�Þ ²�­�¯s¬I«!²�­ÎµÚÑÒªG«!µ � � ÓGÓGÓGÓ ã Âi¯l¯ û ¬ ð Þ ¯�Ö äg�

�u«fªG°G«f¬GÐ� þ ò�æ ×�ª ­�ªG±�±j¬IÌG¯ÿ¬G×�³I¬G­�±j¬I°G¯�ªGÑ�±j·>¯�Â Ï ¯sØl²�¬ Þ ÑÒªG«fÐ�ªGÑ�©�¯l«fÐ�¬I±�­�ç>Ð ð ¯l«fÂ)Í�·�¯s­ ×�ª�²�­�°¬I«f²e±j·;Ð�¯l±j²�Ø´Ð�ª9×&" � AmªvÍw¯l³G¯l«s¶q±j·�¯�Ð�ª�×�ª Ï ¯l«g¬I±j²eª�­�²�Â�²�Ð Ï Þ ¯sÐ�¯s­�±i¯s× ¯���Øl²e¯s­�± ÞeóG� ©�ªG« Ï «jªG°G«g¬GÐ� 0�ò�æ

W ,�k T�XUTbdgaWa��Îd�p r'RÒTWXUP/r��36�|i}g}õyLT�XUZ �l8 �õk P/M\R�RQab[lR��®RQ�vR\aWME@LXUZ TWP`]!dgR�Z TWk M�y�dfP�rO_ �fS�d^NOM\N®TWk�]`TbNs��|\�^�s|�XU[)d�p r'RÒT�XUP/r�36��^�f}^}��s|i}�yLTWXUZ!~v8 }uk P/MQR��eabdfXUMESL�v8 �Bk P/M\Rj�>R\aW[lRE�wRQ�vR\aWMYdfkINtXUZ M\[fSUMQXUTWRidgaG_GM\dg�tP/_GM\M\N![fc�|\}^}g}t�m�I[g_´�ea�djXUMQSY|i�^�f}t�m�I[g_G�E8?qTW]`M\P��lr [gXUM\N�cx[fSLXUZ M��36��^�f}^}`djSUM3cx[gSLXUZOM3cedgP¨XUMES5n^MQSUP/TW[^k�8

Page 17: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� ��î

±j·�¯�ª Ï ¯l«f¬I±j²eª�­������ =��ÛÐ�ª9×/" ²�Â`Øoª�×>¯s×&¬GÂ`¬�Âj²�­>° Þ ¯m«fª�ç�±j²�­�¯ �BÝ Â�� ²�Â`Ð�ç Þ ±j² Ï Þ ²e¯s× ð ó ¯s¬GØf·�×>²e°�²e±����ªGÑ� ã Âj±j¬I«j±j²�­�°´Ím²e±j·�±j·�¯�Ð�ª�Âi±�Âj²e°�­;²xø�Øl¬G­�±u×>²e°�²e± ä ¬G­;×�±j·�¯�Âjç;Ð ¬GØlØlç>Ð�ç Þ ¬I±i¯s×�²�­� ¶�Í�¯�¬ Þ Âiª Ï «j¯s×>²�Øo±¬ ��ç>ªG±j²e¯s­�±�×>²�°�²e± � È ¶�Ð�ç Þ ±j² Ï Þeó " ð ó � È ¶�¬G­>× Âjç ð ±i«f¬GØo± � û ·�¯ Ï «j¯s×;²�Øo±i¯s× ��ç�ªG±j²�¯s­�±�×>²e°�²�±�Øl¬G­ ×>² Gq¯l«ÑÒ«jª�Ð ±j·>¯�ØoªG«j«f¯sØo±�³I¬ Þ ç�¯ ð ó ª�­�¯G¶ ð ¯sØl¬Gç;Âi¯�¬�Â Þ ²�°�·�± Þeó «j¯s×;ç>­>×>¬G­�±�«j¯ Ï «j¯sÂi¯s­�±j¬I±j²eª�­�ªGÑ�±j·�¯�²�­�±i¯l«fÐ�¯s×>²�¬I±i¯«j¯sÂjç Þ ±jÂ!¬ Þ�Þ ªOÍ�Â!¬�ÂjÐ�¬ Þ�Þ ¯l«j«fªG«�±iª ð ¯õØoªG«j«f¯sØo±i¯s×�¬I±m±j·�¯õ­�¯� 9±�Âi±i¯ Ï ã Í�·�¯s­&ÍwªG«jÌ9²�­�°�²�­ ð ¬GÂi¯/��¶;×>²�°�²e±jÂ`²�­±j·�¯�«f¬G­�°G¯�è Ó9ú ��ê�¬I«f¯ Ï ¯l«fЮ²e±i±i¯s× äg�3ÝmÞ Âjª�¶�±j·>¯m«j¯sÂfç Þ ±�ªGÑ-±j·�¯mª Ï ¯l«f¬I±j²�ª�­�²�Â�ª�­ Þ�ó °�ç>¬I«f¬G­�±i¯l¯s×αiª Þ ²�¯m²�­�±j·�¯²�­�±i¯l«j³I¬ Þ è Ó9úfÆ " ä ¬G­>×�±iª ð ¯õØoªG«f«j¯sØo±�Ð�ª�×8" � � ²e±j·&±j·>¯sÂi¯õ«j¯oø1­�¯sÐ�¯s­�±jÂl¶���� � =�� Ð�ª9×/"�Øl¬G­ ð ¯Ï ¯l«jÑÒªG«fÐ�¯s×�¬ Þ Ð�ª�Âj±u¬GÂBÑÒ¬GÂj±�¬GÂ������ =�� � ©�ªG« � ¸ ð ²�±3­�ç>Ð ð ¯l«fÂl¶ Ï «jªG°G«g¬GÐ 0 Ï ¯l«jÑÒªG«fÐ� � ã � �IÆGé ä ÈqÊ � ã � ä� ª Ï Â Ï ¯l«!Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­�Ð�ª9× " !�±j·>²�Â`±j¬IÌG¯sÂ�Ö ã � �IÆGé ä È Ê � ã � ä Ø Þ ª9ØjÌ Ø ó Ø Þ ¯sÂ`ª�­&±j·>¯ M �wÆGÆIÓGÓ'F � Ó �©�ªG«�±j·�¯*0w«gç>­>Øf·>¯l« Ï «jªG°G«f¬GÐ�Â�©5¸ K ¶BÍw¯ ¬s³Gª�²�× ±j·�¯&Ð�ª�× ª Ï ¯l«f¬I±j²eª�­ ¬GÂ�Ð�ç>Øf· ¬GÂ Ï ª�ÂjÂj² ð Þ ¯ �ÕË Ñm±j·�¯

­�ç>Ð ð ¯l«(" ±iª ð ¯�Ѩ¬GØo±iªG«j¯s×&²�Â`¬)Øoª�Ð Ï ª�Âj²�±i¯´×>²e³9²�ÂiªG«wªGÑ3Æ Ä . � ¶�±j·�¯s­Î±j·>¯�¯ Þ�Þ ² Ï ±j²�Ø�Ølç�«j³G¯�ª Ï ¯l«f¬I±j²�ª�­>Â`¬I«j¯Ï ¯l«jÑÒªG«fÐ�¯s× Ð�ª9×�Æ Ä . � «f¬I±j·>¯l«�±j·>¬G­&Ð�ª�× " ��Ý ±m±j·�¯´¯s­>×�ªGÑ3¯s¬GØf· Ï ·>¬GÂi¯�Íw¯�Øoª�Ð Ï ç�±i¯õ¬ K 0 @#Í�²e±j·" ã ­�ªG±�Í�²e±j· Æ Ä . � ð ¯sØl¬Gç>Âi¯®±j·;²�´Íwª�ç Þ × °�²e³G¯�Ì9­�ªOÍ�­ Ѩ¬GØo±iªG«fÂ�ªGÑ ã Æ Ä . �vä � " äg�ÕÝ Ð�²�­>ªG« Ï ¯s­>¬ Þ ± ó²�Â�±j·>¬I±�Íw¯ Í�ªG«fÌ Í�²e±j· µ ªG« ã µÿÊ �vä ¸ ð ²e±�­�ç;Ð ð ¯l«fÂ�«f¬I±j·�¯l«�±j·;¬G­ Í�²e±j· � Þ ªG° È "&�o¸ ð ²e±�­�ç>Ð ð ¯l«f � û ·�¯¬G×�³I¬G­�±j¬I°G¯�²�ÂB±j·>¬I±�Í�¯�Øl¬G­ Ï ¯l«fÑ�ªG«fÐ#±j·>¯!«j¯s×>ç>Øo±j²�ª�­>ÂBÐ�ª9×®Æ Ä . � ç>Âj²�­>° ð ²�­>¬I« ó Âf·>²eÑÒ±3¬G­;×�¬G×>×�F Âjç ð ±i«f¬GØo±ª Ï ¯l«f¬I±j²eª�­>Âs¶�Í�·>²�Øg·Ú¬I«j¯�Ð�ç;Øf·ÚѨ¬GÂi±i¯l« ã ÑÒªG« Þ ¬I«j°G¯�µ ä ±j·;¬G­ÚÐ�ç Þ ±j² Ï Þeó ªG«õ×;²e³�²�×�¯õª Ï ¯l«f¬I±j²eª�­; ��û ·�ç;Âl¶�±j·�¯0w«fç;­>Øf·�¯l« Ï «fªG°G«f¬GÐ�Â3©5¸ K «fç>­)¬ ð ª�ç�±3±�Í�²�Øo¯t¬GÂ3Ѩ¬GÂi±Bª�­�©�¯l«gÐ�¬I±BªG« ��¯l«gÂi¯s­>­�¯`­�ç>Ð ð ¯l«fÂ3¬GÂ3ª�­ �\°G¯s­�¯l«g¬ Þ �­�ç>Ð ð ¯l«f �

Ù � Æ ��* . 9;4E4�� ��9��82541=@?�þwó ±j·�¯�¯s­>×�ªGÑ � ���GÙ�¶;¬I± Þ ¯s¬GÂj± �vå Ѩ¬GØo±iªG«fÂtªGÑ-Ö�Ó�ªG«`Ð�ªG«j¯�×�¯sØl²�Ð�¬ Þ ×;²e°�²e±jÂt·>¬G×ð ¯l¯s­�ÑÒª�ç>­>× ð ó æ�0\� �1û ª�² Þ�Þ ç>Âi±i«g¬I±i¯´±j·�¯ Ï ¯l«jÑÒªG«fÐ�¬G­;Øo¯õªGÑ�æ�0\� ¶1Íw¯�Ð�¯s­�±j²eª�­�±j·�«j¯l¯�ªGÑu±j·�¯sÂi¯�ÑÒ¬GØo±iªG«g·�¯l«j¯ � @m¯l±j¬G² Þ Â)¬I«j¯&¬s³I¬G² Þ ¬ ð Þ ¯�²�­ è � Æ ê � þ ¯sØl¬Gç>Âj¯ ªGÑ�±j·�¯ Âi¯ Þ ¯sØo±j²�ª�­ Øo«f²e±i¯l«f²eª�­L¶B±j·�¯sÂi¯Î¯� 9¬GÐ Ï Þ ¯sÂ)×�ªÚ­�ªG±² Þ�Þ ç>Âi±i«f¬I±i¯�± ó Ï ²�Øl¬ Þ ð ¯s·>¬v³�²�ª�ç�«�ªGÑ�æ�0\� !B«f¬I±j·�¯l«s¶w±j·�¯ ó ² Þ�Þ ç>Âj±i«f¬I±i¯Î±j·�¯&¬G ó Ð�Ð�¯l±i« ó ­�ªG±i¯s×7¬I±�±j·>¯&¯s­>תGÑ�$jÖ � Ö �û ·�¯ Þ ¬I«f°G¯sÂi±!Ѩ¬GØo±iªG«mÑÒª�ç>­>× ð ó ±j·>¯´¬Gç�±j·�ªG«m²�Â

��÷��wÅëÆGÙGÆGÜGÜ Ö�ÙIÓ �vå éGé � Ù���ñGéGÙGÓGÓ�ÆGÜGÖ�Ó�éGÜ�ñGÆ Ö�ÆIÓ�ñ�� � Ù�Ù å9� Æ � Ü&ú¬ Ѩ¬GØo±iªG«õªGÑ!ÙGÙ � È ý Ê �I��Ë ±´Íw¬GÂõÑ�ª�ç;­>×Úç>Âj²�­>° Ï «jªG°G«f¬GÐ�æ Í�²e±j· ���õÅ Æ Ö�ÓGÓGÓGÓ9¶ � Å � Ù�¶�®Å �I��û ·�¯Ølç�«j³G¯�²�Â`×�¯oø�­�¯s× ð ó $�ÅëñIÓ�ÙGñ � é��Gñ���¶L¬G­>×αj·�¯õ°G«jª�ç Ï ªG«f×>¯l«�²�Â

� ÅÇÆ È ìvÜ È ìsÖ å ìvé � ìOñ���ìvÆGÜGÜ�ìvñGÆ���ìvÜGéIÓ å ì � ñ Ö�Ù � ì Ù Ö�ñGé���ì å ÙGÆGÆGÜ´ì � Ö �GñGÆ å ìvÜ Ö�ÙGÙGÙ ��âû ·�¯)°G«jª�ç Ï ªG«f×�¯l« � ²�Â�¯� 9Øo¯ Ï ±j²�ª�­>¬ Þ�Þeó ÂjÐ�ª�ªG±j· � ©�«jª�Ð $jÖ ��� ¶5±j·�¯ Ï «jª ð ¬ ð ² Þ ²e± ó ±j·>¬I±´¬Î«f¬G­;×�ª�Ð�²�­�±i¯l°G¯l«­�¯s¬I« � � � Æ®·>¬GÂ Þ ¬I«j°G¯sÂi±!¬G­>×�Âi¯sØoª�­>×9¸ Þ ¬I«f°G¯sÂi± Ï «f²�Ð�¯�ÑÒ¬GØo±iªG«fÂm¬GÂ�ÂjЮ¬ Þ�Þ ¬GÂ`±j·�ª�Âi¯õªGÑ � ²�Â`¬ ð ª�ç�±

� ) Þ ªG° ã � � � Æ äÞ ªG° � Ö �GñGÆ å úÞ ªG°wÜ Ö�ÙGÙGÙ �Þ ªG° � Ö �GñGÆ å , 4 Ü â é/= � Ó # í â

Ý ­�ªG±j·�¯l« Þ ¬I«j°G¯ÎѨ¬GØo±iªG«�Ñ�ª�ç;­>× ð ó ±j·>¯Î¬Gç>±j·�ªG«s¶�ç>Âf²�­�° Ï «jªG°G«f¬GÐ 0 Í�²e±j· � � Å Ü å ÓGÓGÓGÓ9¶ � Å ÆGÙGÙ�¶�Åëé�¶;²�Ât±j·�¯õÖ�Ó ¸�×>²e°�²e±`Ѩ¬GØo±iªG«

� ÷jù Å � Ö�Ó>�GñGÙGÜGÆIÓ�ÙGé��GéGéGéGÖ � é�ñ � Ö ��é å9� Ö�ÜGÆ���ÙGÙGÓ å � å ÖGÖ�Ü�� åªGÑ � È î È % � ¶qÍ�·�¯l«j¯B� È î È ²�Â!±j·�¯ Þ ¬I«j°G¯sÂj± Ï «g²�Ð�¯´Ñ¨¬GØo±iªG«�ªGÑ�à ��ù �� ¯l¯/$'�®ÑÒªG«�±j·�¯�¬ Ï;Ï Þ ²�Øl¬I±j²eª�­&±iª Ï «jªv³9²�­�°Ï «f²�Ð�¬ Þ ²e± ó ªGÑ:� È î È ��û ·�¯´Ølç�«f³G¯õ²�Â`×�¯oø�­>¯s× ð ó $�Å Ö�ñ����GñGÜ��Gñ�¶�¬G­>× ±j·>¯�°G«jª�ç Ï ªG«f×>¯l«�²�Â

� Å Æ È ìvÜ�ìvÙ�ì �vå ìOÜ � È ì ÙGÜ�ìvé å ìOÆGÜGÜ�ì å Ü���ìvÙGÙGéGÜ�ì å �IÓ � ìOÆIÓ�ÆIÓ � ì Ù å�� éGÜ�ìOÜIÓ>�GÜGÜGÙ � Ü å®âû ·�¯ Þ ¬I«j°G¯sÂj± Ï «f²�Ð�¯mÑÒ¬GØo±iªG« � �tªGÑ � ²�Ât¬ ð ª�ç�±`ñGÜGé �5�!Ím·>²�Øf·�Âj·�ªOÍ�Âw±j·�¯�³�²�«j±jç�¯�ªGÑ-±j·�¯ ð ²e«j±j·;×>¬ ó Ï ¬I«f¬G×�ª� Øoª�­�±j²�­�ç;¬I±j²eª�­ ��ö ªG±i¯´±j·>¬I± K 0�@ ã Æ Gú � � % �vä Å � Æ �û ·�¯ Þ ¬I«j°G¯sÂj±mѨ¬GØo±iªG«´Ì9­�ªvÍ�­ÿ±iª ·>¬v³G¯ ð ¯l¯s­ÿÑÒª�ç>­>× ð ó æ�0\� ç Ï ±iªÎ±j·�¯)¯s­>×ÿªGÑ � ���GÙ&²�Â�±j·>¯�Ö å ¸�×;²e°�²e±

Ѩ¬GØo±iªG«�;÷ í Å � ÆIÓ�ÆGÙ å Ó�ÆIÓGÓGÓGÓ�éGÙ � ñ�ÜGñGÓ�Ù å Ù � Ù � Ü å ÜGÆ�é � éGÆ å é�Ù � é � ñ � ñGÓGÓ �Gé �

Page 18: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

� ý ������������� �

ªGÑ`ÙIÈ î ý Ê � ¶LÑÒª�ç>­>×Õ²�­ ö ªv³G¯sÐ ð ¯l« � ���GÙ ð ó �Y¯l±i¯l«5��ª�­�±i°Gª�Ð�¯l« ó è�é � êU¶-ç>Âj²�­�°�����Å ÜIÓGÓGÓGÓGÓGÓ&Ím²e±j·Õ·>²�©3© û Øoª�­�±j²�­�ç;¬I±j²eª�­ � ©�ªG«�Ð�ªG«j¯õ×�¯l±j¬G² Þ Âl¶>Âi¯l¯�è � ÆIê ��û ·�¯�°G«jª�ç Ï ªG«f×�¯l«�²�Â

� ÅëÆ ý ìOÜ�ìOÙ�ì å ìvÆGÜ�ì/��� å ìsÖ�ÜGÆGÜ å ìOÙGÙ Ö�Ù å ÜõìvéGÙ�� å ÆGÜ´ìvÆGÆGÆIÓIÖ å ��ìvÆ Ö�Ù�� Ö � å ìT�IÓ�ÜGÜGÜGÙ��Gé�� âé � � ��¦1¡�£-¢�§�� �>¡1§Ò£-  £ � à ��ù

� ·�¯s­�æ�0�� Ít¬GÂw²�Ð Ï Þ ¯sÐ�¯s­�±i¯s×)ª�­®±j·�¯�©>ç��j²e±jÂjç M � � ÓGÓ�²�­ ��¬I«fØg· � �GñGñ�¶�Âiª�Ð�¯mªGÑq±j·�¯`ø;«fÂj±u­�ç>Ð ð ¯l«fÂÍ�·>²�Øf·&Íw¯)¬I±i±i¯sÐ Ï ±i¯s×Õ±iª�ÑÒ¬GØo±iªG«�Íw¯l«j¯�±j·�¯�©�¯l«gÐ�¬I±�­�ç>Ð ð ¯l«fÂ�à �Iújà ��ùGújà ����¬G­>×ÿà � È ¶qç>Âj²�­�°�³I¬I«f²�¬G­�±jÂmªGÑÏ «jªG°G«g¬GÐ 0 � �Ú¯�Íw¯l«j¯�Âiª�ª�­�Âfç>ØlØo¯sÂjÂiѨç Þ Í�²e±j·�à ��� ã Âi¯l¯ $ åGä ¶ ð ç>±�­�ªG±tÍ�²e±j·�±j·�¯�ªG±j·�¯l«`©�¯l«fЮ¬I±!­�ç>Ð ð ¯l«fÂl¶¬ Ï ¬I«j±mÑ�«fª�Ð «j¯s×;²�ÂjØoªv³G¯l«g²�­�°�Ì9­�ªvÍm­�Ѩ¬GØo±iªG«f � �Ú¯�Øoª�­�±j²�­�ç�¯s×�¬I±i±i¯sÐ Ï ±jÂm±iª�Ѩ¬GØo±iªG«�à � ð ó æ�0�� ã ç>­�±j² Þ ²e±Ít¬GÂ�×�ª�­>¯ ð ó �9ö © � ¶5Âj¯l¯�$ �vä ¶Bà ��ù�¬G­>× Ã � È ¶3×�¯l³GªG±j²�­�°�ªGÑw±j·>¯®ªG«f×�¯l«�ªGÑ!ÆGÙ��ªGÑ`¬v³ ¬G² Þ ¬ ð Þ ¯®Øoª�Ð Ï ç�±i¯l«±j²�Ð�¯�±iª�±j·>¯sÂi¯�©>¯l«fÐ�¬I±�­�ç;Ð ð ¯l«fÂm¬G­>×�±j·>¯�«j¯sЮ¬G²�­>×�¯l«!±iªÎªG±j·�¯l« Ï «jª��i¯sØo±j ��û ·�¯ Ï ·>¬GÂi¯ ��Þ ²�Ð�²e±2� � Ít¬G²�­>Øo«f¯s¬GÂi¯s×άGÂw²�± ð ¯sØl¬GÐ�¯�¬ Ï>Ï ¬I«j¯s­�±t±j·>¬I±w±j·�¯�ç>­�Ì9­�ªvÍ�­�Ѩ¬GØo±iªG«fÂtÍ�¯l«f¯�×>² ��Ølç Þ ±u±iª�ø�­;× � A�ªOÍ�¯l³G¯l«s¶;±j·�¯l«j¯Íw¯l«j¯`Âiª�Ð�¯tØoª�­>Âi±i«f¬G²�­�±jÂ3ª�­-��� �Bþ ¬I±jØf· �\ª ð Â5Íw¯l«j¯ Þ ²�Ð�²e±i¯s×õ±iª�¬I±3Ð�ª�Âi±3±�Íwª´·�ª�ç�«fÂY¬G­>×�¶I±iª�Ð�¬IÌG¯t¯���Øl²e¯s­�±ç>Âi¯�ªGÑ�±j·>¯�³G¯sØo±iªG«�ç>­>²e±jÂl¶1Íw¯)·;¬G×�±iª Øoª�Ð Ï Þ ¯l±i¯�Âi¯l³G¯l«f¬ Þ Ølç�«j³G¯sÂ�²�­�±j·>¬I±�±j²�Ð�¯ �´û ·�¯�±j²�Ð�¯�Ñ�ªG« � Ølç>«j³G¯sÂ×�ª�­�¯�Âj²�Ð�ç Þ ±j¬G­�¯lª�ç> Þeó Ít¬GÂ Ï «jª Ï ªG«j±j²eª�­>¬ Þ ±iª � Ê � � � È ¶5Ím·�¯l«j¯ � � � È ×�¯ Ï ¯s­>×�¯s× ª�­ ±j·�¯�Âi±j¬I«j±jç Ï Øoª�Âi±�ªGѳG¯sØo±iªG«�ª Ï ¯l«g¬I±j²eª�­> �

�Ú¯�«f¬G­�¬ ð ª�ç�±�ÆIÓGÓGÓÎØlç�«j³G¯sÂ�Í�²e±j· �5��Å(éIÓGÓGÓGÓΪ�­�±j·>¯ M � � ÓGÓ�²�­�±j·�¯ Ï ¯l«f²eª9× �ÿ¬I«fØf· � �GñGñ�±iª Þ ¬I±i¯� ���IÓ � æ�¬GØg·)«fç>­õª�­�±j·>¯ M � � ÓGÓ�±iª�ªGÌõÂ Þ ²�°�·�± Þeó´Þ ¯sÂjÂ-±j·;¬G­�±EÍ�ª�·�ª�ç>«fÂ-ÑÒªG«uéGÜ�Ølç�«j³G¯sÂl¶GÍ�²�±j· � � � È 4 � Ó �Bû ·�¯M � � ÓGÓ´Ít¬GÂwç Ï °G«f¬G×�¯s×�±iª�¬ M ��ÆGÆIÓGÓ�²�­ � ��� � ¶�¬G­>×�Íw¯�«g¬G­�¬ ð ª�ç�± �vå ÜGéIÓ)Ølç�«j³G¯sÂ�Í�²e±j·�����ÅëÆIÓGÓGÓGÓGÓ�ª�­±j·�¯ M �wÆGÆIÓGÓ�²�­&±j·�¯ Ï ¯l«f²eª9× Ý ç�°�ç>Âi± � ��� � ±iª Ý ç�°�ç>Âj± � ���GÙ � æ�¬GØg·�«gç>­ ª�­�±j·�¯ M ��ÆGÆIÓGÓ�±iª�ªGÌ�Â Þ ²e°�·�± ÞeóÞ ¯sÂjÂ3±j·;¬G­�±EÍ�ª�·�ª�ç�«fÂYÑÒªG«�éGÆ�Ølç�«j³G¯sÂl¶�Ím²e±j· � � � È 4 � Ö �uû ·�¯w²�Ð Ï «jªO³G¯sÐ�¯s­�±u²�­�Â Ï ¯l¯s×�ªO³G¯l«�±j·�¯�M � � ÓGÓ�Ít¬GÂÏ ¬I«j± Þ�ó ×;ç�¯´±iª�«j¯lÍ!«f²e±j²�­�°�±j·�¯�²�­>­�¯l« Þ ª�ª Ï ªGÑ Ï «jªG°G«f¬GÐ 0 ±iª�«j¯s×>ç>Øo¯�Ð�¯sÐ�ªG« ó «j¯lÑÒ¯l«j¯s­>Øo¯sÂ�¬G­>×�²�Ð Ï «fªv³G¯±j·�¯õç;Âi¯�ªGÑ5³G¯sØo±iªG«�«j¯l°�²�Âj±i¯l«f �Ë ­ � ¯ Ï ±i¯sÐ ð ¯l« � ��� Ö�Í�¯´Âj±j¬I«j±i¯s×&«fç;­>­>²�­�° Ï «jªG°G«f¬GÐ @ ª�­&ª�­�¯�ªG«!±�Íwª�éIÓ �ÿ·�ü � ç Ï ¯l« � Ï ¬I«fØ Ï «jª9Øo¯sÂ\¸

ÂiªG«f � 6mÂfç>¬ Þ�Þeó Íw¯�ç>Âj¯s× ª�­�¯ Ï «jª9Øo¯sÂjÂiªG«�Ñ�ªG«)à ��ù�¬G­>×�ª�­�¯�Ñ�ªG«)à � È �&Ë ­ &�ç Þeó � ���GÙ�Âf²2 Ð�ªG«j¯ éIÓ`�ÿ·�ü� ç Ï ¯l« � Ï ¬I«fØ Ï «fª�Øo¯sÂjÂjªG«f ð ¯sØl¬GÐ�¯�¬v³ ¬G² Þ ¬ ð Þ ¯�ÑÒªG«`¬ Þ ²�Ð�²e±i¯s× Ï ¯l«g²eª�× � �Ú¯�¬I±i±i¯sÐ Ï ±i¯s×αiª)Ѩ¬GØo±iªG«!à ��ùmª�­Î¬ Þ�Þ¯s²e°�·�± � ç Ï ¯l« � Ï ¬I«fØlÂ3ç>Âj²�­>°´Øoª�Ð Ï ç�±i¯l«�±j²�Ð�¯`Í�·>²�Øf·�Ít¬GÂ�­�ªG±�«j¯���ç;²e«j¯s× ð ó ªG±j·>¯l«�ç>Âi¯l«g � @m¯l±j¬G² Þ Â�¬I«j¯�°�²e³G¯s­²�­ û ¬ ð Þ ¯�Ü �´Ý ±�±j·�¯�Âj¬GÐ�¯)±j²�Ð�¯�Í�¯)Íw¯l«j¯�¬I±i±i¯sÐ Ï ±j²�­>°�±iª Ѩ¬GØo±iªG«´Ã � È ±iª à ��î ç>Âj²�­�°�æ�0\� ª�­ @�ç ð ­�¯l«0w«fç;­>Øf·�¯l«g ã Âi¯l¯ $gñ äg�Ë ­ û ¬ ð Þ ¯�Ü�¶-à ²�Â�¬G­Õ¯sÂj±j²�Ð�¬I±i¯�ªGÑ�±j·>¯�¯� Ï ¯sØo±i¯s×Ú­�ç>Ð ð ¯l«�ªGÑw±j²�Ð�¯sÂ�±j·>¬I±�±j·�¯�Ѩ¬GØo±iªG«�� ÷jù Âj·�ª�ç Þ × ð ¯

ÑÒª�ç>­>×�Í�²e±j·�±j·�¯�°�²�³G¯s­+�5�)¬G­>× ­�ç;Ð ð ¯l«�ªGÑ�Ølç�«j³G¯s ã Âi¯l¯1$jÖ � Ö äg� � ²�Â�¬G­ ¯sÂi±j²�Ð�¬I±i¯ÎªGÑ!±j·�¯Î¯���Øl²e¯s­>Ø óØoª�Ð Ï ¬I«f¯s×�±iªÚ±j·>¯�ª Ï ±j²�Ð�¬ Þ Øg·�ª�²�Øo¯ ªGÑ ��� 4 Ü Ö�ÓGÓGÓGÓGÓ � û ·�¯ Ï «fªG°G«f¬GÐ�Â)ç>Âi¯s× ±j·�¯ ð ²e«f±j·>×>¬ ó Ï ¬I«f¬G×�ª� Øoª�­�±j²�­�ç;¬I±j²eª�­�¶ ð ç�±´±j·�¯�¯sÂj±j²�Ð�¬I±i¯s´ªGÑ � ¬G­>× Ã ¬GÂjÂjç>Ð�¯®±j·�¯®²�Ð Ï «jªv³G¯s× Âj±j¬G­>×>¬I«f× Øoª�­�±j²�­�ç>¬I±j²eª�­ Í�²e±j·� È Å � ÓGÓ&���s¶>Âiª)±j·�¯ ó ¬I«f¯�ª�­ Þeó ¬ Ï>Ï «fª� �²�Ð�¬I±i¯ ã Âi¯l¯;$jÖ � Ö äg��û ·�¯ Þ ¬GÂi±t«jªvÍ ªGÑ-±j·>¯�±j¬ ð Þ ¯�°�²e³G¯sÂt±iªG±j¬ Þ Â ã ÑÒªG«­�ç>Ð ð ¯l«`ªGÑBØlç�«j³G¯sÂ�¬G­>×&Ã ä ¬G­;×ÎÍ�¯s²e°�·�±i¯s×�Ð�¯s¬G­> ã ÑÒªG«3� � ¬G­>× ��äg�û ·�¯ �;÷jù)Ѩ¬GØo±iªG«õªGÑ`à ��ù)Íw¬GÂõÑ�ª�ç;­>× ð ó ¬ «fç;­ÕÍ�·>²�Øg·ÚÂi±j¬I«j±i¯s×Úª�­ � Øo± � Ö�¬G­>×Õø1­>²�Âj·�¯s×ÿª�­ � Øo±�ÆIÓ9¶

� ���GÙ ��û ·�¯�«fç>­�±i«f²e¯s× � ÓÎØlç�«f³G¯sÂmÍm²e±j· ����Å#ÆIÓGÓGÓGÓGÓGÓ&²�­�¬ ð ª�ç�± �G� Ö ·�ª�ç�«fÂmªGÑ�0�� 6#±j²�Ð�¯G¶ � Ö�ñ�·>ª�ç�«f¯ Þ ¬ Ï Âj¯s× Íw¬ Þ�Þ ¸�Ø Þ ª9ØjÌ�±j²�Ð�¯ �©�«jª�Ð ±j·�¯�ÑÒ¬GØo±iªG«g²eüs¬I±j²eª�­>ÂmªGÑ �;÷jù . � °�²e³G¯s­ÿ²�­=$'��¶q²�±m²�Â�¯s¬G ó ±iª Ï «jªv³G¯�±j·>¬I± �;÷jùõ²�Â Ï «f²�Ð�¯G¶;¬G­>×ÿ¬ Þ Âiª

±iª®Âi¯l¯õÍ�· ó ±j·�¯-�Yª Þ�Þ ¬I«f× �;. � Ð�¯l±j·>ª�×>Â�×>²�× ­�ªG±tø�­;× � ÷jù �û ·�¯wÆGÙGÆO¸�×>²e°�²e±YØoªGÑÒ¬GØo±iªG« È �E� � �;÷jùBÍt¬GÂY«f²e°GªG«jª�ç; Þeó Ï «jªv³G¯s×�±iª ð ¯ Ï «f²�Ð�¯ ð ó ±�Íwª�²�­>×�¯ Ï ¯s­>×�¯s­�±LÐ�¯l±j·�ª9×>Âã Âi¯l¯ $'� äg�

é ���I��, �+4$<'�H.�=�� 41(@< ? ©>«jª�Ð Øoª Þ ç>Ð�­�à ªGÑ û ¬ ð Þ ¯�Ü�¶q²e±m¬ Ï;Ï ¯s¬I«gÂ!±j·>¬I±�Í�¯�Í�¯l«f¯õÑÒªG«j±jç>­>¬I±i¯�±iª�ø�­;× � ÷jù¬GÂ�Âiª�ª�­�¬GÂ�Íw¯�×>²�× ��û ·�¯ Ï «jª ð ¬ ð ² Þ ²e± ó ²�Âm¬ ð ª�ç>± � % ¯� Ï ã % Ó â Æ Ö�Ü Ö ä 4 Ó â ÆGÆ � �Ú¯)Ì�­�ªOÍ ªGÑ�Âiª�Ð�¯�ªG±j·�¯l«¬I±i±i¯sÐ Ï ±j �Îþ «fç;Øo¯/@�ª�×;Âiª�­ ±i«f²�¯s× ¬ ð ª�ç�± � ÓGÓÿØlç�«j³G¯s´Ím²e±j· ����Å Æ.= � Ó ý ¶#�Y¯l±i¯l« ��ª�­�±i°Gª�Ð�¯l« ó ±i«f²e¯s׬ ð ª�ç�± � ÓGÓGÓÚØlç�«j³G¯sÂ)Í�²e±j·�����Ô � Ó�í ã Ð�¯s¬G­ ³I¬ Þ ç�¯Îç>­�Ì9­�ªvÍ�­ ä ¶ 8 ª ð ¯l«j± � ² Þ ³G¯l«fЮ¬G­ ±i«f²e¯s× ¬ ð ª�ç�±�ÙIÓGÓØlç�«j³G¯sÂõÍ�²e±j· ����Å � Ó ý ¶Y¬G­>× � ¬GÐ�ç�¯ Þ � ¬I°�Âi±j¬HG ±i«f²�¯s× �i¬&ÑÒ¯lÍ ×�ªGül¯s­ ��Ølç�«j³G¯sÂ�Í�²e±j· � ÙIÓGÓGÓGÓ Ô ����ÔÖ�ÓGÓGÓGÓGÓ ��û ·�¯l«j¯tÍ�¯l«f¯ Ï «jª ð ¬ ð Þeó ªG±j·�¯l«�¬I±i±i¯sÐ Ï ±jÂuªGÑ1Í�·>²�Øg·)Í�¯!¬I«f¯�ç>­>¬vÍw¬I«j¯G¶ ð ç�±3±j·�¯!Âj²�ül¯`ªGÑ�±j·�¯`Ì9­�ªOÍ�­

Page 19: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� � í� � ������� � æ�0\� «fç>­>Âwª�­�à ��ù

��� Ølç>«j³G¯s à � Ð�¬GØf·>²�­�¯ ã Â ä ¬G­>×&×;¬I±i¯sÂé;= � Ó ÷ ÆIÓGÓGÓ Ó � ÓGÓ � Ó Ó ��� Ö M � � ÓGÓ9¶��ÿ¬I« � �GñGñ�ò ö ªO³ � ���IÓÆ;= � Ó î �vå ÜGéIÓ Ó � Ó>� � Ó Ó � Ö�Æ M �wÆGÆIÓGÓ9¶ Ý ç�° � ��� � ò Ý ç�° � ���GÙÙ;= � Ó î å ÓGÓ Ó � Ó � ÙGÆ Ó � é�� � Ï ¬I«fØ.= Æ�¶ � ¯ Ï � ��� Ö�ò�&�ç ÞB� ���GÙ� Ó ý Ö�ñIÓ Ó � Ó�ÆGéGÆ Ó � ñ å � Ï ¬I«gØ = ñ�¶ &�ç ÞB� ���GÙ�ò Ý ç�° � ���GÙ

Æ;= � Ó ý �IÓGÓ Ó ���G� ÓGÓ Ó � �Gñ � Ï ¬I«fØ = ñ�¶ Ý ç�° � ���GÙ�ò � Øo± � ���GÙÆ â �;= � Ó î Æ � ÖGÖ�Ó Ó � Æ Ö�Ü Ö Ó � éGÜ

Øoª�Ð Ï ª�Âj²e±i¯ Ѩ¬GØo±iªG«®ªGÑ�à ��ù ã Æ�� � ×�¯sØl²�Ð�¬ Þ ×;²e°�²e±jÂ ä «j¯s×>ç;Øo¯s× ±j·�¯&­�ç;Ð ð ¯l«�ªGÑ�¬I±i±i¯sÐ Ï ±j � ©�ªG«�¯� 9¬GÐ Ï Þ ¯G¶��ª�­�±i°Gª�Ð�¯l« ó ïW 0w«f¬ ó Ï «jªG°G«f¬GÐ Ít¬G´«j¯sÂj±i«f²�Øo±i¯s×Ú±iª�²�­ Ï ç�±jÂ�ªGÑt¬I±�Ð�ª�Âj±�ÆGÙGÙ ×>²e°�²�±jÂl¶-¬G­>× � ¬I°�Âj±j¬HGÛ×>²�×­�ªG±�ç;Âi¯�±j·�¯M�ÿ¬GÂ Ï ¬I«�è�ÜGÜIê ð ¯sØl¬Gç;Âi¯õ²e±`Íwª�ç Þ × ·>¬v³G¯õ«j¯���ç>²e«f¯s×�¬®Â Ï ¯sØl²�¬ Þ �iÂj²eül¯õÜGÜ � Ï «jªG°G«f¬GÐ �©�«jª�Ð Øoª Þ ç>Ð�­ � ªGÑ9±j·�¯�±j¬ ð Þ ¯G¶I²e±-²�Â-Ø Þ ¯s¬I«-±j·>¬I±5±j·�¯u«fç;­>Â�ª�­´±j·>¯ M � � ÓGÓ�¬G­;× M ��ÆGÆIÓGÓmÍ�¯l«j¯�²�­�¯���Øl²e¯s­�± �

�Ú¯�Âf·�ª�ç Þ × ·>¬v³G¯�²�Ð Ï Þ ¯sÐ�¯s­�±i¯s× ¬�ÑÒªG«fÐ ªGÑBØf·>¯sØjÌ Ï ª�²�­�±j²�­>°®Âiª�±j·>¬I±3� � Øoª�ç Þ × ð ¯´²�­>Øo«j¯s¬GÂj¯s×&±iª®¬I± Þ ¯s¬GÂi±� Ó ý ¶�¬ Þ�Þ ªvÍ�²�­>°�ç>Â3±iª�±j¬IÌG¯mÐ�ªG«j¯t±j·>¬G­)±�Íwªõ·�ª�ç>«fÂ Ï ¯l«BÂi¯l±BªGÑqØlç�«j³G¯s �YÝ ±u±j·>¯w±j²�Ð�¯tÍ�¯!×>²�×�­�ªG±BÌ9­�ªOÍ ±j·>¬I±±j·�¯�ç>­�Ì9­�ªvÍ�­ ÑÒ¬GØo±iªG«�·>¬G× Ö�Ó�×�¯sØl²�Ю¬ Þ ×>²e°�²�±jÂl¶-±j·�ª�ç>°�· ð ó Ð�²�×9¸ � ���GÙ&Í�¯�Í�¯l«f¯®«j¯s¬GÂiª�­;¬ ð Þeó Øoª�­9ø�×�¯s­�±±j·>¬I±�²e±)·>¬G× ¬I± Þ ¯s¬GÂi±®ÜGÙÿ×>²�°�²e±j � � ç>«�°G¯s­�¯l«g¬ Þ Âi±i«f¬I±i¯l° ó Íw¬GÂ)±iª ²�­>Øo«j¯s¬GÂi¯ ����°G«g¬G×>ç>¬ Þ�Þeó ¶B°�ç;²�×�¯s× ð ó¯sÂi±j²�Ю¬I±i¯sÂ�ªGÑ3±j·�¯�ª Ï ±j²�Ð�¬ Þ ���m¬G­;× ±j·�¯õ¯� Ï ¯sØo±i¯s×�­�ç>Ð ð ¯l«!ªGÑBØlç�«j³G¯sÂmÑ�ªG«mÑÒ¬GØo±iªG«fÂmªGÑuÜIÓOò�Ö�ÓÎ×>²e°�²e±j � �Ú¯×>²�× ­>ªG±`¯� Ï ¯sØo±!±iª�ø1­>× ¬G­ ó Ѩ¬GØo±iªG«fÂ�Ð�ç>Øg· Þ ¬I«j°G¯l«!±j·;¬G­&Ö�Ó�×>²�°�²e±j �é � Æ ��� /4&( .�9�� �-<���<! 5.�,���2 0&."�H* ? æ�¬GØf·�Ølç�«f³G¯wª�­)¬�éIÓO��·>ü � ç Ï ¯l« � Ï ¬I«fØu±j¬IÌG¯sÂu¬ ð ª�ç�±3Ù â�å = � Ó # ý ���·�ª�ç�«gÂmªGÑ 0�� 6ô±j²�Ð�¯ ��Ë Ñw¬&éIÓ ��·>ü � ç Ï ¯l« � Ï ¬I«fØ�²�Â�Øoª�ç>­�±i¯s× ¬G´¬&éIÓ ¸��ÿ² Ï Â�Ð�¬GØg·>²�­�¯G¶�±j·�¯s­Õª�ç>«´Øoª�Ð)¸Ï ç�±j¬I±j²�ª�­Ú±iª�ªGÌ ¬ ð ª�ç�±�Æ Ö�Ó �ÿ² Ï Â\¸ ó ¯s¬I«f ��û ·>²�Â�²�´Øoª�Ð Ï ¬I«g¬ ð Þ ¯�±iª�±j·�¯�Ü Ö�Ó �ÿ² Ï Âi¸ ó ¯s¬I«f´¯sÂi±j²�Ð�¬I±i¯s× ÑÒªG«Âj²e¯l³9²�­�°�±iª�Ѩ¬GØo±iªG«mà � ð ó �9ö © � è�ÙGÜ ê ��ã3�9ö © � ·>¬GÂ!Âj²�­;Øo¯ ð ¯l¯s­ ²�Ð Ï «jªv³G¯s× ð ó ��ª�­�±i°Gª�Ð�¯l« ó ¬G­>× ªG±j·>¯l«fÂl¶Âiª�±j·>¯õÜ Ö�Ó �ÿ² Ï Â\¸ ó ¯s¬I«fÂ�Øoª�ç Þ × Ï «fª ð ¬ ð Þeó ð ¯�«j¯s×;ç>Øo¯s× ð ó ¬G­ ªG«g×�¯l«!ªGÑ3Ð�¬I°�­>²e±jç>×>¯ ��ä

� ²�­;Øo¯õ±j·�¯�²�­>­>¯l« Þ ª�ª Ï Â�ªGÑ Ï «fªG°G«f¬GÐ�Â-0�ò�æëç>Âj¯ � ª�¬I±j²�­�°I¸ Ï ª�²�­�±�¬I«f²�±j·>Ð�¯l±j²�ØI¶ � � ª Ï Â�¬I«j¯�¬ÎÐ�ªG«j¯�¬ Ï ¸Ï «jª Ï «f²�¬I±i¯�Ð�¯s¬GÂjç�«f¯�±j·>¬G­ ��² Ï Â �)û ·�¯ M ��ÆGÆIÓGÓ'F � Óÿ²�Â�«f¬I±i¯s× ¬I± � ÆGÙIÓ`� � ª Ï ã Ï ¯s¬IÌ Ï ¯l«jÑÒªG«fÐ�¬G­>Øo¯ äg��Ë Ñª�ç�«tѨ¬GØo±iªG«f²eüs¬I±j²eª�­ ªGÑ5à ��ù ·>¬G× ð ¯l¯s­ Ï ¯l«fÑ�ªG«fÐ�¯s×®¯s­�±j²e«j¯ Þeó ª�­Î±j·�¯ M �wÆGÆIÓGÓ9¶;²e±wÍ�ª�ç Þ ×�·>¬v³G¯�±j¬IÌG¯s­&¬ ð ª�ç�±é�Íw¯l¯lÌ�Â`ªGÑYÐ�¬GØf·;²�­�¯�±j²�Ð�¯G¶9ªG« � Ö�Ó � � ª Ï ¸ ó ¯s¬I«f � 0w« ó Ï ±iªG°G«f¬ Ï ·>¯l«fÂtÂj·�ª�ç Þ ×έ�ªG±i¯�±j·;¬I±w±j·;²�Â`¬GÐ�ª�ç>­�±jÂt±iª¬ ð ª�ç�± å Ù�Ð�²�­�ç�±i¯sÂ!ª�­&¬ �´û ¯l«f¬ � ª Ï Ð®¬GØf·>²�­>¯ �û ·�¯�­�ç;Ð ð ¯l«õªGÑmÐ�ç Þ ±j² Ï Þ ²�Øl¬I±j²�ª�­> ã Ð�ª9× " ä ²�Âõ¬ÕÐ�¬GØg·>²�­�¯o¸�²�­;×�¯ Ï ¯s­>×�¯s­�±´Ð�¯s¬GÂjç�«j¯�ªGÑ!±j·�¯�Í�ªG«fÌ ±iª

Ѩ¬GØo±iªG« " � æ�¬GØg·ÎØlç�«j³G¯�±j¬IÌG¯sÂ`¬ ð ª�ç�±tÆGÆ â � ���tÂjç;Øf·�Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²�ª�­> ��� ³G¯l«f¬ Þ�Þ ¶�ª�ç�«�Ѩ¬GØo±iªG«f²eüs¬I±j²�ª�­®ªGÑ-à ��ù±iª�ªGÌ �Iâ Ö = � Ó ��� Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­; ã Ð�ª9×." ä ¶�Í�·>¯l«j¯�" Å È �E� ��ã¨û ¬ ð Þ ¯´Æ Ï «j¯s×>²�Øo±jÂtÜ â Ü = � Ó ��� Í�²e±j·�±j·�¯ª Ï ±j²�Ю¬ Þ Øf·�ª�²�Øo¯�ªGÑ Ï ¬I«f¬GÐ�¯l±i¯l«f ��ä)ö ç>Ð ð ¯l«gÂ`Ð�ª�× È �E�!Íw¯l«j¯�«j¯ Ï «f¯sÂi¯s­�±i¯s×&Í�²e±j· ÜGñ�×>²e°�²�±jÂw¬G­;× ð ¬GÂi¯õÆ È

ýã ª�­Õ±j·�¯&M � � ÓGÓ'F+M ��ÆGÆIÓGÓ ä ªG«�Í�²e±j·ÿÖ � ×>²e°�²e±jÂ�¬G­>× ð ¬GÂi¯®Æ È ÷ ã ª�­Õ±j·�¯ � Ï ¬I«fØ ä ¶LÂiªÎ¯s¬GØf· Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­ã Ð�ª9× " ä ±iª�ªGÌ Ð�ªG«f¯�±j·>¬G­ � Ó ÷ � ª�¬I±j²�­�°I¸ Ï ª�²�­�±`ª Ï ¯l«f¬I±j²�ª�­> �é � Ü ��� /4 7 �+. ��� ."�H684��$?´û ·�¯ÿÂjç>ØlØo¯sÂjÂiѨç Þ ¯ Þ�Þ ² Ï ±j²�Ø Ølç�«j³G¯Õ¬G­>× Âi±j¬I«j±j²�­>° Ï ª�²�­�±�¬I«j¯Õ×�¯oø�­�¯s× ð ó ã ñ ä¬G­>× ã\�G�vä ¶YÍ�²e±j· $ Å � Ö � ÙGÆGÆGé å ã ×�¯l«g²e³G¯s×ÚÑÒ«jª�Ð ±j·�¯®Âi±j¬I«f±j²�­�° ×>¬I±i¯®¬G­>×Õ±j²�Ð�¯ � Øo±iª ð ¯l« � Ö ¶ � Ù �bÆGÆ �bÙ Ö äg�æ Ï Þ ²�Øl²e± Þeó ¶�Í�¯õØl¬G­�±j¬IÌG¯õ±j·�¯�¯ Þ�Þ ² Ï ±j²�Ø�Ølç�«j³G¯�²�­Î±j·�¯�ÑÒªG«fÐ ãUåGä ¬GÂ

7 1 È Å32 � Ê � Ù�� å ÖGÖ å ÜIÓ�ñGÆ��IÓ�Ü � ñGÜ�ÙGÆ�ÆGñIÖ ��Ù å Ü Ö�Ü �vå ÆGñ�ÙGñGÖ�Ó�ÜGé � ñ 2 È Ê 2 Ð�ª9×O� ÷jù âû ·>²�Â`Ð�¬ ó ¬ Þ Âiª ð ¯�Í!«g²e±i±i¯s­ ¬G 7 1 È Å 2 ã 2 %�à äoã 2 % � � à ä Ð�ª9×O�;÷jù ¶>Í�·�¯l«f¯

à Å � Ó�ÜGéGñGÆGÆGÆGÆGÙGÙ � Ü å Ó å�å Ö�é � Ù�ÜGÙ�ÆGÜ �vå ÜGÙ �våGå�å ñ å ñ�ÙIÓ�Ö åÎâ7�²�Â!¬G­ ó ��ç>¬G×�«g¬I±j²�Ø�­�ª�­9¸U«j¯sÂf²�×>ç�¯ ã Ð�ª9× �;÷jù ä ¶>¯ � ° � 7tÅÇÙ ��û ·�¯�°G«jª�ç Ï ªG«f×�¯l«�²�Â

� Å � ÷jù Ê � % ÜGé å Ö�ñ å ÆGÆGÙ�� � Æ�� Ö ���GÓ�Ü�ñÅ Æ È ìvÜ È ìOÙ�ì � Ö ��ì � éGÜ�ì � � å ì å�� ñ å ì � ñGÜ �G� ì � ÆGÜGé åGå ìvÆGÆGé � ÜGÜõìOÜ � Ö�ÆGéGÜõìlÖ�é åGå ñGÙGÜ â

Page 20: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

� � ������������� �

û ·�¯ Ï «jª ð ¬ ð ² Þ ²e± ó ±j·;¬I±�¬�«f¬G­>×>ª�Ð ²�­�±i¯l°G¯l«m­�¯s¬I« � � � Æ®·>¬GÂ Þ ¬I«j°G¯sÂi± Ï «f²�Ð�¯�Ѩ¬GØo±iªG«�¬I±�Ð�ª�Âi±!Ö�é åGå ñGÙGÜ�¬G­>×Âi¯sØoª�­>×�¸ Þ ¬I«j°G¯sÂi± Ï «f²�Ð�¯&ÑÒ¬GØo±iªG«&¬I±ÎÐ�ª�Âi± Ü � Ö�ÆGéGÜ ²�Â�¬ ð ª�ç>±ÎÙ â ñ = � Ó # ý � û ·>¯ Ï ·>¬GÂj¯ �ÿÞ ²�Ð�²e±�ÑÒªG«Î±j·�¯Âjç>ØlØo¯sÂfÂiÑÒç Þ «fç>­�Íw¬GÂ3����ÅÇÆ = � Ó ý ¶ ð ç�± Ï «jªG°G«g¬GÐ� 0�ò�æÛ¬G­>× K ø�­>× ��÷jùmÍm²e±j· ���`¬GÂmÂjÐ�¬ Þ�Þ ¬GÂ�Ü � Ö�ÆGéGܲeÑ5±j·�¯õÂj¬GÐ�¯�Ølç�«j³G¯´¬G­>×&Âj±j¬I«j±j²�­�° Ï ª�²�­�±!¬I«j¯´ç>Âi¯s× �

å�� � ��¦1¡�£-¢�§�� �>¡1§Ò£-  £ � à ���Ý Ñ�±i¯l«�±j·�¯®Ñ¨¬GØo±iªG«f²eüs¬I±j²eª�­�ªGÑ`Ã��®²�­ � �GñIÓ9¶u­�ª&ª�­>¯ Ï «j¯s×;²�Øo±i¯s×Ú±j·>¬I±�à ���õÍwª�ç Þ × ð ¯®±j·�¯®­�¯� 9±�©�¯l«gÐ�¬I±

­�ç>Ð ð ¯l«m±iª ð ¯�Øoª�Ð Ï Þ ¯l±i¯ Þeó Ѩ¬GØo±iªG«j¯s× � �u«jªG°G«f¬GÐ 0�¶1×�¯sÂjØo«f² ð ¯s×�²�­ $gÙ�¶qÍt¬GÂ�²�Ð Ï Þ ¯sÐ�¯s­�±i¯s×&ª�­ÿ¬�©>ç��j²e±jÂjçM � � ÓGÓ�²�­ �ÿ¬I«fØf· � �GñGñ �wÝ ÑÒ±i¯l«tÑÒ¬G² Þ ²�­�°´±iª�ø�­>×�¬G­ ó ­�¯lÍ ÑÒ¬GØo±iªG«fÂwªGÑ-à ��¬G­>×�à ��ùG¶�Í�¯�Øoª�Ð Ï ² Þ ¯s× � Þ ¬I«j°G¯+�³G¯l«fÂj²�ª�­>Â�ªGÑ Ï «fªG°G«f¬GÐ 0 Âjç>²e±j¬ ð Þ ¯�ÑÒªG«´Ã ���´¬G­>×Õà � È � �Ú¯�Ì9­�¯lÍ#ÑÒª�ç�« Ï «f²�Ð�¯õѨ¬GØo±iªG«fÂ�ªGÑ�à � È ÑÒ«jª�Ð�è Ö � ê �� ­ �ÿ¬ ó é�¶ � �GñGñ�¶1¬�«fç>­ÎÑÒª�ç>­>× ¬�ø;Ñ�±j·ÎѨ¬GØo±iªG«

� � ý�Å � ÆGÙGé � ÜGÆ � Ü Ö � ÆGÙGÙGé>��ÅëÆ �U÷ ì å È ìOÙGÜ�ìvÆ��GÙGÆ � ñ Ö � Ê �ªGÑtà � È � �-¬I±i¯l«�Í�¯ Þ ¯s¬I«g­�¯s×ÿ±j·>¬I± 8´�Lþ ¬G² Þ�Þ ²e¯�·;¬G×Ú¬ Þ «j¯s¬G× ó Ñ�ª�ç>­;×Õ±j·>²�Â�ÑÒ¬GØo±iªG«�²�­ &�ç Þ�ó � �GñGé�¶Yç;Âj²�­�°�±j·�¯�Yª Þ�Þ ¬I«f× � % � Ð�¯l±j·�ª9× � �Ý(Þ ¬I«j°G¯´³G¯l«fÂf²eª�­ÎªGÑ Ï «jªG°G«f¬GÐ 0 ±iª�ªGÌ Â Þ ²�°�·�± Þeó�Þ ¯sÂjÂt±j·>¬G­&ª�­�¯õ·�ª�ç�«`ÑÒªG«mÆIÓ�Ølç�«j³G¯sÂ`Í�²�±j· �5�wÅ � éIÓGÓGÓ

ª�­ ±j·�¯�­�ç>Ð ð ¯l« ý ù ý Å Ã ���?� ã Ü � � Ö�ñ��´ì/� å Ö�ñ Ö � äg� � ­ �ÿ¬ ó � Ü�¶ � �GñGñ�¶�¬®ÆGÆO¸�×�¯sØl²�Ð�¬ Þ ×>²�°�²e±tÑÒ¬GØo±iªG«� ÈiÈ ÅëÜGÙGéIÓ�ñ Ö � �IÓ�é ÖGÖ�ÙGñ�ÜGÜ>�GÆIÓ�Ù � Ü®ÅÇÆ

��� ì å ìvé åGå ìQ���U÷�Ê �Ít¬G®ÑÒª�ç>­>× ð ó Ï ·>¬GÂi¯�Æ � �Ú¯ÿ·>¬G× Ï «f¯l³�²eª�ç; Þeó ±i«f²�¯s×ÛéGñ Ølç>«j³G¯s®ç>­;Âjç>ØlØo¯sÂjÂiѨç Þ�Þ�ó Í�²e±j· ��� 4 � ÙIÓGÓGÓ �� ³G¯l«f¬ Þ�Þ ²�±�±iª�ªGÌ�¬ ð ª�ç�±´Æ â ñ = � Ó í Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­; ã Ð�ª9× ý ù ý ä ¬G­;×ÿÂ Þ ²e°�·�± Þeó�Þ ¯sÂjÂ�±j·>¬G­ÕÑÒª�ç�«�·�ª�ç�«gÂmªGÑÐ�¬GØg·>²�­�¯�±j²�Ð�¯�±iª�ø�­>× � ÈiÈ � @�²e³9²�×>²�­�° ý ù ý ð ó � ÈiÈ °�¬s³G¯�¬®ÙGñ ÖI¸�×>²e°�²e±mØoª�Ð Ï ª�Âj²e±i¯´­�ç>Ð ð ¯l« î �\÷ �û ·�¯�­�¯� 9±!«fç;­ÎªGÑ Ï «fªG°G«f¬GÐ 0�¶�ª�­ ��¬ ó �vå ¶ � �GñGñ�¶;ÑÒª�ç>­>×&¬�Æ � ¸�×;²e°�²e±!Ѩ¬GØo±iªG«

� È ��Å � é å �GñGñGÙGÙGéGÜ Ö �vå éGÓIÖ å Ù � Ü å ÅëÆ�U÷ ìOÜ�ìOÜ å Ü�ìOé å ÓGÓ�Ü�ì � ÜGé å ÙGÆGÙ Ö å Ê �

ªGÑ î �\÷G¶w¬I°�¬G²�­ ç>Âj²�­�°ÕÆIÓÚØlç>«j³G¯sÂ�Í�²e±j· ����Å � éIÓGÓGÓ � Ë ±)Íw¬GÂ�Âfç�« Ï «f²�Âj²�­�°&±j·;¬I±�¬ Þ ¬I«j°G¯l«�ÑÒ¬GØo±iªG« ã � ÈiÈ ä·>¬G× ð ¯l¯s­�Ñ�ª�ç>­;×�ø;«gÂi±l¶ ð ç�± ð ¯sØl¬Gç>Âj¯mªGÑ5±j·�¯ Ï «jª ð ¬ ð ² Þ ²�Âi±j²�Ø`­;¬I±jç�«j¯�ªGÑ5æ�0�� ±j·�¯l«j¯�²�Ât­�ª�°�ç>¬I«f¬G­�±i¯l¯�±j·>¬I±ÂjÐ�¬ Þ�Þ ¯l«�Ѩ¬GØo±iªG«fÂ`Ím² Þ�Þ ð ¯�Ñ�ª�ç;­>× ð ¯lÑÒªG«j¯ Þ ¬I«j°G¯l«tª�­�¯s ��� ³G¯l«f¬ Þ�Þ ¶9²e±w±iª�ªGÌά ð ª�ç�±tÜ â é = � Ó í Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²eª�­>Âã Ð�ª9× ý ù ý�ªG« î �\÷ ä ¬G­>× Þ ¯sÂjÂ�±j·>¬G­�ø;³G¯Î·�ª�ç�«fÂõªGÑ�Ð�¬GØf·;²�­�¯®±j²�Ð�¯�±iª�ø�­>× ð ªG±j· Ѩ¬GØo±iªG«f ð ó æ�0\� �5Ë ±Íwª�ç Þ × ·>¬v³G¯ ð ¯l¯s­ÎÑ�¯s¬GÂf² ð Þ ¯�±iª�ø1­>× � È � ã ð ç�±!­�ªG± � ÈiÈ ä ð ó ±j·�¯-�Yª Þ�Þ ¬I«f× � % � Ð�¯l±j·�ª�× �û ·�¯O��ç�ªG±j²e¯s­�±!·>¬G× ÙGé Ö®×>²e°�²e±jÂt¬G­>×L¶�±iª�ª�ç>«!Âjç�« Ï «g²�Âi¯ ã Âj¯l¯;$ � Ó ä ¶;²e± Ï ¬GÂjÂi¯s× ¬ Ï «fª ð ¬ ð ² Þ ²�Âi±j²�Ø Ï «g²�Ð�¬ Þ ²e± ó

±i¯sÂi±Îè Ö å ¶ Ï � Ü å �Oê � Ë Ñw±j·>²�Â�±i¯sÂj±�²�´¬ Ï;Ï Þ ²e¯s×Õ±iªÿ¬&Øoª�Ð Ï ª�Âj²e±i¯®­�ç>Ð ð ¯l«s¶L±j·�¯�Øf·>¬G­>Øo¯�±j·>¬I±õ²e±�²�­>ØoªG«j«j¯sØo± Þ�óØ Þ ¬G²�Ю®±j·�¯Õ­�ç>Ð ð ¯l«�²�Â Ï «f²�Ð�¯�²�Â Þ ¯sÂf®±j·>¬G­ � �OÖ � �Ú¯Õ«f¬G­ Ð�¬G­ ó ²�­>×�¯ Ï ¯s­>×>¯s­�±�±i«f²�¬ Þ Âl¶!Âiª Í�¯ÿÍw¯l«j¯Øoª�­9ø�×>¯s­�±´±j·;¬I±´±j·�¯ ��ç�ªG±j²e¯s­�±õÍt¬GÂõ²�­;×�¯l¯s× Ï «f²�Ð�¯�¬G­>×Ú±j·>¬I±´±j·>¯�ÑÒ¬GØo±iªG«g²eüs¬I±j²eª�­�ªGÑtà ��� Ít¬GÂõØoª�Ð Ï Þ ¯l±i¯ �û ·>²�Â�Íw¬GÂ5³G¯l«f²xø�¯s× ð ó ��ªG«f¬G²�­L¶ ¬GÂY×�¯sÂjØo«f² ð ¯s×�²�­;$'� �Bû ·�¯�Øoª�Ð Ï Þ ¯l±i¯�Ѩ¬GØo±iªG«f²eüs¬I±j²eª�­�ªGÑ>à ���5Ít¬GÂ5¬G­>­>ª�ç>­>Øo¯sײ�­ è�ñ ê �

à ���wÅÇÜ � � Ö�ñ��´ìT� å Ö�ñ Ö �õì � é å �GñGñGÙGÙGéGÜ Ö �vå éIÓGÖ å Ù � Ü å ìvÜGÙGéIÓ�ñ Ö � �IÓ�é ÖGÖ�ÙGñGÜ�Ü���ÆIÓ�Ù � Ü)ì���î ý ÷

ñ � �   ��� � ��¦1¡�£-¢ £�� à ���û ·�¯ @�ç ð ­�¯l«�0w«gç>­>Øf·>¯l«mè�ÆGÆ9¶�Ü ÖGêq²�Âu¬ ð ª�¬I«g×�Í�·>²�Øg· Ï Þ ç>°�Âu²�­�±iªõ¬G­ Ë\þ �&¸�Øoª�Ð Ï ¬I±j² ð Þ ¯ ��0 ��û ·�¯ ð ª�¬I«f×

·>¬G´¬�×>²e°�²e±j¬ Þ Âj²�°�­>¬ Þ Ï «jª9Øo¯sÂjÂj²�­>°ÎØg·>² Ï ã � ��Ë �-ªG°�²�Ø �Yé Ö�Æ Ö�Ó �ÿ© Ë 8�ä Í�·>²�Øg·�¶LÍ�·�¯s­Úç;Âi¯s×ÚÑÒªG«õÐ�ç Þ ±j² Ï Þ ¯o¸Ï «j¯sØl²�Âj²eª�­ÿ²�­�±i¯l°G¯l«�¬I«f²�±j·>Ð�¯l±j²�ØI¶�Øl¬G­ÚÐ�ç Þ ±j² Ï Þ�ó ±EÍ�ª�Ù � ÆO¸ ð ²e±�­�ç>Ð ð ¯l«fÂ�²�­ÿé � Ö � Âi¯sØ ��Ý ÂiªGÑÒ±�Ít¬I«j¯ Þ ² ð «g¬I« ó·>¬G ð ¯l¯s­ÚÍ�«f²e±i±i¯s­ ð ó A�¬I«j³G¯ ó ¬G­>× 8 ª ð ¯l«f± @�ç ð ­�¯l«�è�Ü ÖGê ��û ·>²�Â Þ ² ð «f¬I« ó ¬ Þ�Þ ªvÍ�´¬*0 Ï «jªG°G«f¬GÐ�Ð�¯l«´±iªç>Âi¯�±j·>¯ 0w«fç>­;Øf·�¯l«`ÑÒªG«�Ð�ç Þ ±j² Ï Þ ¯o¸ Ï «j¯sØl²�Âj²�ª�­®²�­�±i¯l°G¯l«m¬I«f²�±j·>Ð�¯l±j²�Ø � � ª�Ð�¯ Þ ²�Ð�²�±j¬I±j²eª�­>Ât¬I«j¯ �

Y ?qZOMLSUMQP/r a�XUTWk �BR\[gkOcxrOP/TW[^k`M��v_ abdfTbkOP�yLZs��XUZ Mq�Oc�XUZwcxdgRQXU[fS1[fc S UXW TWP�kO[gX;R\[gS/SUMQRQXUa��`dfX/X/SUTWVOrOXUMiN�XU[3:qdgTWaWaWTbM-TWk�+ �^�O{^_�8v|\~oz�-�8

Page 21: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� ���

�I� 0tª�Ð�Ð�ç;­>²�Øl¬I±j²eª�­ ð ¯l±�Íw¯l¯s­ ±j·�¯*0w«fç>­;Øf·�¯l«�¬G­>× ±j·�¯ ��0 ã ³9²�¬ÿ±j·�¯ ��0�ïW ËC�9Ý ð ç>Â ä ²�Â�«j¯ Þ ¬I±j²e³G¯ Þ�óÂ Þ ªvÍ´¶LÂjª Ï ¯l«jÑÒªG«fÐ�¬G­;Øo¯)²�Â�Ð�ç>Øg· Þ ¯sÂjÂ�±j·>¬G­Õ±j·�¯�±j·�¯lªG«f¯l±j²�Øl¬ Þ Ï ¯s¬IÌ�ÑÒªG«´­�ç>Ð ð ¯l«fÂ�ªGÑ Þ ¯sÂfÂm±j·;¬G­ÚÂj¬ ó� ÓGÓGÓ ð ²e±j � ©�ªG«´¯� �¬GÐ Ï Þ ¯G¶�Í�·>¯s­ÚÐ�ç Þ ±j² Ï Þeó ²�­�°�Ù � ÆO¸ ð ²e±´­�ç;Ð ð ¯l«fÂ�ª�­Ú¬*0w«fç;­>Øf·�¯l«�«fç;­>­>²�­�°�²�­Õ¬G­ñIÓ�ÜGñGé FOÖ�Ó���0�¶�ª�­ Þeó ¬ ð ª�ç>± � Ó� ªGÑ3±j·�¯ Ï ¯s¬IÌ Ï ¯l«fÑ�ªG«fЮ¬G­>Øo¯�²�Â!¬GØf·;²e¯l³G¯s× �

Æ ��þ ¯sØl¬Gç>Âj¯�ªGÑ�±j·�¯�Â Þ ªvÍ Øoª�Ð�Ð�ç;­>²�Øl¬I±j²eª�­�²e±L²�Â-×>¯sÂj²e«f¬ ð Þ ¯Y±iªmÌG¯l¯ Ï ª Ï ¯l«f¬G­;×>ÂL²�­�±j·>¯uª�­9¸ ð ª�¬I«f×´Ð�¯sÐ�ªG« ó ¶ªGÑYÍ�·>²�Øf·Îª�­ Þeó ÆGÙGé D þ�ó ±i¯õ²�Â`¬GØlØo¯sÂfÂj² ð Þ ¯�±iª®±j·�¯ 0 Ï «fªG°G«f¬GÐ�Ð�¯l« �

�u«fªG°G«f¬GÐ © ã Âj¯l¯=$gÙ ä Íw¬GÂ�²�Ð Ï Þ ¯sÐ�¯s­�±i¯s×7¬G­;× ×�¯ ð ç>°G°G¯s×7¯s¬I« Þeó ²�­ @m¯sØo¯sÐ ð ¯l« � ��� Ö �Çþ ¯sØl¬Gç;Âi¯�ªGÑÐ�¯sÐ�ªG« ó Þ ²�Ð�²e±j¬I±j²�ª�­>Âl¶-²e±�Øoª�ç Þ ×�­�ªG± 𠯮ç>Âj¯s× ÑÒªG«�©�¯l«gÐ�¬I±�­�ç>Ð ð ¯l«fÂ Þ ¬I«j°G¯l«�±j·>¬G­ à ��î ã ¬�­�ç>Ð ð ¯l«�±j·�¯Âj²eül¯mªGÑ-à ��î�«j¯���ç>²e«f¯sÂuÖ&D þwó ±i¯�ªGÑLÂi±iªG«f¬I°G¯ äg��Ë ­�±j·�¯ Ï ¯l«f²eª9×7&�¬G­�ç>¬I« ó ò7&�ç>­>¯ � ���GÙ�Í�¯�ç>Âi¯s×�¬ 0w«fç>­>Øg·�¯l«²�­ ¬G­ ñIÓ�ÜGñGé FOÖ�Ó ��0 ±iª ¬I±i±i¯sÐ Ï ±�±iª ÑÒ¬GØo±iªG«�à ��� ¬G­>× Ã �U÷ ã ¬G­>× Âiª�Ð�¯�­�ª�­�¸�©�¯l«fÐ�¬I±®­�ç>Ð ð ¯l«f äg� �Ú¯Ð�¬G²�­ Þeó ç>Âi¯s× Ï ·>¬GÂj¯ �)Þ ²�Ð�²e±3����Å � ÓGÓGÓGÓGÓ �7� ­Õà ����¯s¬GØf·�Ølç�«j³G¯)±iª�ªGÌ � Ü å Ð�²�­�ç�±i¯s ã � � Ð�²�­�ç�±i¯sÂ�ÑÒªG«Ï ·>¬GÂj¯ � ¬G­>× Ö�éÕÐ�²�­�ç�±i¯sÂ�ÑÒªG« Ï ·>¬GÂi¯&Æ ä !tª�­ à �U÷ίs¬GØg·7Ølç�«j³G¯Î±iª�ªGÌ Ü�� � Ð�²�­�ç�±i¯s � ©�ªG«�Øoª�Ð Ï ¬I«f²�Âiª�­�¶Ï «jªG°G«g¬GÐ @ «fç>­;­>²�­�°´ª�­Î¬�éIÓ ��·>ü � ç Ï ¯l« � Ï ¬I«fØ`ª�­�±j·�¯�Ð�ç>Øf·ÎÂjЮ¬ Þ�Þ ¯l«t­�ç>Ð ð ¯l«wà ��ù�±j¬IÌG¯sÂ�Ü Ö�Ð�²�­�ç�±i¯sÂÏ ¯l«õØlç�«j³G¯G¶3¬G­>× ±j·�¯>0w«fç;­>Øf·�¯l«õ±j¬IÌG¯s �vå Ð�²�­�ç�±i¯sÂ Ï ¯l«õØlç�«f³G¯ �Îû ·�¯/0w«fç>­;Øf·�¯l«sïWÂõ¬G×�³ ¬G­�±j¬I°G¯�ªv³G¯l«)±j·�¯� Ï ¬I«gØ�«g¬G­�°G¯sÂ)Ñ�«fª�Ð ¬ÕѨ¬GØo±iªG«�ªGÑ�ÆÕ­�¯s¬I«�à ��ù�±iª ¬ÿѨ¬GØo±iªG«�ªGÑ�ÆGÆÿÑÒªG«�à �U÷ � ©�ªG«�­�ç>Ð ð ¯l«gÂ�¬GÂ Þ ¬I«j°G¯&¬GÂà �U÷�±j·�¯ � Ï ¬I«gØ Ï «jªG°G«g¬GÐ Øoª�ç Þ × ð ¯�²�Ð Ï «jªO³G¯s× ð ó ç>Âf²�­�°�¬�Ѩ¬GÂi±i¯l«�Ð�ç Þ ±j² Ï Þ ²�Øl¬I±j²�ª�­ ¬ Þ °GªG«f²e±j·;Ð Âjç;Øf· ¬GÂD´¬I«f¬I±jÂfç ð ¬9ïWÂ)¬ Þ °GªG«f²e±j·;Ð ã Âi¯l¯1$gÙ ���väg� AmªvÍw¯l³G¯l«s¶�Âj²�­>Øo¯�Ü�� � � � Ü å � Ü�¶ D´¬I«f¬I±jÂfç ð ¬9ïWÂ�¬ Þ °GªG«f²e±j·>Ð Íwª�ç Þ ×­�ªG± ð ¯�·�¯ Þ Ï ÑÒç Þ ª�­ ±j·>¯ 0w«fç>­;Øf·�¯l« �û ·�«f¯l¯ Ï «f²�Ð�¯�Ѩ¬GØo±iªG«fÂ�ªGÑ3à ��� Í�¯l«f¯�Ì�­�ªOÍ�­ �

à ����ÅëÆ å�� Ó>�GÙ Ö�éGÜ��GÜGé � ìOÆGéGéGÜGñ Ö�ñGñ åGå�� ÙGÆ � Ö � Ü � Ü)ì ÜGéIÓ�Ü � Ó>�Gñ ÖGÖ�Ù Ö�ÆGÆ�� � ��é���ì È ÷�� í âû ·�¯�ø;«fÂi±�ÑÒ¬GØo±iªG«`Ít¬GÂwÑ�ª�ç;­>× ð ó A�¬ Þ�Þeó ð ç�«j±iª�­�¬G­;× þ «f² Þ�Þ ·>¬I«j±�è Ö � ê �uû ·�¯mÂj¯sØoª�­>×άG­>×®±j·;²e«f×�Ѩ¬GØo±iªG«fÂwÍw¯l«j¯ÑÒª�ç>­>× ð ó 0w«g¬G­>×>¬ Þ�Þ è�ÆGÙ êLª�­��q² Þ�Þ ¬�­�¯l± ã ¬�­�¯l±�ÍwªG«jÌΪGÑB¬ ð ª�ç�± � ÓGÓ�ÍwªG«jÌ9Âi±j¬I±j²eª�­>Â ä ²�­�&�¬G­�ç>¬I« ó ¬G­>× �ÿ¬ ó� ��� � ¶�ç;Âj²�­�°�æ�0�� �

� ­�&�ç>­�¯ � é�¶ � ���GÙ�ª�ç�« 0w«fç;­>Øf·�¯l« Ï «jªG°G«f¬GÐ © ÑÒª�ç>­>× ¬�ÑÒª�ç�«j±j·ÎѨ¬GØo±iªG«� Èjí ÅÇÜ � �GÙ Ö�éIÓ�ÆIÓ�ñGÆIÓ�ÙGÙ � éIÖ�Ü�ÆGÆGÓ�é å ÆGÙ � Ü�ÅëÆ

��� ìOÙ � ÜIÓ>�Gé�� å ì �G� ñ å ñGÙGéGéGñGÙ � ÆGé å Ê �¬IÑÒ±i¯l«õ¬Î±iªG±j¬ Þ ªGÑ�Ö �GÜ Ølç�«j³G¯sÂ�Ím²e±j· � � Å � ÓGÓGÓGÓGÓ9¶ � � ÅôÖ�ñGÙGÜIÓ � ¶3¬G­>×Õ±j¬ ð Þ ¯�Âj²�ül¯ � Å ñGÜ ã Âi¯l¯�$gÜ � Ü äg�û ·�¯)ªO³G¯l«f¬ Þ�Þ Ð�¬GØg·>²�­�¯�±j²�Ð�¯)Íw¬G´¬ ð ª�ç�±�Ö å ×>¬ ó  � �Ú¯®­�ªG±i¯�±j·>¬I±\� Èjí Ê � Å Æ�ìGÜõì å

� ìGÙ���ì'� ÈiÈ �)û ·�¯Ñ¨¬GØo±iªG«f²eüs¬I±j²eª�­>ÂmªGÑ:� Èjí . � ¯� Ï Þ ¬G²�­ Í�· ó �5ª Þ�Þ ¬I«f×�ïW � . � Ð�¯l±j·>ª�×>ÂmØoª�ç Þ ×&­>ªG±!ø�­>×αj·>¯õѨ¬GØo±iªG« � Èjí ²�­&¬«j¯s¬GÂiª�­;¬ ð Þ ¯�±j²�Ð�¯ �û ·�¯�Âjç;ØlØo¯sÂjÂiѨç Þ Ølç�«j³G¯�Ít¬GÂ!ªGÑ3±j·�¯�ÑÒªG«fÐ ã é ä Í�²e±j· ²�­;²e±j²�¬ Þ Ï ª�²�­�±ã 2���� 1 ���� �� ä Å ã\� ÙIÓIÖ�ÓGÓ�ÙGñGñ � ñGñ å ÜGÜGÖ�Ó�Ó>��Æ��GñIÖ å ÙGÜ � ��Æ åGå�� � Ö �IÓ�ñGÙ � Ó�é å éIÖ�é�ÆGÙ�ñ å ñ�ñIÓ�ÆIÓ å � �vä Ð�ª9×O� Èjí¬G­>× °G«fª�ç Ï ªG«f×�¯l« ã ­>ªG±�×>²e³9²�Âj² ð Þ ¯ ð ó Ö�²�­Î±j·>²�Â`Øl¬GÂi¯ ä

� ÅÇÜ È ì å È ì � Ü�ìvÜ � ìOÜGñIÓ�Ü´ìvéIÓ�Ü å ì/�GñGñ å ìvÆGñGñGÙ��´ìvÆ å ÖGÖ å���â6�Âj²�­�°õ©>¯l«fÐ�¬I±lïWÂ Þ ²e±i± Þ ¯�±j·�¯lªG«j¯sÐ è�ÆIÓ�¶ Ï ��Þ ³9²�²�²eêU¶GÍ�¯mÑ�ª�ç>­;× È ÷�� í � � Èjí ±iª ð ¯mØoª�Ð Ï ª�Âj²�±i¯ �3û ·�ç>Âs¶�Íw¯�­�ªOÍÌ9­�ªvÍDZj·;¬I±

à ��� Å Æ å�� Ó>�GÙ Ö�éGÜ��GÜGé � ì ÆGéGéGÜGñ Ö�ñGñ åGå�� ÙGÆ � Ö � Ü � Ü�ìÜGéIÓ�Ü � Ó>�Gñ ÖGÖ�Ù Ö�ÆGÆ>� � �Gé���ìOÜ � �GÙ Ö�éIÓ�ÆIÓ�ñGÆIÓ�ÙGÙ � éIÖ�Ü�ÆGÆGÓ�é å ÆGÙ � Ü�ì È ���E� â

Ý ±®¬ ð ª�ç�±�±j·�¯ ±j²�Ð�¯ ±j·>¬I±�±j·�¯ � Èjí ÑÒ¬GØo±iªG«�ªGÑ�à ��� Ít¬GÂ�ÑÒª�ç>­>×�¶ Ï «jªG°G«f¬GÐ ©#Ít¬GÂ�Ð�ª9×>²xø�¯s×�±iª °�²e³G¯Ï «jªG°G«g¬GÐ K ã Âi¯l¯ $gÙ äg�wû ¯sÂi±j²�­�° Ï «jªG°G«f¬GÐ K ¶�Í�¯��\«j¯s×;²�ÂjØoªv³G¯l«f¯s×��±j·�¯\� Èjí Ѩ¬GØo±iªG«�¬IÑÒ±i¯l«mñ å�� Ølç�«j³G¯sÂ!Í�²e±j·����Å � ÓGÓGÓGÓGÓ9¶�� È Å ÜGÙ Ö�ÆIÓGÓGÓ ��Ý Ñ�±i¯l«´²�­>Øo«j¯s¬GÂj²�­>° ����±iª&ÙIÓGÓGÓGÓGÓ9¶5Í�¯)ÑÒª�ç>­>× � Èjí ¬I°�¬G²�­ ã ±j·>²�Â�±j²�Ð�¯ ð óÏ ·>¬GÂj¯ �vä ¶-¬IÑÒ±i¯l«õÆ � éÎØlç�«j³G¯s �õû ·�¯�¯� Ï ¯sØo±i¯s×Ú­�ç>Ð ð ¯l«gÂ�ªGÑwØlç�«j³G¯sÂl¶ Ï «f¯s×>²�Øo±i¯s×ÿ¬GÂ�²�­=$jÖ � Ö�¶Y¬I«j¯�ÙGÜ å ¬G­>×� Ü å «j¯sÂ Ï ¯sØo±j²e³G¯ ÞeóG�`û ·�¯�Âjç>ØlØo¯sÂjÂiѨç Þ Ølç>«j³G¯sÂm¬I«f¯�×�¯oø�­>¯s× ð ó ã\�G�vä Í�²e±j· $ÕÅÇÖ�ÓGÓ>� � ñ���¬G­>× $ÚÅ ñIÓ�ÆIÓ�Ü Ö�Ù

Page 22: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

È ù ������������� �� ��� �6� � � æ�0\� «gç>­>Âtª�­&Ã5Ä�¶�µ�Å � Æ�ú âlâlâ ú � Ù

µ �5� Ølç�«j³G¯s ·�ª�ç�«gÂ�F Ølç�«j³G¯� Æ � Ó ý ÙGÙIÓ Ù � �� Ü Ù;= � Ó î Ö�ÜIÓ é � é� Ö Ù;= � Ó î ÜIÓGÓ �vå�� Æ� Ù Æ;= � Ó î ÜGéIÓ Æ �I� �

«j¯sÂ Ï ¯sØo±j²e³G¯ Þ�ó ¶>¬G­>× ±j·�¯´°G«fª�ç Ï ªG«f×�¯l«gÂ`¬I«j¯� Å Æ í ìvÜ�ì � Æ å ìvÜIÓ�ñGÜ´ìOÜGÙGÜ���ìT�Gé Ö �´ì � ñGÜGÆ���ìvÜGÜ��GÙGéGÙGÜ ¬G­>×� Å Æ � ìvÜ�ì �vå ìOÆGÜ�ìlÖ � ì �G� Ü�ìOÆ å�� ìOÜIÓ�Ü å ì � Ó�éGñ å ì � ÆGÆGÙ � ìvéGñGÆIÓ>� â

û ·�¯�Ѩ¬GØo±�±j·;¬I± Ï «jªG°G«g¬GÐ�Â!© ¬G­>× K ÑÒª�ç>­>×αj·�¯�Âj¬GÐ�¯�Æ å ¸�×>²�°�²e±!Ѩ¬GØo±iªG«m±j·�«f¯l¯´±j²�Ð�¯sÂ!Âfç�°G°G¯sÂi±j ã ð ç�±!×�ª�¯s­�ªG± Ï «jªv³G¯ ä ±j·;¬I±!±j·�¯õç>­>Ì�­�ªOÍ�­ÎѨ¬GØo±iªG«fÂ!ªGÑ3à ���õ¬I«j¯ Þ ¬I«j°G¯l«�±j·>¬G­ � Èjí �ñ ���I������( <>."� ��'� <! 5.�, �/=� ", 7�� ���8, ( 4��H=@?´û ·�¯�Øoª�Ð ð ²�­>¬I±j²eª�­�ªGÑL¬�Øf·�¯s¬ Ï ��07¬G­>×�¬ 0w«gç>­>Øf·>¯l« ð ª�¬I«f×ã�� 6 � Æ�¶bÙIÓGÓ ä ²�Â)³G¯l« ó Øoª�Âi±\¸U¯�Gq¯sØo±j²e³G¯�Ñ�ªG«�Ѩ¬GØo±iªG«f²�­>° Þ ¬I«j°G¯&²�­�±i¯l°G¯l«f ð ó æ�0�� � K ²�³G¯s­ ±j·�¯&²�­>²e±j²�¬ Þ ³I¬ Þ ç�¯$�Å � Ö � ÙGÆGÆGé å ¶q±j·�¯-0w«fç;­>Øf·�¯l« Ï «fªG°G«f¬GÐ K «fç>­>­;²�­�°�²�­�¬G­&ñIÓ�ÜGñGé FOÖ�Ó>��0ÛÍ�²�±j·�����ÅëÜGÆIÓGÓGÓGÓ�Øl¬G­�ø�­>×±j·�¯ �;÷jùmÑÒ¬GØo±iªG«tªGÑ5à ��ù�²�­�ª�­ Þ�ó ÙGé�Ю²�­�ç>±i¯s 㠱j·>²�Âw²�Â�«j¯s×;ç>Øo¯s×®±iª�ÜGñ�Ð�²�­�ç�±i¯sÂw²�­�¬G­ ñIÓIÖ�ñGé FIÙIÓ9��0 äg�� ­�¯0w«fç;­>Øf·�¯l«�²�­ÿ¬G­ÕñIÓ�ÜGñGé FOÖ�Ó*��0 Øoª�ç Þ ×ÿ·>¬s³G¯�×�ª�­>¯õª�ç�«´Æ Ö�Ó ��² Ï Â\¸ ó ¯s¬I«�Øoª�Ð Ï ç�±j¬I±j²�ª�­�±iª�Ѩ¬GØo±iªG«´Ã ��ù�²�­¬ Ï>Ï «fª� �²�Ð�¬I±i¯ Þeó ±�Íwª ó ¯s¬I«f � ©>ªG«m­�ç>Ð ð ¯l«fÂt¬ Ï>Ï «jª� �²�Ð�¬I±i¯ Þeó ¬GÂ Þ ¬I«j°G¯´¬GÂ�à �U÷G¶�±j·�¯ 0t«fç>­>Øg·�¯l«t«fç>­>Ât¬ ð ª�ç�±·>¬ Þ ÑY¬GÂ!Ѩ¬GÂi±�¬GÂ`±j·>¯ M �wÆGÆIÓGÓ'F � Ó �Ë ­ÿØoª Þ�Þ ¬ ð ªG«f¬I±j²eª�­�Í�²e±j· Am¬I«j³G¯ ó @�ç ð ­�¯l«v¶1Íw¯�¬I«j¯�ç>Âj²�­>°�Âi¯l³G¯l«f¬ Þ 0t«fç>­>Øg·�¯l«fÂ�²�­Õ¬G­ÿ¬I±i±i¯sÐ Ï ±�±iª�ø�­>×

Ð�ªG«j¯�Ѩ¬GØo±iªG«fÂ�ªGÑ�à � È ú âlâlâ újà ��î�ð ó æ�0\� � à �U÷ ²�Â Ï ¬I«j±j²�Ølç Þ ¬I« Þeó ²�­�±i¯l«j¯sÂi±j²�­>° ð ¯sØl¬Gç>Âi¯)²e±�²�Â�Ì9­�ªOÍ�­ÿ±iª ð ¯Øoª�Ð Ï ª�Âj²e±i¯�è å Ü êU¶ ð ç>±�­�ª Ï «g²�Ð�¯�Ѩ¬GØo±iªG«�²�Â!Ì�­�ªOÍ�­ �wË ­ û ¬ ð Þ ¯´Ö®Íw¯õ°�²e³G¯õ±j·�¯�Ølç�«f«j¯s­�± Ï ·>¬GÂi¯ �õÞ ²�Ð�²e± ���s¶±j·�¯�±iªG±j¬ Þ ­�ç>Ð ð ¯l«�ªGÑwØlç�«j³G¯s ã Íw¯s²e°�·�±i¯s× ²�­ Ï «jª Ï ªG«j±j²�ª�­�±iª ±j·�¯ Ï ·>¬GÂi¯ ��Þ ²�Ð�²�±mç>Âi¯s× ä ç Ï ±iª&©>¯ ð «fç>¬I« ó� ���Gé�¶�¬G­>×�±j·�¯õ±j²�Ð�¯õ«j¯���ç>²e«j¯s× Ï ¯l«�Ølç�«j³G¯ ��û ·�¯õ±j²�Ð�¯sÂm¬I«j¯�Ñ�ªG« Ï «jªG°G«g¬GÐ K «fç>­>­;²�­�°�ç>­>×�¯l« @ �M� ª�­ÿ¬0w«fç;­>Øf·�¯l«!²�­>Âi±j¬ Þ�Þ ¯s×β�­ ¬G­ÿñIÓ�ÜGñGé FOÖ�Ó���0�¶�Í�²�±j· � È Øg·�ª�Âi¯s­�Âiª Ï ·>¬GÂj¯õÆ)±j¬IÌG¯sÂm·;¬ Þ Ñ5¬GÂ Þ ª�­�°�¬GÂ Ï ·;¬GÂi¯ �ã Âi¯l¯ $jÖ � Ö äg�

� ���`¢�§ #����§�¡�� *>¢�£Y£ � �Ë ­ è�éOêBÍw¯�°�²e³G¯ Ï «f²�Ю¬ Þ ²e± ó Øo¯l«f±j²xø�Øl¬I±i¯sÂ�ÑÒªG«�±j·�¯ Ï «f²�Ð�¯�ÑÒ¬GØo±iªG«gÂ�ªGÑ�ÃYîIú âlâlâ újÃ#�G¶Lç>Âj²�­>°®¯sÂjÂi¯s­�±j²�¬ Þ�Þ�ó ±j·�¯

Ð�¯l±j·�ª9× Ï ²�ª�­�¯l¯l«j¯s× ð ó �5ç>Øl¬GÂ�è�ÙGé>¶ Ï � ÜIÓ�ÆOêU¶:D�«f¬G²e±jØg·>²eÌÎè Ö �9¶ Ï �>� ÜGÙOêq¬G­>× �-¯s·>Ð�¯l«mè�Ù � ¶ Ï � ÜGÜIÓ ê ��û ª Ï «jªO³G¯� Ï «f²�Ð�¯G¶;Í�¯)Øoª�Ð Ï Þ ¯l±i¯ Þ�ó ÑÒ¬GØo±iªG«G� % � ¬G­;× ø�­>×�¬ Ï «g²�Ð�²e±j²e³G¯�«fª�ªG± ã Ð�ª9× � äg��û ·�¯õÐ�¯l±j·>ª�×ÿ²�Â�¬ Ï;Ï Þ ²e¯s׫j¯sØlç�«gÂj²e³G¯ Þeó ±iª Þ ¬I«j°G¯ Ï «f²�Ð�¯�ÑÒ¬GØo±iªG«gÂ�ªGÑ�� % �I�

� ²�Ю² Þ ¬I«`Øo¯l«j±j²xø�Øl¬I±i¯sÂ�Øl¬G­ ð ¯´°�²�³G¯s­&ÑÒªG«�±j·�¯´Ñ¨¬GØo±iªG«f �;÷C�´¬G­>× ������ªGÑBÃ��I¶1ç>Âj²�­>°/0w«f¬G­>×>¬ Þ�Þ ïWÂtÑÒ¬GØo±iªG«g²eüs¬ ¸±j²eª�­>Â�è�ÙGÜ êLªGÑ:�;÷C� % � ¬G­>× �:��� % � �

� ÷C��% � Å Æ ��� ì � ��ìlÖ å ì ñGÆ Ö�ñGñ å ñ � ì �G� Ö�ÜGÆ��IÓ�ÆGÆGñ � é � Ü�Æ � ìlÖ�ÜGÆGÆGé Ö �IÓ�ÜGÙ��GÙGÙ åGå Ó�é�éGÆ �Îú� ����% � Å Æ ��� ì �G� Æ���ìvÆGéGñ � ÜõìlÖ�Ó�é ÖGÖ�Ü åGå ì �vå ÜGÜGñ Ö�Ü å Ù åGå�� Æ � ìQ��ý � ú ¬G­>×

� ý � % � Å Æ�ìvÜ � ì � Ü�ì � ÙGÜ � ì �vå ÜGñ�� å ì �vå Ö�é å Ù � ì � ÆIÓ�ñGñGÜGé � �GñGÜ�ì� Ü��GÆGÙ Ö�ÆGÆIÓ�ñIÓIÖ�ÆIÓ ��� ÆGÓ>��ìvÜIÓ�ñGñGñGñGñGÙIÓ�Æ Ö�éGñGÜGÓ�Ù å ñGÆ�ÙGÙ>� â

Ë ­ ±j·�¯sÂi¯�±j·�«f¯l¯õØl¬GÂi¯sÂ!±j·>¯ Þ ¯s¬GÂi± Ï «f²�Ð�²e±j²�³G¯m«jª�ªG±jÂ!¬I«j¯�Ü �©�ªG«m±j·�¯ Ï ¯s­�ç Þ ±j²�Ð�¬I±i¯�ÑÒ¬GØo±iªG« �;÷jù�ªGÑBà ��ùI¶;±j·�¯ Þ ¯s¬GÂi± Ï «f²�Ð�²e±j²�³G¯m«jª�ªG±�²�Â!Ù�¶�¬G­>× Íw¯õ·>¬s³G¯

� ÷jù�% � Å Æ � È ìvÜ�ìvÙGéGÜ���ìOñGÆGÜ � ìlÖ�ÜGÜGéGÜ��´ì � ñGñ Ö�Ó�ñGéGÆ å ��� � éGÙ�ÜGñ�éIÓ�Ó�Ü>�Gé å ú� ÷jù Ê � Å Æ�ìvÆGñGñ å ìOÙGÆ Ö å�� Ö åGå ìvÜ �G� ñGé � Ù å Ù��GÜ�ìsÖ �GÜ �våGå ÜIÓIÖ �våGå Ó �G� ÙIÓ å â

Page 23: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� È �

û ·�¯�Âj¬GÐ�¯�Ð�¯l±j·�ª9×ÚØl¬G­ ­�ªG± ð ¯�¬ Ï>Ï Þ ²e¯s×ÿ±iª Ï «fªv³G¯ Ï «g²�Ð�¬ Þ ²e± ó ªGÑ�±j·>¯ Þ ¬I«j°G¯sÂi± Ï «g²�Ð�¯�ÑÒ¬GØo±iªG«fÂõªGÑwà ��ù¬G­>×&à ���s¶ ð ¯sØl¬Gç>Âi¯�Íw¯õ·>¬v³G¯´ª�­ Þeó ²�­>Øoª�Ð Ï Þ ¯l±i¯�ÑÒ¬GØo±iªG«g²eüs¬I±j²eª�­> �

� È î È % � Å Æ ��� ìvÜ�ì � Ü�ìOÆGÜ�ìvÆ���ìOéGÜGÆ���ì å éIÓ�Ü Ö å�� Ó>��ìÆ �G� ñ��GñGÙGÆIÓ�ñGÜGÆGñGÙ � é�ÙGÆIÓ � ñ å Ó�ñ�� � Ü>� Ö�Ü�Ü �Oå ì� Ö�Ó>�GñGÙGÜGÆIÓ�ÙGé��GéGé�é Ö � éGñ � Ö^�Gé å�� Ö�Ü�Æ��GÙ�ÙIÓ å � å Ö�Ö�Ü>� å ì ��î ��ú

� È î È Ê � Å Æ�ìvÆ ÖGÖ�Ó å ìvÙIÓ åGå Ó�Æ � Ù�� Ö�é��)ì È �iî�ú��î ý ÷ % � Å Æ ��� ì � Ü���ì � ñ Ö å ì ÜGÆ Ö�ñGñGéGÆGñGÙIÓ�Ü�ì � ñ Ö å Æ å ÆGÆGñGÙGñGÜ � ñ�ñGÜ�ì

�GÆ � Ö å Ü Ö�Ù��Gñ Ö�ÆIÓ�ñ � � � ìOÆGÜ�� � ñ å éIÓ>�GÆ Ö � é Ö�ÆGÙ�ñ Ö�ñGñ�ÆGé � ì ÷,����ú��î ý ÷�Ê � Å Æ�ìvÜ È ìOéGÙGÆGÜ � ñGÜGÜ�ì îiîiî â

�Ú¯`Øl¬G­)¬ Ï>Ï Þeó5� ¯ Þ ÑÒ«f²�×>°G¯GïW �<0tç ð ¯ 8 ª�ªG± û ·>¯lªG«j¯sÐ9�)è�ÆIÓ�¶ û ·�¯lªG«j¯sÐ �G� ê9±iª � È î È ¶�Âf²�­>Øo¯�� È î È % � Å ÃÎì ��î � ¶Í�·�¯l«f¯)à %ôÆ = � Ó ��� ²�Â�Øoª�Ð Ï Þ ¯l±i¯ Þ�ó Ѩ¬GØo±iªG«j¯s×�¶ � È î È � ÆIÃ� Ê ÆIÃ�¶-¬G­>× ±j·�¯�ªG±j·�¯l«õØoª�­>×;²e±j²eª�­>Â�ªGÑw±j·�¯

0tç ð ¯ 8 ª�ªG± û ·�¯lªG«f¯sÐ ¬I«j¯ ¯s¬GÂj² Þ�ó ³G¯l«f²xø�¯s× � û ·�ç;Âl¶�� È î È ²�Â Ï «f²�Ð�¯G¶�¬G­;× ±j·�¯ Ѩ¬GØo±iªG«f²eüs¬I±j²eª�­ ªGÑ�à ��ù ²�ÂØoª�Ð Ï Þ ¯l±i¯ �û ·�¯ Þ ¬I«j°G¯�ÑÒ¬GØo±iªG«G��î ý ÷�ªGÑ�à ����Íw¬GÂ Ï «fªv³G¯s× Ï «f²�Ð�¯ ð ó ��ªG«f¬G²�­ ã ²�­ &�ç>­�¯ � �GñGñ ä ç;Âj²�­�°®¬�×>²�Âj±i«f² ð ç�±i¯s×

³G¯l«fÂj²�ª�­�ªGÑm·>²�Â�º � �9� Ï «jªG°G«f¬GÐ è�éGÆ9¶ Ï �t� ÜOê � �Ú¯ ·>¬s³G¯&ç>Âj¯s× ±j·�¯ Ï ç ð Þ ²�Ø Þeó ¬s³I¬G² Þ ¬ ð Þ ¯®³G¯l«fÂj²�ª�­�ªGÑ)º�� �6��¶Í�·>²�Øf·®²�Ð Ï Þ ¯sÐ�¯s­�±jÂB±j·�¯>�\¯ Þ�Þ ² Ï ±j²�Ø`Ølç�«j³G¯+��Ð�¯l±j·�ª9×�ªGÑ Ý ±iÌ9²�­�¬G­>× ��ªG«f¬G²�­Õè � ¶�ÆOêU¶�±iª)Øoª�­9ø;«fÐ ±j·>²�Âu«f¯sÂjç Þ ± �MB¯l«fÂj²�ª�­�M�Ü � Ö ��� ªGÑL¯sØ Ï>Ï ¶�«fç>­>­;²�­�°�ª�­�¬)éIÓ ��·�ü � ç Ï ¯l« � Ï ¬I«fØI¶G¯sÂi±j¬ ð Þ ²�Âj·�¯s×®±j·�¯ Ï «g²�Ð�¬ Þ ²e± ó ªGÑ�� î ý ÷ ²�­�ÆGñ·�ª�ç�«g ��Ë ±!±iª�ªGÌ ª�­ Þeó ª�­�¯õ·�ª�ç�«m±iª Ï «jªO³G¯ � È î È Ï «f²�Ð�¯ ð ó ±j·�¯õÂj¬GÐ�¯õÐ�¯l±j·�ª9× � �u«f²�Ю¬ Þ ²e± ó �iØo¯l«j±j²xø�Øl¬I±i¯sÂ<�¬I«j¯õ¬v³ ¬G² Þ ¬ ð Þ ¯®è � Ü ê ��û ·�¯ ó Øl¬G­ ð ¯´Øf·�¯sØfÌG¯s×�ç>Âf²�­�°�¬�Âi¯ Ï ¬I«g¬I±i¯ Ï «jªG°G«f¬GÐ � ��4�º����@�gºl»^¾�� � �

� Ó ��� (��>  ¡q£7¥ � �+�������$�L L¤ *>¢�£$� * ��¦1¡�� ��£-¢ à � È� ·�¯s­7Ѩ¬GØo±iªG«f²�­�° Þ ¬I«j°G¯�²�­�±i¯l°G¯l«f ð ó æ�0\� Í�¯ÿ×�ª ­�ªG±�ç>Âjç>¬ Þ�Þeó Ì9­�ªvÍ ±j·�¯�Âf²eül¯&ªGÑ�±j·�¯�Ѩ¬GØo±iªG«fÂ�²�­

¬G×�³I¬G­>Øo¯ �uû ·�ç>Âl¶�²�±�²�Â�²�Ð Ï ª�ÂjÂj² ð Þ ¯�±iª�¯sÂi±j²�Ð�¬I±i¯�·�ªvÍ Þ ª�­�°õæ�0\� Í�² Þ�Þ «j¯���ç>²e«f¯ �5Ë ­�Øoª�­�±i«f¬GÂi±l¶9±j·�¯�«gç>­>­>²�­>°±j²�Ð�¯sÂuªGÑ�±j·�¯ �1��) � ¬G­>×)°G¯s­�¯l«f¬ Þ ­�ç>Ð ð ¯l«3ø;¯ Þ ×�Âj²�¯l³G¯ ã K ö © �>ä Ð�¯l±j·�ª9×>ÂuØl¬G­ ð ¯ Ï «j¯s×>²�Øo±i¯s×)ÑÒ¬G²e« Þeó Íw¯ Þ�Þ ¶ð ¯sØl¬Gç>Âi¯m±j·�¯ ó ×�¯ Ï ¯s­>×�Ð�¬G²�­ Þeó ª�­®±j·�¯mÂf²eül¯!ªGÑ�±j·�¯�­�ç>Ð ð ¯l« ð ¯s²�­�°�Ѩ¬GØo±iªG«j¯s×�¶�¬G­>×�­�ªG±uª�­�±j·�¯mÂj²�ül¯!ªGÑ�±j·�¯ã ç>­�Ì9­�ªOÍ�­ ä Ѩ¬GØo±iªG«fÂ)è�éGñ ê �`Ý ­�²�Ð Ï ªG«j±j¬G­�± ��ç�¯sÂj±j²eª�­�²�Â�·�ªvÍ Þ ª�­�°�±iª�Â Ï ¯s­>× ª�­�æ�0\� ð ¯lÑ�ªG«j¯�ÂiÍ�²e±jØg·>²�­�°±iª®¬�Ð�ªG«j¯ Ï «j¯s×>²�Øo±j¬ ð Þ ¯�Ð�¯l±j·�ª9×&Âjç>Øg·&¬GÂB�1��) � F K ö © �q�û ·�¯lªG«f¯sÐ ÜÿªGÑ MB¯l«fÂj·>²eÌëè�ñ � ê�Ð�¬ ó ð ¯&·>¯ Þ Ï Ñ¨ç Þ/� 8 ª�ç�°�· Þ�ó ¶�²e±�Âj¬ ó ®±j·>¬I±�±j·�¯�«f¬I±j²eª�Â Þ ªG° ��� Þ ªG°�� ªGÑ

Þ ªG°�¬I«f²�±j·>Ð�ÂtªGÑ3­�¯s²e°�· ð ªG«g²�­�° Þ ¬I«j°G¯ Ï «g²�Ð�¯�×>²e³9²�ÂiªG«f ��¶�� ã � � � ä ªGÑ Þ ¬I«j°G¯õ«f¬G­>×�ª�Ð ²�­�±i¯l°G¯l«fÂm¬I«j¯´¬G ó Ð Ï ¸±iªG±j²�Øl¬ Þ�Þ�ó ²�­;×�¯ Ï ¯s­>×�¯s­�±)¬G­;×7ç>­>²eÑÒªG«fÐ Þ�ó ×>²�Âi±i«g² ð ç�±i¯s× ²�­ ã Ó9ú �väg� 6�Âj²�­�°Ú±j·>²�Â�±j·�¯lªG«j¯sÐ ã ¬GÂjÂjç>Ð�²�­�°Ú±j·�¯­�ç>Ð ð ¯l«fÂ�±iª ð ¯)ÑÒ¬GØo±iªG«f¯s× ð ¯s·>¬s³G¯ Þ ²�ÌG¯�«g¬G­>×�ª�Ð ²�­�±i¯l°G¯l«fÂ ä ªG« Ï ¬GÂj±�¯� Ï ¯l«f²e¯s­;Øo¯�°�¬G²�­�¯s× ð ó ÑÒ¬GØo±iªG«g²�­�°&¬Ø Þ ¬GÂjÂuªGÑ1­�ç>Ð ð ¯l«g ã Âjç>Øf·®¬G 0tç>­>­;²�­�°�·>¬GÐ(­�ç;Ð ð ¯l«fÂ ä ¶�Í�¯!Øl¬G­�Ð�¬IÌG¯�¬�«jª�ç�°�·�¯sÂi±j²�Ð�¬I±i¯!ªGÑ1±j·�¯ Ï «fª ð ¬ ð ² Þ ¸²e± ó �#±j·>¬I±!æ�0\��Í�² Þ�Þ Ñ¨¬GØo±iªG«�¬)°�²e³G¯s­ ­�ç>Ð ð ¯l«`²�­�ª�­�¯�ç>­;²e±�ªGÑ5±j²�Ð�¯ ã Âj¬ ó ª�­�¯´×;¬ ó ªGÑ3Øoª�Ð Ï ç>±i¯l«`±j²�Ð�¯ äg�û ·>²�Ât¯sÂi±j²�Ð�¬I±i¯õÂf·�ª�ç Þ ×�±j¬IÌG¯�²�­�±iª�¬GØlØoª�ç>­�±�±j·>¯õ²�­�ÑÒªG«fÐ�¬I±j²eª�­�±j·>¬I±mæ�0\� ·>¬GÂ!¬ Þ «f¯s¬G× ó Â Ï ¯s­�±!Âiª�Ð�¯ ã Âj¬ ó�Yù ä ç;­>²e±jÂtªGÑY±j²�Ð�¯´ç>­>Âfç>ØlØo¯sÂjÂiѨç Þ�Þeó Âi¯s¬I«fØf·;²�­�°�ÑÒªG«�¬�Ѩ¬GØo±iªG« �G� ² Þ ³G¯l«fÐ�¬G­ ¬G­>× � ¬I°�Âi±j¬HG è å �9¶%$ å ê5Âjç>°G°G¯sÂi±¬ þ ¬ ó ¯sÂj²�¬G­&¬ Ï>Ï «jª�¬GØg· ��Ý Â`¬�Âj²�Ð Ï Þ ¯m¬ Ï>Ï «jª� �²�Ð�¬I±j²eª�­L¶�Íw¯´Øoª�ç Þ × ç>Âj¯m±j·�¯�«f¯sÂjç Þ ±jÂwªGÑ $jÖ � Ö®±iª�¯sÂi±j²�Ð�¬I±i¯)�Âjç>Øg· ±j·>¬I±`±j·�¯õ¯� Ï ¯sØo±i¯s× ±j²�Ð�¯�Ñ�ªG«�æ�0\� ±iª)ø�­>× ¬)Ѩ¬GØo±iªG«mØ Þ ª�Âi¯õ±iª ��²�Â��YùI¶>¬G­>× ±j·>¯s­ ¬GÂjÂjç>Ð�¯´¬ Þ ªvÍw¯l«ð ª�ç>­>× �õª�­®±j·>¯�ç>­>Ì�­�ªOÍ�­�Ѩ¬GØo±iªG« �mã¨û ·>²�Â�¬GÐ�ª�ç>­�±jÂw±iª)¬ Ï>Ï «jªT 9²�Ð�¬I±j²�­�°´±j·>¯mѨç>­>Øo±j²eª�­�²�­ ã Æ�� ä ð ó ¬�Âi±i¯ ÏѨç>­>Øo±j²eª�­ ��ä©�ªG«�¯� �¬GÐ Ï Þ ¯G¶�²eÑmÍ�¯&¬I«j¯ Ѩ¬GØo±iªG«f²�­>°+" 4 � Ó ��ùiù ¬G­>×�3ù�²�Â�Âjç>Øf· ±j·;¬I±"� 4 � Ó �iù ¶�±j·�¯s­ Íw¯&Øoª�ç Þ ×

¬GÂjÂjç;Ð�¯`±j·>¬I±u±j·�¯!ç;­�Ì�­>ªvÍ�­)Ѩ¬GØo±iªG«�� Þ ²e¯sÂB²�­)±j·�¯!²�­�±i¯l«j³I¬ Þ�E� Ó �iù ú � Ó îiù � ¬G­;×�±j·>¬I± � � Þ ªG° ��ù ��²�ÂBç>­;²eÑ�ªG«gÐ Þeó×>²�Âj±i«f² ð ç�±i¯s×�²�­�±j·�¯�ØoªG«j«f¯sÂ Ï ª�­>×>²�­�°´²�­�±i¯l«j³I¬ Þ5ã\� �IÙIÓ9ú � �IÜIÓ äg�`û ·>¯ Ï «jª ð ¬ ð ² Þ ²e± ó � Øl¬G­�­�ªOÍ ð ¯`¯sÂi±j²�Ю¬I±i¯s×�¶ç>Âj²�­�°�±j·�¯�«j¯sÂjç Þ ±jÂ�ªGÑ $jÖ � Ö�¶L²�ÑYÍw¯�¬GÂjÂjç>Ð�¯õ±j·>¬I±�±j·�¯ Ï ¬I«f¬GÐ�¯l±i¯l«f ����¬G­>× � È ¬I«j¯�Øg·�ª�Âi¯s­ÿª Ï ±j²�Ю¬ Þ�Þeó ±iªø�­>×ÎѨ¬GØo±iªG«fÂ�ªGÑ3Âf²eül¯´Ø Þ ª�Âi¯�±iª � �uû ·�¯�¯sÂi±j²�Ð�¬I±i¯�Ð�²e°�·�±l¶�Ñ�ªG«m¯� 9¬GÐ Ï Þ ¯G¶ ð ¯ � ��� 4 �YùI¶�Ím·�¯l«j¯ 4 � �Ë Ñw±j·>¯ Ï «j¯s×>²�Øo±i¯s× «fç>­;­>²�­�°�±j²�Ð�¯�ªGÑ �1��) � F K ö © � ¯� �Øo¯l¯s×> � ��� ç>­>²�±jÂl¶-±j·�¯s­�²e±õ²�´Í�ªG«j±j·�Í�·>² Þ ¯�±iª

Øoª�­�±j²�­�ç>¯�Í�²e±j·Ûæ�0\� ÑÒªG«�¬ Þ ²e±i± Þ ¯ Þ ª�­�°G¯l« �ëË Ñ�æ�0\��²�Â�ç>­>Âjç;ØlØo¯sÂjÂiѨç Þ ¶wÍ�¯�«j¯ Ï ¯s¬I±®±j·�¯ Ï «jª9Øo¯s×>ç�«j¯&ªGÑ

Page 24: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

ÈiÈ ������������� �� ��� �6��� �M� ¯sØoª�­>×9¸ Þ ¬I«j°G¯sÂj± Ï «f²�Ð�¯mѨ¬GØo±iªG«fÂmªGÑYÃ5Ä

µ å ñ � � Ó �G�� È Ó � �Gñ Ó � Ö�Ù Ó � ñGÆ Ó � Æ å Ó � Ó�é

¯sÂi±j²�Ю¬I±j²�­�°�� � æu³G¯s­�±jç>¬ Þ�Þeó ¶m¯s²e±j·�¯l«Î¬�Ѩ¬GØo±iªG« Í�² Þ�Þ ð ¯�ÑÒª�ç>­>× ð ó æ�0\��ªG« ±j·>¯�¯sÂi±j²�Ð�¬I±i¯ÕªGÑ � Í�² Þ�Þð ¯sØoª�Ð�¯�Âjª�ÂjÐ�¬ Þ�Þ ±j·>¬I±�¬�ÂiÍ�²e±jØg· ±iª �(��) � F K ö © � ²�Â�²�­>×>²�Øl¬I±i¯s× �®Ë Ñt¬�ÑÒ¬GØo±iªG« � ²�Â�ÑÒª�ç>­>× ð ó æ�0\��¶±j·�¯s­Õ±j·�¯ ��ç�ªG±j²e¯s­�±,"7� ��²�Â�¯s²e±j·>¯l« Ï «f²�Ð�¯ ã Âiª ±j·�¯�Ѩ¬GØo±iªG«f²eüs¬I±j²�ª�­ ²�Â�Øoª�Ð Ï Þ ¯l±i¯ ä ªG«õÐ�ç;Øf·ÚÂjÐ�¬ Þ�Þ ¯l«�±j·>¬G­±j·�¯!ªG«g²e°�²�­>¬ Þ Øoª�Ð Ï ª�Âj²�±i¯�­�ç;Ð ð ¯l« "�¶�¬G­>×�·>¯s­>Øo¯�Ð�ç>Øf·�Ð�ªG«j¯!¯s¬GÂj² Þeó Ѩ¬GØo±iªG«j¯s× ð ó �1��) � F K ö © �1��ã¨û ·>²�²�Â�­�ªG±�±i«fç�¯®ÑÒªG« �9ö © � ¶ ð ¯sØl¬Gç>Âi¯�Ì�­>ªvÍ Þ ¯s×>°G¯�ªGÑ�¬ÿ­�ª�­9¸�¬ Þ °G¯ ð «f¬G²�Ø�ÑÒ¬GØo±iªG«�×�ª�¯sÂ)­�ªG±�°G«j¯s¬I± Þeó Â Ï ¯l¯s× ç Ï�9ö © � ¶>Âi¯l¯�è�ÙGÆ9¶1ÙGÜ ê ��äö ªG±i¯�±j·>¬I±�ª�ç�«´¬ Ï>Ï «fª�¬GØf· ²�Â�«j¯s¬GÂjª�­>¬ ð Þ ¯�²�­ÿ±j·�¯ Þ ²�Ð�²e±j²�­�°�Øl¬GÂi¯ " � � ¶ ð ¯sØl¬Gç>Âi¯)±j·�¯�¬GÂjÂjç>Ð Ï ±j²eª�­

±j·>¬I± � � Þ ªG°���²�ÂBç>­>²eÑÒªG«fÐ Þeó ×;²�Âi±i«f² ð ç�±i¯s×)²�­ ã Ó9ú � � Þ ªG°�� ä °�²e³G¯sÂ�¬ Ï ª�Âj²e±j²�³G¯`¯sÂi±j²�Ð�¬I±i¯!ÑÒªG« � � ©�ªG«�¯� 9¬GÐ Ï Þ ¯G¶«j¯ Ï Þ ¬GØl²�­�° " 4 � Ó ��ùiù ð ó " 4 à ��î�²�­&±j·�¯�¯� 9¬GÐ Ï Þ ¯�¬ ð ªO³G¯)Ð�ç Þ ±j² Ï Þ ²e¯sÂt±j·�¯�Øoª�­>Âi±j¬G­�± ð ó ¬�Ѩ¬GØo±iªG«�ªGѬ ð ª�ç�±�Æ â Ù�Å ã\� �IÜIÓ ä � ã\� �IÜIÓ % � �IÙIÓ äg�©�ªG«�±j·�¯®©�¯l«fЮ¬I±�­�ç>Ð ð ¯l«fÂ�Ã Ä ¶ � ÆÿÔ µ Ô � Ù�¶5±j·>¯ Ï «j¯s×>²�Øo±i¯s× Ï «jª ð ¬ ð ² Þ ²e± ó ªGÑ`Âjç>ØlØo¯sÂjÂõÑ�ªG«õæ�0\� ²�Â

Þ ªOÍ´¶ ð ç>±�±j·�¯ Ï «j¯s×>²�Øo±i¯s×&«fç>­>­>²�­�°�±j²�Ð�¯´ªGÑBªG±j·>¯l«�Ð�¯l±j·�ª�×;Â�²�ÂmÂiª Þ ¬I«f°G¯õ±j·>¬I±�²e±m²�Â�«f¬I±j²eª�­>¬ Þ ±iª�Øoª�­�±j²�­�ç�¯±i« ó ²�­�° æ�0�� ��û ·�¯l«f¯�²�Â�­�ª Ï «f¬GØo±j²�Øl¬ Þ ¬ Þ ±i¯l«f­>¬I±j²e³G¯�¯� �Øo¯ Ï ±´±j·�¯)ª Þ ×ÚÐ�¯l±j·�ª�×ÕªGÑw±i«f²�¬ Þ ×;²e³�²�Âj²eª�­ ã Âj¯l¯�±j·�¯×>²�ÂfØlç>ÂjÂj²eª�­Î¬I±!±j·>¯�¯s­>× ªGÑ�$jÖ � Ü äg�Ý�Þ ±j·�ª�ç�°�·�±j·�¯�©�¯l«fÐ�¬I±`­�ç>Ð ð ¯l«fÂw¬I«j¯�­�ªG±w«f¬G­>×�ª�Ð ²�­�±i¯l°G¯l«fÂl¶�²e±w²�Â�²�­�±i¯l«j¯sÂj±j²�­�°�±iª�Øoª�Ð Ï ç�±i¯ � È ã � ä ÑÒªG«� Å Þ ªG°�Ã Ä � Þ ªG°�� È ¶�Ím·�¯l«j¯ � È ²�Â3±j·�¯!Âi¯sØoª�­;×9¸ Þ ¬I«j°G¯sÂi± Ï «f²�Ð�¯wѨ¬GØo±iªG«�ªGÑ�Ã Ä ¬G­>× � È ã � ä ²�ÂB×�¯oø�­>¯s× ð ó ã\� Ö äg�û ·�¯�³ ¬ Þ ç�¯sÂ�ÑÒªG«�µ Å å ú âlâlâ ú �G� ¬I«j¯�°�²e³G¯s­Õ²�­ û ¬ ð Þ ¯)Ù � ©>ªG« Þ ¬I«j°G¯�«g¬G­>×�ª�Ð ²�­�±i¯l°G¯l«gÂl¶qÍ�¯�¯� Ï ¯sØo± � È ã � ä±iª ð ¯�ç>­>²eÑÒªG«fÐ Þ�ó ×>²�Âi±i«f² ð ç>±i¯s× ã Âi¯l¯>$jÖ ���väg� �Ú¯�Âi¯l¯�±j·>¬I±�à ����·>¬GÂ�¬�Âfç�« Ï «f²�Âj²�­�° Þeó ÂjÐ�¬ Þ�Þ Âi¯sØoª�­>×9¸ Þ ¬I«f°G¯sÂi±

Ѩ¬GØo±iªG«s¶-¬G­>×ÚÃ í ·;¬GÂ�¬ Âjç�« Ï «g²�Âj²�­�° Þ�ó�Þ ¬I«j°G¯�Âi¯sØoª�­;×9¸ Þ ¬I«j°G¯sÂi±�Ѩ¬GØo±iªG« �)û ·�¯)Âi¯sØoª�­>×9¸ Þ ¬I«j°G¯sÂi±�Ѩ¬GØo±iªG«fÂ�ªGÑ�Ã#�I¶Ã ��¬G­>×&à ��ù�¬I«j¯õ­�ªG±!¯� �Øo¯ Ï ±j²eª�­;¬ Þ�Þeó ÂfÐ�¬ Þ�Þ ªG« Þ ¬I«j°G¯ �û ·�¯ Ï «fª ð ¬ ð ² Þ ²�± ó ±j·>¬I±´¬Î«f¬G­;×�ª�Ð ²�­�±i¯l°G¯l«�" Ø Þ ª�Âi¯�±iª&à � È ·;¬GÂ�Âi¯sØoª�­>×9¸ Þ ¬I«f°G¯sÂi± Ï «f²�Ð�¯�Ѩ¬GØo±iªG« � È �� Ó ÷jù ²�Â Ó â Ó�Ù�� � à � È ·>¬GÂÎø;³G¯ÚÌ9­�ªOÍ�­ Ï «f²�Ð�¯�Ѩ¬GØo±iªG«f ã Âi¯l¯ $ �vä ¶´¬G­>× K �!þ�� K ª�Âi±j²�­ ·>¬G Âj·�ªOÍ�­ ð ó

¯� �·>¬Gç>Âi±j²e³G¯ÎÂi¯s¬I«fØg· ±j·>¬I±�±j·>¯sÂi¯�¬I«f¯�²e±jÂõø;³G¯ÎÂjÐ�¬ Þ�Þ ¯sÂi± Ï «g²�Ð�¯�Ѩ¬GØo±iªG«f �Õû ·�¯®ø;Ñ�±j·�¸�ÂjÐ�¬ Þ�Þ ¯sÂj±õ²� þ ¬G² Þ�Þ ²e¯GïWÂ��� ý Å � ÆGÙGé � ÜGÆ � Ü Ö � ÆGÙGÙGé�� ã Âi¯l¯*$ åGäg�&Ý ­�²�­>×>²�Øl¬I±j²�ª�­ÚªGÑt±j·�¯ Þ ²�ÌG¯ Þeó Âj²�ül¯®ªGÑt±j·�¯�Âf²2 �±j·�¸�ÂjÐ�¬ Þ�Þ ¯sÂj± Ï «f²�Ð�¯Ñ¨¬GØo±iªG«�Øl¬G­ ð ¯�ª ð ±j¬G²�­�¯s×�Ñ�«jª�Ð MB¯l«fÂj·>²�Ì�ïWÂ`«f¯sÂjç Þ ±�è�ñ � ¶ û ·>Ð � ÜOêU¶;Í�·>²�Øf·&²�Â Ï ¬I«f¬ Ï ·>«f¬GÂi¯s×&¬ ð ªv³G¯ � � ­�±j·�¯ªG±j·�¯l«�·>¬G­>×�¶�A�¬I«j³G¯ ó @�ç ð ­>¯l«�¬G­>× ±j·�¯�¬Gç�±j·�ªG«m·;¬s³G¯õ±i«g²e¯s×&Ð�ªG«j¯�±j·>¬G­�ÙIÓGÓ�Ølç�«j³G¯sÂ�Í�²e±j· ���wÅ � Ó ý ¶;²�­¬G­ÿ¬I±i±i¯sÐ Ï ±�±iª�Ѩ¬GØo±iªG«�à � È ¶;Ím²e±j·�ª�ç�±`ø1­>×>²�­�°�Ð�ªG«f¯õ±j·>¬G­�±j·�¯´ø;³G¯�Ì�­>ªvÍ�­ Ï «f²�Ð�¯õÑÒ¬GØo±iªG«f �!û ·�ç;Âl¶>ÑÒ«jª�Ðû ¬ ð Þ ¯�Æ®¬G­>× ã Æ�� ä ¶1Í�¯�Øl¬G­ ð ¯´«f¯s¬GÂiª�­>¬ ð Þeó Øoª�­9ø�×>¯s­�±m±j·>¬I±�±j·�¯�Âj²2 9±j·9¸�ÂjÐ�¬ Þ�Þ ¯sÂi± Ï «f²�Ð�¯�ÑÒ¬GØo±iªG«�ªGÑBà � È ²�¬I± Þ ¯s¬GÂj± � Ó �iù !;¬�ÂjЮ¬ Þ�Þ ¯l«`Ѩ¬GØo±iªG«�Íwª�ç Þ × ·>¬v³G¯ ð ¯l¯s­ÎÑ�ª�ç;­>×ÎÍ�²e±j· Ï «fª ð ¬ ð ² Þ ²�± ó °G«j¯s¬I±i¯l«�±j·>¬G­&Ó â � �û ·�¯�Øoª�Ð Ï Þ ¯l±i¯�ÑÒ¬GØo±iªG«g²eüs¬I±j²eª�­ÿªGÑuà � È Ð�¬ ó ·>¬s³G¯)±iª�Íw¬G²e±mÑ�ªG«�±j·�¯ Ï · ó Âj²�Øl¬ Þ Øoª�­>Âi±i«fç;Øo±j²eª�­�ªGÑ�¬ ��ç>¬G­9¸

±jç>Ð Øoª�Ð Ï ç�±i¯l«�Øl¬ Ï ¬ ð Þ ¯�ªGÑB«gç>­>­>²�­>° � ·�ªG«sïWÂ�¬ Þ °GªG«f²e±j·;Ð è å é êU¶qªG«�¬�Âjç�« Ï «g²�Âj²�­�°�­�¯lÍ#×�¯l³G¯ Þ ª Ï Ð�¯s­�±�Âjç>Øg·¬G ¬ Ø Þ ¬GÂjÂj²�Øl¬ Þ�ã ×�¯l±i¯l«fÐ�²�­;²�Âi±j²�Ø�ªG« «f¬G­>×�ª�Ð ä Ï ª Þeó ­�ª�Ð�²�¬ Þ ¸U±j²�Ð�¯�²�­�±i¯l°G¯l«&Ѩ¬GØo±iªG«f²�­�° ¬ Þ °GªG«f²�±j·>Ð � Am¯l«j¯G¶� Ï ª Þeó ­�ª�Ð�²�¬ Þ ¸U±j²�Ð�¯+��Ð�¯s¬G­>Â�±j·>¬I±u±j·�¯m¯� Ï ¯sØo±i¯s×�«gç>­�±j²�Ð�¯mÂj·�ª�ç Þ × ð ¯ ð ª�ç>­;×�¯s×�¬ ð ªv³G¯ ð ó ¬ Ï ª Þeó ­�ª�Ð�²�¬ Þ²�­Î±j·�¯��eºl¿$!�¾54 ã ­�ªG±!±j·�¯´²�­;×�¯� ä ªGÑ3±j·�¯õ­�ç>Ð ð ¯l«`±iª ð ¯�Ѩ¬GØo±iªG«j¯s× �

� � �)�>¢��> 5¦ � �+�| -w2u8���8 a;8�2qXU�sTWk�dgk N�p18���[gS�dfTbk�{���(2<�K (2<��B+3.�(2����0Q6 #\*,.�J'#&+��Q;'J �"!$#�#,626 ( N$�)( *\*,.�J '# %�#&�"!/;�KO;����Q��*,��;'J�(����'�)( ; <��9��dfXUZ�8.1[g]`_�8����)�¨|\�^�g�o�E{G�g�^���l~o}^�s8

+ � -w2u8���8Ca>8j2qXU�sTbkudfkINBp18f��[fS�dgTWk�{�� 626 ( N �)( *:*,.TJ'#&+ � <�K:N J�(2%��'6 (2�)H�N$JL;��(2<����v��dfXUZ�8f.1[^]`_�8����w�¨|\�^�g�o�E{o�g���l�^zs8^6�SU[^�fS�dg]`PdingdgTWabdgVOabMuVl�`c�XU_mc�SU[^] ��!�"$#&%�'�()%�*+#,��(.-�/�0�1�23054�/�6�7�"�"$#98�:;#=<.# >�#=!�*�(.#�?�8

+ ��-�Kÿ8�:�[^P/]!d�dfkINõ2�8�@t8^a�M\kOP¨X/S�dv{$A�< ("% N�6 #&%�#&<$���'�)( ; < ;��G�"!$#B#,626 ( N �)( * *,.�J'# (2</�X#9� #&JB�,��*,��;'J�(����'�)( ;'< % #&�"!T;�K��1.1[^]t�_OrOX�djXUTb[gkIdfa>25aW�^M\VvS�dmdfkIN�=YrO]wVGMQSY?qZ MQ[gS/���eMiNvT�XUMiN�Vs��KÕ8�:1[^P/]!d!dfkINõ2�8GngdgkõNOMESB69[l[gS/XUMQkG�E{�@BaWrsy1MQSu2-Rid^NvM\]`TWR6>r VOaWTbP/ZOMQSUP\{O43[gS�NsSUM\REZlXi{�|i�g�o�s{�|^|\���I|\�^�s8

+ ~ -�+u8�698�:�SUM\klXi{;A�6 ��;'J&("�"!�%DC�E�FHG �JIK���L�^;'J&�)JP�'< %G.T6 �)( N�6 #NM8N JL#3*,("+�();'< �'J�(2�"!�% #&�)()* NT��*OE�P� # �;25.��#?�S�dgk P\8�[gk)��djXUZ MQ�]!djXUTWRidga Av[fc�XÒyqdfSUMRQ)�¨|\�o�fzo�E{��l| �szs|^8

Page 25: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� È �

+ � -�+u8O698I:�SUMQksXi{ A�<M(2% N$JL;�'#LKG��;'<$�X#�4 �'J&68; �Q�E*,��;'J&( �C� �)( ;'< �E6 ��;'J�(2�"!�% ��:�,¨? � �)�¨|\�^z^}^�E{�|i�g��� |izf~O8+ ��-�+u8G698I:�SUM\klXi{�-$./*3*,(2< *,� N$JL;�;��&+ ;�� N J�(2%��'6 (2�)H �Q;'J �"!$# �Q��*,��; J&+G;�� +,; % # ��#&J�%��'� </.�%�0�#&J&+ ����dfXUZ�8G.1[^]`_�8����)�¨|i�gzo�g�E{�g�g���v�^�g�s8

+ � -�+u8O698I:�SUMQksXi{>- ;'% # (2<$�X# � #&J��Q��*,��;'J�(����'�)( ; < �'6 ��;'J&("�"!�%G+ .�+3(2<�� #,626 ( N$�)( * *,.�J'#&+ ��25r P¨X/S�dfaWT�dfk´.1[^]`_OrOXUMQS�AsR\TWM\k RQM�.1[^]t�]�r kOTbR\dfXUTW[^kOP����¨|i�gz^�o�E{�|\~^���I|\�^�s8 + <�S/S�dfX�d��-���v8 �^��P/Z [gr abN�SUMidgN�� W ����Y�������W���� �e]`[sN ���E8 -92-aWP/[`+5M\_G[fS/X5.���2-�+5�o�i��zo�s{I.1M\klX/SUM3cx[fSB��djXUZ M\]!djXUTWRidga�25k dga��vP/TbP\{I25rOP¨X/S�dgaWTbdgk�=YdfXUTW[^k dga��Yk TWnoMESUP/TWX��l{ .qdgksVGMQS/S�dv{�AvM\_vXi8�|i�^z^�s{��f}w_ _�8

+ z�-�+u8�698�:�SUM\klXi{+� ��*,��;'J�(����'�)( ;'< ;�� �"!$# #,6 # '#&</�"! ��#&J&%\�'��<$.�%�0�#&J��8N JL#,6 (2%G(2< �'J�H J3#XNT; J&������23�A´25VOP¨X/S�dgRQXUP �����¨|i�gz^�^�E{zg�^?1�U|^|E�Ò�g�v8

+ ��-�+u8�6981:;SUM\klXi{ � ��*,��; J�G �'< (2</�X#9� #&J �,��*,��;'J�(����'�)( ;'< N JP; �'JL�'% �Q; J �"!$#��! �� I 4;�q+-M\_G[gS/X�?q+L�Ò. As��z^�j�¨�f�v{-.1[g]`_ rvXUMQSAsR\TWM\k RQM\P�a�dgVG[fS�dfXU[fS/�l{I2-r P¨X/S�dgaWTbdgk�=YdfXUTW[^k dga��YkOTbnG8W{ .qdgksVGMQS/S�ds{��YRQXi8�|i�gz^�v{G��_ _�8

+�|i}�-�+u8�698�:�SUM\klXi{+I��'JP�E626 #,67�E6 ��; J&(2�"! %G+K�,;'JG(2</�X# � #&J �Q��*,��;'J�(2+&�'�)( ; <��>=YrO]wVGMQSY?qZ MQ[gS/��dgk N�.�S/�v_vXU[^�gS�df_ Zl���eMiNvTWXUM\NõVl�#"O80�8/a�[ �sXU[^kG�E{I.qdg]�VOSUTbNO�gM$�YkOTWnoMQSUP/T�XÒ�m6�SUM\P/P\{�|\�^�^}s8

+�|^| -�+u8O698I:�SUMQksXi{&%$#L*,��;'J �'<�K NT�'JL�'626 #,6��E6 ��; J&(2�"! %G+.�Q; J (2<$�X# � #&J$�Q��*,��; J&(2+&�'�)();'<���6�SU[lR^8G?qZOTWS�N�25r P¨X/S�dfaWT�dfkMAvrO_GMQSUR\[g]`_ rOXUMES.1[gkOc�MQSUM\k RQM^{���M\aWVG[grOSUk Mg{�|i�g�^}s8

+�|j� -�+u8O698I:�SUMQksXi{�' �'J � # �Q��*,��; J&+$�Q; .T< K 0,H � 4 � �GdjngdgTWabdgVOaWM�Vl�!c�XU_�c�SU[g] ')%�(�)+*-,�#�*5'+*$#,6/./*$#�*0*$-�/5"+*/))/1 (�6�'�!�/�702)*3()"�, #=6�7�(�8

+�|i��-�+u8 698Î:�SUM\klXi{ I�J&("%��E6 (2�)H *�#&J&�)( 9�*3�'�X#&+ �Q;'J �Q��*,��;'J�+ ;�� +&;'%�# � #&J&%\�'� </.�%�0�#&J&+ �Ûdin^dfTWa�dfV aWM(Vl� c�XU_ c�SU[^]')%�(�)+*-,�#�*5'+*$#,6/./*$#�*0*$-�/5"+*/))/ 1 (�65'�!�/54 >765"�8:9+8+#&756�(<;=4H>�> "�9/>�<.#�7�6�(�8

+�|\~ -�+u8O698I:�SUMQksXi{�?�<5*3;'%�N .��)(2<�� �,��*,��;'J�+ ;�� *,H�*Q68;'��; %G( *�NT;E6 H'< ;'%G()�E6 + ����dfXUZ�8�.1[^]`_�8����´�¨|i�g�^�^�E{�|i�s| � |\~o�s8+�|j� -�+u8s698s:�SUMQksXqdgk NA@�8Ta;8v.1[gZ M\k�{�A <�#&I�68;'I:#&J 03;'.�<�KK�Q;'J ;�K�K N/#&J ��#L*,�7<$.�%�0�#&J�+����dfXUZ�8s.1[^]`_�8�B��´�¨|i�^zg�o�E{v~o�s| �o~o�o�l8

Asr _O_ aWM\]`M\klXi{ ( 0,()Kg{�AO����A �f~v8+�|i��-�+u8^6�8o:�SUM\klXi{�@�8�a;8v.1[gZ M\k!dfkINw0�8:"O8+"O8oXUM�+-TbMQabMg{C��% N JP;�'#LK �X#L*�!�<$(ED,.$#&+ �Q;'J�68;'I:#&J�03;'.�<�K'+��Q;'J�;�K�K�N/#&J ��#L*,��</.T%\0�#&J&+ ���djXUZ�8I.1[^]`_�8FBHG��¨|i�g�v|j�E{Gzo�^�P�szg�^zv8

+�|j� -�+u8s698v:�SUM\klXqdgkINA"O8O��8O69[gababdjS�N�{�� ��*,��;'J�(����'�)( ; < ;=���"!$# #&(��C!��"!R��#&J�%��'�7</.T%\0�#&J ����djXUZ�8O.1[g]`_�8��H�õ�¨|\�^zv|i�E{I�^�^�i�Ò�g�^}s86�SUM\aWTW]`TWkIdfS/��dgkOk [gr k RQM\]`M\klXqTWk�2Y�Am25VOP¨X/S�dgRQXUP �´�¨|\�^zg}o�E{��f�o�l8

+�|iz�-�+u856983:�SUMQksXõdfkIN�0�8I"O8J"O85XUM�+-TWM\aWM^{ � ��*,��; J&(����'�)();'<$+ ;�� � ÉLK |���|i�NM �PO |i}g})�`+-M\_G[gS/X´=3�m��+Y�^�s|j�l{�43MQ�_ dfS/XU]`M\klX�[gct=5r ]`MQSUTWRidfa���dfXUZOM\]!dfXUTWR\P\{�.1MQksX/SUrO] no[l[fS�K TWP/�sr kINvM�MQk�,�kOcx[fSU]!dfXUTWRids{�25]`P¨XUMQS�NOdg]�{J"^r kOM�|i�g�o�l82-aWP/[��xyLT�XUZ�6�8 a;8���[^klXU�g[^]`MQS/� �E{�QTNTK��'�X#SR ��; � ��*,��;'J�(����'�)( ; <$+ ;=� � É K |��õ|i�TM �UO |i}^}���+-M\_G[gS/X�=B�m�+5�g~v|i�s{�.1MQksX/SUrO] n^[l[gS�K TWP/�sr k NOM®M\k�,�kOc�[gSU]!dfXUTWRids{�25]`P¨XUMQS�NOdg]�{�AvM\_vXUM\]wVGMQS®|i�g�g~v8t2-n^dfTbabdfV aWM�Vl�Õc�XU_ c�SU[^]')%�(�)+*-,�#�*5'+*$#,6/./*$#�*0*$-�/5"+*/))/ 1 (�65'�!�/�(�"+)H>�:�<�V #!V�#�?�8

+�|i��-W"v8I:�SUTWaWaWZIdjS/Xi{�-�;'%�# % (2+&*�#,6268�'<>#L;'.�+$�Q�E*,��;'J&( �C� �)( ;'<$+ ����dfXUZ�8G.1[g]`_�8;�-G��¨|\�^�g�o�E{I~g~l�P�l~l�f}v8+ �g}�-W"v8�:�SUTWaWabZ dfS/Xi{�4w8O0�8 a�M\ZO]`MQSi{H"v8>a;8�AvM\a�c�SUTbNv�^M^{�:58I?�r R��oMQSU]!dfk�{ dgk NMA�8>AG8 K)dg�^P¨X�d0X�{Y"gSi8W{�� �E*,��;'J&( �C� �)( ;'<$+\;��Z� É K |��

� � ��[���[���[���[���[i|\}�[i|g|/[i|j�.CN ��; !�(��C!\NT;'I:#&J&+U{��fk N�MiN�8W{�25]`MESi8G��djXUZ�8^Av[lRg8b{�6�SU[jnsT�NvM\k RQM^{�+5,�{�|\�^z^zs8�25aWP/[!r _�N djXUM\PXU[W�Y_�N djXUMu�s8 �s{G25rO�^rOP¨XB|i�v{�|i�g�o�s8

+ �s| -�=�8H@�8INOM�:�SUrOT RÒk�{]\^!$#G�'+3H % N ��;'�)()*�0&#3!T���( ;'.�JG;�� �B�&.�<�*,�)();'< ;�*3*,.�J�J&(2<�� (2<M�"!$# �"!$#L;'J�H ;���N$J&(2%�#&+ �^"v8I,Òk NOTbdgk���dfXUZ�8As[lR^8 ��B��¨|i�^�s|j�E{G�^�P�s�o�l8

+ �^� -t.581.qdfa�NsyqM\aWae{_\^!$#a` .$0,<�#&JLI 4 4�J&.�< *�!$#&JLb � % ( *,JL;�*3;'%�N .��X#&JM*3;&N JL;�*�#&+3+&;'J�*3�'JLKR�Q; JOK�; (2<�� (2</�X#9� #&J�'J&("�"!�% #&�)( *Q{SUMQnvTWMQy�TWkL"O8O+-M\R^8I��dfXUZ�8 � B��¨|i�E{�|\�^�g�v8

+ �g��-�4�8 ��8odfkINW@�8���8l.1ZsrINvk [jnsP/�l�l{ ->#!D,.$#&<�*�#&+ ;=� </.�%�0�#&J&+�� #&<�#&JL�'�X#LK�0,H ��KEK'(2�)( ;'< (2< �Q; J&%\�E6��'JP;'.CN +��'<�K <>#&I N J&("%��E6 (2�)H� <�KB�Q��*,��;'J�(����'�)( ;'<5�X#&+3�)+ ��25NOnG8vTWk�25_O_ ae8 ��dfXUZ�8<G��¨|\�^zg�o�E{9|\zo���s�g�^�s8

+ �f~ -�0�81.1[^Z MQk�{B� 626 ( N �)()**,.TJ'#&+ �3pvSU[^]#=5r ]wVGMES`?qZ MQ[gS/�®XU[)6>Zs�vP/TWR\P��eMiNvT�XUMiN�Vl����8>K�dfa�NvP/REZO]`TbNvXi{�6981��[^rOP/PUdv{&"O8 ���8$a�rORE��dfkIN�.58 ,�XUhQ�v�sP/[^kI�E{$Av_OSUTWkO�^MQS/� �;MQSUabdg�O{O=YMEy ?1[fSU��{�|\�^�o�l{��s|i���s�g�o�l8

+ �^� -�+u8O<L8I.�S�dgk N dfabae{�I�JL;X1C#L*,�)+ (2<+&*,(�#&<$�)( 9�*G*3;'%�N .����'�)( ;'<3��As_OSUTWk �gMQS/� ��MESUa�df�O{O=YMQy ?�[gSU�G{�|i�g�g~O8+ �g��-�+u8O<L8I.�S�dgk N dfabae{ \ ;&N ( *,+ ("<5��K� �'< *&#LK +,*,( #&<$�)( 9�*\*3;'% N .����'�)();'<�� Av_vSUTWk �^MES/� �;MQSUabdg�v{v=YMQy ?�[gSU�G{�|i�^�g�v8+ �^� -�+u89.�S�dgkINOdgaWae{�"O8�43[lM\k TbdfP\{;.589=5[gS/SUTWM^{>dfkIN�"O8 ?�[^r kO�O{J\^!$# �)I:#&<$�)H�M +Q#L*3;'< K ��#&J�%��'� </.�%�0�#&J�(2+�*3;'%�NT;'+3(2�X# �q��dfXUZ�8.1[g]`_�8��3Q��¨|\�^�^�^�E{Gzg�^���sz^�gzv8

+ �gz�-�+u8g.�S�dgk N dfabaOdgk Nw:58gpIdf�^TWk�{�` (2+&*,JL#&�X#�I:#&(��C!��X#LK �)JP�'<$+ �Q; J&% +��'<�K 68�'J � #NM�(2</�X# � #&J �'J�(2�"!�%�#&�)( *N�O��dfXUZ�8l.1[g]`_�8�� � �¨|i�g�g~o�E{�g}o�P�s�o�j~O8

+ �g��-w2u8�"O8v.58O.1rOk kOTbkO�^Z dg]�dgkIN`2u8s<�8oK)M\P¨XUMESUk�{<?�<R��#&J&%\�'�0c +�<$.�%�0�#&J�+ �G6�SU[lR^8Ta�[gkINO[gkm��dfXUZ�8�Av[lR^8����g� �m�¨|i�g}g~o�E{�|i�^�l8+ �^}�-w2u8�"O8 .58O.1r kOk TWk �gZIdf] dfkIN!0�8�"O8lK�[l[sN dfabae{H� ��*,��;'J�(2+&�'�)( ; <5;��d ÉZe |��Y � ��[/��[E��[/��[E��[i|\}�[\|g|/[i|i� . N��; ! (��C! NT;'I:#&J&+

� </����05[sNO�gP/[^k�{$a�[gkINv[^k�{�|i�^�^�l8+ �v| -t��8s4YM\rOSUTWk �v{C` (�#f\$H�N/#&<MK #&J��M.T6 �)( N�6 ( OE� ��;'J3#&</J&(2<�� # #,6"6 ( N �)(2+&*�!$#&JB�>.T<�O �)( ;'<�#&<)O<g;'J�N/#&J ��2-V Z�8v��dfXUZ�8/AvM\]�8l0YdgkOP/TbP/R�Z MQk

�5k TWnG8;��Q��¨|i�f~O|i�E{�|i�^���s�^�^�l8+ �o� - @t8943TWRE�s]!dgk�{$?�< �"!$# �&JL#hD,.$#&< *,H ;�� <$.�%�0�#&J�+ *3;'</���'(2</(2<�� N J�(2%�# �Q��*,��;'J�+ ;�� ��*&#&J����'(2< J3#,68�'�)( '# %����'</(2�)./K # �-2-SU�G8��djXi8W{G25P¨X/SU[gk [g]`T�[lR�Z�p��vP/TW� ���Ci {�|\}��¨|i�g�^}^�E{9| � |\~O8

+ �^��-w:-8G43T �v[^k�dfkIN�2�8�@t8>a�MQk P¨X/S�dv{ ���'+3+3( '#,6 H NT� JL�E626 #,6�#,626 ( N �)( * *,.�J '# �Q�E*,��;'J&("<�����6>SU[lR^8�<>rOSU[lRES/�O_vXkj �^�s{ a�MQRQXUrOSUMu=Y[fXUM\PTWk�.1[^]`_ rvXUMQS�AvRQTbMQk R\M ��BC��{�As_OSUTWk �gMQS/� ��MESUa�df�O{ :1MQSUaWTWk�{�|i�^�g�v{9|\z^���I|i�g�v8

Page 26: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

È ÷ ������������� �

+ �g~ -�0�8j43r VOk MQS�dgk N�+u8j43rOV k MESi{Y\^!$#�` ./0,<>#&J�I 4 4�J&.�<�*�!/#&J�G$I JP; �'JL�'% % #&J�+ � .�( K # �'< K �>.T< *,�)( ;'<���# ��#&J3#&< *�# � pOM\VOSUr dfS/�|i�s{�|i�^�g�v8

+ �o� -t.58�<>abNOMQSUP/Z diy dfkIN´+�8�698�:�SUMQksXi{+� ��*,��; J&(����'�)();'< ;��\68�'J � #\("<$�X# � #&J&+�;'< +&;'% # '#L*,��;'JB�'< KGNT�'JL�'626 #,6 *�; % N .��X#&J�+��6�SU[g�RQM\MiNOTWkO�^PL[gc>=5M\rOS�dfa�{ 6�dfS�dfaWabMQa�dgk NOAvRQTbMQksXUT�� Ru.1[^]`_OrOX�dfXUTW[gk P �´�¨|i�g�o�g�E{�|\~^�f�U|\~^zv8

+ �^��-Ga>8O<;rOabMESi{�? 0,+Q#&J �'�)( ; <>#&+ K # �"!$#L;'J3#&%\�'�X#WD,./;�K��'% � #&J&%\�'�)( �'< ;B�'6 (2(2+ D,.$# ��KG</.�% #&JL; +�N J�(2%�;'+�+ N/#3*,��� <$�)( 0,.�+ ��.1[^]`]�82-Rid^N�8 AvR\Te8�69MQX/SU[^_G[ga�8K��{�dgN�dfk k [gP`|j�g�^���l�^���¨|j�f�^zo�E{�|\}^���I|i}^��@:a�M\[gk Z dfS�NOT;<>r aWMQSUT �Y_GMQS�d �Y]`kOT�ds{�AsMQSi8�,E{�no[^ae8�,/,E{?�M\r VOk MQSi{/a�MQTb_Oh\TW�O{�|i�v|i�s{9|�v�l8

+ �o� -Ga>89<>r aWMQSi{�\^!/#3; J3#&%�� ��� *,(2JL*3�5K'( �(2+&;'JL#&+�<$.�% #&JP;'J�.T% �1=5[jnvTe8�.1[^]`]�8�25RidgN�8�AvR\Te8�69MQX/SU[^_G[ga�8 �I{>d^N�dfk kO[^P`|j�j~l���o~oz�¨|i�^�f}o�E{��f}��o~ozv8

+ �^z�-w6989NOM!p MQSU]!djXi{J?�#&.��JL#&+ K�# ��#&J�%��'� �1no[^ae8�,/, � 4 ; J&JL#&+�NT; <�K��'< *�# �L698�?9dfk k MES/�)dfkIN�.58�05M\kOS/�®�eMiNvT�XU[gSUP��E{�@3dgrvXUZ TWMQS/��5TWababdjSUP\{G6�dfSUTWP\{�|iz^�f~O8

+ �^��- ��8/@Y[^k R�ZIdjSU[jn�{F?�< �"!$# 9:#,68K ;���*3;'%\0,(2<��'��; J&H �'< �E6 H'+3(2+ �v,ÒhQn�8g2-�^dgN�8f=3dfr � A$A/Av+ AvMQSi8^��djXi8f���¨|\�g~g~l�E{s���l~oz�@v<;kO�^aWTWP/ZX/S�dfk P/ae8OTbk�25]`MQSi8 ��dfXUZ�8�As[lR^8 ?�S�dgkOP/a�8����^�������¨|\�^�o�g�E{�|�l~o�s8

+ ~o}�-f@�8 :58�@Y[^P¨XUTWk�{�� #&I �Q��*,��;'J�+ ;��B��#&J�%��'��</.�%�0�#&J&+ ����djXUZ�8I.1[^]`_�8 �3Q��¨|\�^�^�^�E{G�g�^���s�^�^�s8+ ~O| -W"v8G.58 03dgaWa��vV rvS/XU[^k�dfkIN�0�8 :�SUTWabaWZ dfS/Xi{d\�I�;�<>#&I �,��*,��;'J�+ ;��B��#&J&%\�'��</.T%\0�#&J&+ ����dfXUZ�8G.1[g]`_�8 � �)�¨|i�^�^�g�E{�|i}g��� |^|j�l8.1[fS/SUTW�^M\k NOr ]�{ ( 0,( KW���)�¨|i�^�g�^�E{�|i�gzv8

+ ~l� -�0�8�0YdgP/P/M^{d #&I:#&(2+�K #&+ A�<��E68; ��; <$+BK #&J���(�#&%\�'<$</+&*�!$#&<N%$#&J�%G.��).�<�� ��g.�JBK'( # A�J��)(2<$+&*�!$#&< . =$� =�FG=:-�*�!�% ( K'�)+&*�!$#&<F ;'<��'J�.$#&<�� �C#&�����&.�<)O �)( ; <>#&< (2< � #&I7(2+3+,#&< #,626 ( N$�)(2+&*�!$#&< �Ag�'626 #&<��Y=3dgR�ZOSi8Z@YM\P/M\aWae8�K TWP/P/M\k�8I@��[fX/XUTbkO�^M\k&, Q � �¨|i�g�^�^�E{�g�g���v�g�^�s8

+ ~o��-�0�8�,ÒP/ZOTWZIdfX�ds{�?Y8>05[gSUTWM�dgk N�?Y8�AsZ TW]`TWh\r�{ A�JP*�!�(2�X#L*,�).�J3# �Q;'JB�"!$# ABI&RM! (��C!T6 HBNT�'JL�E6"6 #,6�*�; % N .��X#&J�YpOr'R¨T�XUP/r�AvR\Te8?�M\REZ�8�"v8 � ���¨|\�^�g�o�E{G���I|Q~O8

+ ~^~ -W"v8 ��+u8�"^[gaW�l{ �ZD,.$� �)( ;'<$+O#&� �'J�(��#&���#&+O�'6 � �#Q0,J&(ED,.$#&+B+3.TJ .�< *3;'J�N + 97<$( � a�j <;kOP/M\TW�^kOM\]`M\klX!��djXUZ �M\]!djXUT �lrOM ��� �¨|i�^�g�^�E{|�I|g|j�s8

+ ~l� -�Kÿ8;@BM\aWaWMQSi{K�^��*,��;'J�+O;�� ��#&J�%��'� </.T%\0�#&J&+ �'<�K�68�'J � # N J�(2%�#&+B;��B�"!$# �,;'J&%����v� É � |��-��djXUZ�8;.1[g]`_�8KQ ���¨|i�gz^�^�E{�g�v|�s�o�f�v8�2-abP/[w_IdfS/XL,/,�{ _vSUM\_OSUTWklXi{��YkOTbn^MQSUP/T�X��dfX-03df]wV rvSU�O{/AsM\_OXi8 �^�l{�|\�^�o����dingdgTWabdgVOabMBc�SU[^] XUZ MudfrOXUZO[gSE�E8

+ ~o��-�Kÿ8�@BM\aWaWMQSi{�� #&I 4�./6"6 #&<�N J&("% #&+ ����dfXUZ�8G.1[g]`_�8��3Q��¨|\�^�o�g�E{9|j�f�^���I|j�j~O|g8+ ~l� -�4�8�<L8�@BksrvXUZ�{ \^!$#\� J&��;��\*3;'%�N .��X#&J N JP; �'JL�'% %G(2<����_%$;'6 .T%�# E)G -�#&%G(2</.T%�#&J&()*��'67�E6 ��;'J�(2�"!�%G+u���jkIN�MiN�8 �E{�25N NOTWP/[gkO�K�M\P/aWMQ�l{ ��M\k aW[!6�djSU��{I.q2�{�|i�gzv|g8

+ ~oz�-�4�8 <�83@BkvrvXUZ�dgkIN a;8O?�S�dfV V�6�djS�NO[v{�A�< �E6 H'+3(2+ ;�� �\+�("% N�6 #.�Q��*,��;'J�(����'�)( ; < �E6 ��; J&(2�"! % ��?qZOM\[gSi8I.1[^]`_�8 AsR\Te8F���¨|i�^�g�^�E{�^�s|�s�g~^zv8

+ ~o��-t��83@3S�dgT�XUR�Z TW�G{]\^!��#Q;'J&( # K #&+ < ;'%�0,JL#&+ ��?�[^]`Mu�l{Y@3dgrvXUZ TWMQS/� �YTWaWabdfSUP\{I6�dfSUTWP\{�|\�o�g�s8+ �g}�-wp18$a�dfkINsS/�l{�� ;'�X# +�.�J 68�BK �#Q*3;'% NT; +�("�)( ;'< K'. < ;'%�0,JL#3����� � |t��< �sX/S�dgT�XE�E{I.58I+u8O25RidgN�8 AvR\Te8 6�dfSUTWP����´�¨|izgz^}^�E{9|i�gzv8+ �s| -�4�8I0�8 a�M\ZO]`MQSi{d\�#&+��)+ �Q;'J�N J�(2%\�E6 (2�)H 0,H �"!$#\*3;'<�'#&J�+,# ;=� ��#&J�%��'�0c + �"!/#3; J3#&% �9:1r aWa�2-]`MQSi8I��dfXUZ�8>Av[lR^8 ���)�¨|i�^�^�g�E{�^�^�P�s�g~^}v8

+ �^� -w2u8�@t8/a�M\kOP¨X/S�dwdgk Nm0�8lKÕ8/a�M\kOP¨X/S�dv{C"gSi8��eMiNOT�XU[gSUP��E{�\^!$# K # '#,68;&N %�#&<$��;�� �"!$# </.�%�0�#&J79�#,68K\+3(�# '# �>a�M\RQXUrvSUM3=Y[fXUM\P�TWk��djXUZ M\]!djXUTWR\P ��B�B3Q�{>Av_vSUTWk �^MES/� �;MQSUabdg�v{I:�MQSUaWTbk�{�|\�^�g�v8

+ �g��-w2u8)@t8 a�M\k P¨X/S�ds{ 0�8vKÿ8 a�M\k P¨X/S�ds{�"gSi8W{G��8�AG8G��dfkIdgP/P/Mg{Gdgk N"O8G��8I69[gaWa�djS�N�{]\^!$#��Q�E*,��;'J&( �C� �)( ;'< ;�� �"!$# </(2<$�"!L��#&J�%��'�</.�%�0�#&J ����djXUZ�8I.1[^]`_�8+���´�¨|i�^�g�o�E{I�v|i���s�f~o�v8

+ �f~ -w2u8 @t8:a�M\kOP¨X/S�d�dgkINÎ��8�A�8���dgkIdfP/P/M^{ � ��*,��;'J�(2<�� 0,H #,6 #L*,�)JP;'<$( *5%��'()6 �Y6�SU[lR^8L<>rOSU[lRQS/�v_OX j z^�s{ a�M\RQXUrOSUM´=5[gXUM\P�TWk.1[g]`_ rvXUMQS�AvR\TWM\kOR\M QH�3Q�{>As_OSUTWk �gMQS/� ��MESUa�df�O{ :1MQSUaWTWk�{�|i�^�g}v{I�o�^�P�s�^�s|^8

+ �^� -�0�8sKÕ8 a�M\k P¨X/S�ds{H"gSi8W{H� ��*,��;'J�(2<��B(2<$�X# � #&J�+ I ("�"!M#,6"6 ( N �)( * *,.�J '#&+ ��25kOkIdgaWP�[fc���dfXUZ MQ]!dfXUTWR\Pu���^� � � ���¨|i�^z^�^�E{I�g~o���s�^�g�v8+ �g��-w<�8^a�rORidgP\{J\^!��#Q;'J�(�# K #&+ �Q;'< *,�)( ;'<$+ <$.�%��#&J�(ED,.$#&+\+3(2%�N�6 #&% #&</��N/#&J�( ;�K'(ED,.$#&+ ��25]`MQSi8�"v8���djXUZ�8 ���¨|iz^�gzo�E{�|izf~5�v�f�^����fz^���s�o�l|^8

+ �^� -W"v81ngdgk�NOMMa�rOk M�dfkIN�<L8;K�djX/XUM\ae{_?�< �"!$#<$.�% #&J�( *3�E6�+,;'6 .T�)();'< ;��� K'( ��#&J3#&</�)( �E6 MXK'( ��#&J3#&<�*�# #!D,./�'�)( ; < � J&(2+3(2<�� ("<� <��E6 H �)( * </.�%�0�#&J �"!$#L;'J�H�����dfXUZ�8�.1[g]`_�8 � ���¨|\�^�^�^�E{I~v|j���o~l�l|^8

+ �gz�-w698Ta>8v��[^klXU�^[g]`MQS/�l{�-EN/#3#LK'(2<����"!/#BI�;E6268� JLK��'< K #,626 ( N �)( * *,.�J'# %�#&�"!/;�K'+ ;��+�Q��*,��; J&(����'�)();'<�����djXUZ�8O.1[g]`_�8HQH�õ�¨|i�gzo�g�E{�j~o���v�g�f~O8

+ �g��-w698�a;8L��[gklXU�^[^]`MES/�l{ A�< �;�J\ #� '�X#&<$+3( ; < ;��O�"!$# #,626 ( N �)( *5*,.�J '#M% #&�"!/;�K ;�� �Q��*,��;'J�(����'�)( ;'<3�u6>Z�8q4�8qNvTbP/P/MES/X�dfXUTW[^k�{��djXUZ M\]!djXUTWR\P\{C�5k TWnoMQSUP/T�X��m[fc;.qdgaWT�cx[fSUk Tbd�dfX�a�[gP52-k �gM\aWM\P\{�|\�^�^�s8

+ �^}�-w698$a;8I��[^klXU�^[g]`MQS/�l{HA +3.�J '#&H ;�� %�;�K #&J�<M(2<$�X# � #&J��Q��*,��; J&(����'�)();'< �'6 ��;'J&("�"!�%G+ ��.�K ,�DYr dfS/XUMQSUa��=G��¨|\�^�g~o�E{G�g�o���l�^�g�v8+ �v| -w698$a;8I��[^klXU�^[g]`MQS/�l{v_GMQSUP/[^k dga�R\[^]`]�r kOTbR\dfXUTW[^kmVs��MQ��]!dfTbae{v=Y[jnoMQ]wVGMQS-�f�v{�|i�^�^�s8+ �o� -wp18l��[gS�dfTbk�{�4 ;'.�JQ0�#&+ #,626 ( N$�)(ED,.$#&+�#&���X#&+��)+�K #�N$J&(2%\�E6 (2���# �O6>Z�8g4w8^XUZOM\P/TWP\{:�YkOTbn^MQSUP/T�X��MY.1abdfrINOME�Ò:�MQSUkIdjS�Nv�Xa��l[gkt,E{opOS�dgkOR\M^{|\�^�g}v8G2-n^dfTWa�dfV aWMBVl�!c�XU_�c�SU[^] ��!�".#&%�'�()%�*+#,��(.-=/�0�1�230�4�/5"+*+)�!�%�7�*�!3%�"5')/�#+2�6:,�6�,�/#%$'&/6�>P<�<.#,!�*�(;#=?´8

+ �^��-wp18 ��[fS�dgTWk�{H` (2+3�)J�( 0,.��X#LK N$J&(2%\�E6 (2�)H�N JL;��("<�� � <�KG�"!$# N J&("%��E6 (2�)H\;��Y��� Y)(PY�* � |j��+g����2YNvn^dfk R\MQP1TWk�.�S/�v_OXU[gaW[^�g� �`<;rvSU[g�RES/�O_vX j �^}v{�Av_vSUTbkO�^MQS/� �;MQSUabdg�v{�|\�^�^}s{�|g|i}�� |j�f�v8

Page 27: Factorization of the Tenth and Eleventh Fermat Numbers …maths-people.anu.edu.au/~brent/pd/rpb161tr.pdf · Factorization of the Tenth and Eleventh Fermat Numbers Richard P. Brent

��������� ����� ���� �� ����������� �* ��� ������� È î

+ �g~ -t��892�89��[gS/SUTWP/[^k®dfkIN�"O8�:�SUTWabaWZ dfS/Xi{ A % #&�"!T;�K ;�� �Q��*,��; J&(����'�)();'< �'< KM�"!/# �Q��*,��;'J�(����'�)( ; < ;�� S � �q��dfXUZ�8>.1[g]`_�8 � ��¨|\�o�g�^�E{�|\z^���v�g}^�s8+ �o� -t��8v6�djXUMQSUP/[^kmdgk N a;8TAlXU[lRE�s]`MQ�lMESi{ ?�< �"!/# </.�%�0�#&J ;���<�;'</+,*3�E68� J %G.T6 �)( N�6 ()*�� �)( ;'<$+�<�#3*�#&+3+&�'J�H���; # �E6 ./�'�X# NT;'6 H'< ;'%G( M

�'6 +�>Av,¨2Y�T"O8O[^k´.1[g]`_ rOXUTWkO� � �¨|\�o�g�^�E{G�g}��s�g�v8+ �^��-W"v8q��8�69[gababdjS�N�{_\^!$#L;'JL#&%G+ (2< �Q�E*,��;'J&( �C� �)( ;'< �'< K N J&("%��E6 (2�)H��X#&+3�)(2<����56�SU[lR^8�.qdf]wVvSUT�Nv�^M�6;ZOTbaW[gP\8 As[lR^8�G3���¨|i�^�f~o�E{�g�s|�v�^�fzv8

+ �o� -W"v8G��8I69[^aWabdfS�N�{�A ��;'</�X#B4 �'J&68;\% #&�"!/;�K �Q;'J.�Q�E*,��;'J&( �C� �)( ;'<���:1,Ò?���B��¨|i�^�^�g�E{G�^�s| �l�^�g~v8+ �^z�-t.58I69[^]`MQS�dfk R\Mg{]\^!$# <$.�%�0�#&J�9�#,68K�+3(�# '# ��6�SU[lR\M\MiNvTbkO�^P5[gc�As�v]`_G[gP/T�d�TWk�25_O_ aWTWMiN���djXUZ M\]!djXUTbRQP QY��{I2-]`MQSi8 ��dfXUZ�8

As[lR^8W{I6�SU[jnsT�NvM\k RQM^{I+-Z [sNvM3,ÒP/abdgk N�{�|i�g�g~O{ ~o�^���l~^z^}s8+ �^��-t.58f69[g]`MQS�dgkOR\M^{/"O8iKÕ8'As]`TWXUZ�dfkINB+�8f?�r aWMQSi{�A N ( N/#,6 (2<># �'JP*�!�(2�X#L*,�).�J3#+�Q;'JH�Q��*,��;'J�(2<�� 68�'J � # ("<$�X# � #&J&+:I ("�"! �"!$#&D,./��K JL�'�)( *

+3(�# '# �E6 ��;'J�(2�"!�% �^Av,¨2Y�T"O8O[^k´.1[g]`_ rOXUTWkO� �-G��¨|\�^zgzo�E{G�gzo�P�l~o}g�v8+ �g}�-�+u8Ea;8o+-TWnoM\P¨Xi{o2�8�AsZIdf]`TWSi{odgk N\a;8o2YNOaWM\]!dfk�{�A`% #&�"!/;�K �,;'J�;�0,���'(2</(2<��GK (��'(2���E6$+3(��'< �'�).�J3#&+ �'<�K�N ./0Q6 ( *PM O�#&H\*,J�H&N$��;'+3H'+M

�X#&% + ��.1[^]`]�8O2-.�� � �´�¨|i�^�gz^�E{9|j�f}��I|i�g�s8+ �s| -�+u8 ��8I+-[^VOTbkOP/[^k�{���#&J&+,#&<$<�#\�'<�K ��#&J�%�� ��<$.�%�0�#&J�+ ��6�SU[lR^8I25]`MQSi8O��djXUZ�8>As[lR^8^B��¨|i�^�f~o�E{�zg~o���szf~o�s8+ �^� -W"v8 a>8�AsM\a�c�SUTbNO�gM^{�� �E*,��;'J&+ ;��B��#&J�%�� ��<$.�%�0�#&J�+ ����?92-. G��¨|i�o�f�o�E{G�^�f~��v�g�^�s8+ �g��-W"v8 a>8�AsM\a�c�SUTbNO�gM�dfkINm2�8O0YrOS/yLT�XUh^{H��#&J�%��'��<$.�%�0�#&J�+ �'<�KG� #&J�+,#&<$<># <$.�%�0�#&J�+����djXUZ�8I.1[^]`_�8;�����¨|i�^�f~l�E{9|Q~o���I|\~^zv8+ �f~ -�4�87AvZ dgkO�vP\{ 4768� +�+�</.�%�0�#&J � �M�"!$#L;'J�H ;��B�Q��*,��;'J�(����'�)( ; <�� �'<�K � #&<>#&JP���L6�SU[lR^8�As�v]`_�8�6>rOSUMm��dfXUZ�8 � ��{12-]`MQSUTWRidgk��djXUZ�8 Av[lR^8W{I6>SU[jnsTbNOM\kOR\M^{O+�8v,E8W{�|i�^�s|^{I~O|j�P�l~g~o}v8

+ �^� -Ga>8v2�8/AsZ M\_O_mdfkIN A�8v698/a�aW[i�ON�{<?�JPK #&J3#LKB*,H�*Q6 # 6 #&<��'�"!�+�(2<M�GJP�'<�K�; % N/#&J&% .T��� �)( ;'<���?�S�dgk P\8O2-]`MQSi8v��dfXUZ�8$Av[lR^8 � � ��¨|\�^�g�o�E{G�f~o}��s�o�g�s8

+ �g��-w698>KÕ8�AvZO[gSi{ A�6 ��;'J�(2�"!�% + �Q; J#D,./�'<$�).�% *3;'% N$.T��� �)( ;'<�G K'(2+&*,J3#&�X#M68; ���'J&("�"!�%G+O�'<�KL�Q��*,��; J&(2<����B6�SU[lR^81�o�jXUZÎ25k ksr dgaAl�v]`_G[^P/TWr ]�[^k�p [gr k N dfXUTW[gk P�[gc�.1[g]`_ rvXUMQS�AvR\TWM\kOR\M^{O,¨<;<�<�6>SUMQP/P\{�a�[gP-25abdg]`T�XU[^P\{I.qdgaWT�cx[fSUk Tbdv{�|\�^�f~O{9|i�f~O8

+ �^� -W"v8�0�8�AsTWabn^MQSU]!dgk�{J\^!$#B�'J�(2�"!�% #&�)()* ;�� #,626 ( N �)( *B*,.TJ'#&+ �]@5S�d^NOr dfXUM!?�M �vXUPuTWk���djXUZ M\]!djXUTWR\P ��� ��{ As_OSUTWk �gMQS/� ��MESUa�df�O{=5MQy ?1[fSU�G{�|\�^zg�v8

+ �gz�-�+u8v4w8 AvTWaWnoMQSU]!dgk�{�\^!$# % ./6 �)( N�6 # NT;'6 H'< ;'%G( �'6^D,./��K'JP�'�)( * +3(�# '# ����djXUZ�8I.1[^]`_�8�QY�)�¨|i�^z^�^�E{I�o�g���s�g�^�v8+ �g��-�+u8v4w8$AvTWaWnoMQSU]!dfk�dfkINOAG8$A�8sK�df�^P¨X�d X�{�"gSi8W{�A N JL��*,�)()*��'6��'<��'6 H'+3(2+ ;�� �"!$#G#,626 ( N �)( * *,.TJ'#��,��*,��;'J�(2<��B�E6 ��;'J�(2�"!�% ����dfXUZ�8.1[g]`_�8����´�¨|\�^�g�o�E{I~g~l�P�l~o�^�s8

+ z^}�-�0�8$AvrO�ldg]!dv{���<P�Q;'J�%��E6�N J3#,6 ("%G(2< �'J&HGJ3#XNT;'J��q��z^�E{ _GMQSUP/[gkIdfa�R\[g]`]wr kOTWRidfXUTW[^k�{3�YRQXU[^VGMES�|\�^z^�s8+ zv| -w2u8���8��;MQSUP/ZOTb�G{&\^!$# �'+3H'%�N ��;'�)( * K'(2+3�)J�( 0,.��)( ;'< ;��K�Q�E*,��;'J&( �C� �)( ;'<$+B;=�G<��'�).�JL�'6 </.T%\0�#&J&+\("<$��;GN J�(2% # K'( �("+,; J&+ �>43[g�vae82-�^dgN�8^=3dgrO��A/A$Av+ � � ���¨|i�gz^�o�E{I�g�g���s�^�^� @G<>k �gabTWP/Z!X/S�dgkOP/a�8sTWk Av[jnsTWMQXL��djXUZ�8l4Y[^�sae8��)Q��¨|i�gzo�g�E{I�g���l�v|^8�+ <�S/S�dfX�d��;[gkXUZOMYSUTW�^ZlX-P/TbNOMB[gc9M �lrIdfXUTW[gk)���^��� 3�� $�� P/ZO[^r abN�VGM�� 3�� $ & U � dgk N�� � P/Z [^rOabN�VGM���U � 8 -

+ zo� -�0�8 .58 K TWaWaWT�df]`P\{ � ;'I�I��'+ S � �,��*,��;'J3#LK��&�9��djXUZ�8G.1[^]`_�8����´�¨|i�^�g�o�E{ ~o�^���l~o�f~O8+ z^��-�0�8s.58lK TWaWaWTbdg]`PqdfkINA"O8TAG8:"^r N N�{ -�;'%�# �E6 ��;'J�(2�"!�% ++�Q; J�N J�(2%�#��X#&+3�)(2<��\.�+�("<�� � #&<>#&JP�E6 (��C#LK�'�#3!�%�#&J;�&.�<�*,�)();'<$+ �I��dfXUZ�8.1[g]`_�8F���)�¨|\�o�f�o�E{Gzg�o�P�sz^zg�v8

+ zg~ -\AG8GK TWk [g�gS�d^N�{.� �E6 ./�'�)(2<��\NT;'6 H'< ;'%G( �'6 + .�+3(2<��OJL� �)( ;'<��'6�� . '(26 ( �'J�H �&.�<�*,�)();'<$+ �>,¨:q�#?�M\R�Z k TWRidfa>43TWP/R\aW[^P/rOSUMt:1r aWaWMQXUTWk�����¨|i�o�f}o�E{�|^|i�g��� |^|i�^�s8

��� ����� D��vF��IJ�������J��OC! >H�A � FIHsD � F#"%$'&(��)*��+,$>@ � CQDGFIH -.��H �0/uHsD�� � �GH -�12���435�vFGC6��D�"7$ � H ��A��OFGFIH'$>@ ��8:9�;<9<9 $@ � CQDGFIH -.��H�KM�%\�'()6^��K�K'JL#&+�+ ��(�"+)<=�7/, !�* )$#=*5'+*$#=65./*$#�* *