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A reconstruction of Nevilles Farey series of order 1025(1950)

Denis Roegel

To cite this version:Denis Roegel. A reconstruction of Nevilles Farey series of order 1025 (1950). [Research Report] 2011.
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A reconstruction of

Nevilles Farey series of order 1025

(1950)

Denis Roegel

22 October 2011

This document is part of the LOCOMAT project:http://locomat.loria.fr

1 Introduction

Eric Harold Neville (18891961) was born in London and entered Trinity College in 1907.There he became acquainted with Bertrand Russell and G. H. Hardy.

In 1914, visiting India, he was instrumental in persuading Srinivasa Ramanujan to goback with him to England.

Neville worked on a variety of topics, and in particular on Jacobian elliptic functions.He chaired the Mathematical Tables Committee of the British Association for the Ad-vancement of Science (BAAS) from 1931 to 1947. He contributed two sets of tables,one on the Farey series of order 1025 (1950) [22] reconstructed here, and another onrectangular-polar conversion (1956) [23].

2 Farey series

2.1 Introduction

The Farey sequence Fn of order n is the increasing sequence of fractionsp

qwith 0 p

q n. If three consecutive terms ai1bi1

, aibi

, ai+1bi+1

, of the sequence are given, the middle onecan be obtained by the mediant property:

ai1 + ai+1bi1 + bi+1

=ai

bi

This property was described by John Farey (17661826) in 1816 [9].An algorithm for the construction of this sequence had already been given by Haros

in 1802 [14], and the sequence is nowadays called the Haros-Farey sequence [8].Farey sequences are related to the construction of the Stern-Brocot tree of the ratio-

nals [6, 27]. A Farey sequence is sometimes called a Brocot table [20].1 Such a table isuseful to find rational approximations of a number.

2.2 The Farey series of order 1025

Nevilles main table [22] gives the fractions 0 < pq

1

1in increasing order and with

q 1025. Each fraction is given as a triplet (p, q, q p), which allows to stop the tableat 1

2. The fractions following 1

2can be read backwards from the end of the table.

Each page contains 400 fractions. The last page contains 282 fractions. This tablecovers exactly 400 pages, and contains therefore 159882 fractions. The total number offractions, when the two parts of the table are separated, is 1598812+1 = 319765, sincethe last fraction counts only for one. Each page also gives the value at the end of theprevious page and the value at the beginning of the following page.

1For an introduction to the construction of the Stern-Brocot tree and Farey sequences, see [15, 16,12, 13].

i

Apart from its use in finding rational approximations, this table also gives solutionsof the Diophantine equation bx ay = 1 for 1 a b 1025, with one fraction being a

b

and the next one in the series xy. For instance, the solution of 657x 134y = 1 is found

in the second column of page 164, right after the entry 134657

, and is x = 103, y = 505.Using this table is however not that convenient, because it appears difficult to locate

a given ratio in the main table. In his introduction, Neville provides an additional table(not reproduced here), giving the location of specific fractions, but one reviewer regrettedthat decimal equivalents were not given at least for the last value on each page.


The main table is followed by three smaller tables. The first one gives the Farey seriesof order 50 over two pages. For each fraction p

q, a triplet (p, q, q p) is given, as well as

five-digit decimal equivalents of pq

and qpq

.


The second appendix gives the Farey series of order 64 over one page. For each fractionp

q, a triplet (p, q, q p) is given.

2.5 The Farey integer-series of order 100

The third appendix gives the denominators of the fractions of the Farey series of order100.

2.6 Nevilles table of factors

This table gives the complete factorization of all integers from 1 to 1049.

3 Reconstruction

For the reconstruction, we have used Patracu and Patracus algorithm. They gave avery simple way for the computation of the next term of a sequence [24, 26]. If a

b< c

d< p

q

are three consecutive terms, we have a+pb+q

= cd

(mediant property) and there is a k such

that kc = a + p and kd = b + q. It is easy to show that k =

n+bd

where n is thegreatest possible denominator. Once k is computed, p and q follow. This algorithm iseasily adapted for a decreasing sequence.

ii

4 Acknowledgements

It is our pleasure to acknowledge the help of Scott Guthery who first drew our attentionto Nevilles table.

iii

References

The following list covers the most important references2 related to Nevilles table. Notall items of this list are mentioned in the text, and the sources which have not been seenare marked so. We have added notes about the contents of the articles in certain cases.

[1] Raymond Clare Archibald. Scarce mathematical tables. Mathematical Tables andother Aids to Computation, 1(2):6667, 1943. [On Brocots original table.]

[2] Raymond Clare Archibald. Scarce mathematical tables. Mathematical Tables andother Aids to Computation, 1(4):132, 1943. [On Brocots original table.]

[3] Raymond Clare Archibald. E. H. Neville: The Farey Series of Order 1025 (review).Mathematical Tables and other Aids to Computation, 5(35):135139, July 1951.

[4] Paul T. Bateman. Review of E. H. Neville, The Farey series of order 1025,displaying solutions of the diophantine equation bx ay = 1. Bulletin of theAmerican Mathematical Society, 57(4):325326, 1951.

[5] Thomas Arthur Alan Broadbent. Obituary: Eric Harold Neville. Journal of theLondon Mathematical Society, 37:479482, 1962.

[6] Achille Brocot. Calcul des rouages par approximation, nouvelle mthode. Revuechronomtrique, 3:186194, 1860.

[7] Achille Brocot. Calcul des rouages par approximation, nouvelle mthode. Paris,1862. [not seen]

[8] Cristian Cobeli and Alexandru Zaharescu. The Haros-Farey sequence at twohundred years: A survey. Acta Universitatis Apulensis, Mathematics-Informatics,5:138, 2003.

[9] John Farey. On a curious property of vulgar fractions. Philosophical Magazine,47:385386, 1816.

[10] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.London: Taylor and Francis, 1873. [Also published as part of the Report of the forty-thirdmeeting of the British Association for the advancement of science, London: John Murray, 1874.

2Note on the titles of the works: Original titles come with many idiosyncrasies and features (line

splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has thereforeseemed pointless to capitalize works according to conventions which not only have no relation with theoriginal work, but also do not restore the title entirely. In the following list of references, most titlewords (except in German) will therefore be left uncapitalized. The names of the authors have also beenhomogenized and initials expanded, as much as possible.

The reader should keep in mind that this list is not meant as a facsimile of the original works. Theoriginal style information could no doubt have been added as a note, but we have not done it here.

iv

A review by R. Radau was published in the Bulletin des sciences mathmatiques et

astronomiques, volume 11, 1876, pp. 727]

[11] James Whitbread Lee Glaisher. Table, mathematical. In Hugh Chisholm, editor,The Encyclopdia Britannica, 11th edition, volume 26, pages 325336. Cambridge,England: at the University Press, 1911.

[12] Scott B. Guthery. A motif of mathematics: history and application of the mediantand the Farey sequence. Createspace, 2010.

[13] Godfrey Harold Hardy and Edward Maitland Wright. An introduction to the theoryof numbers. Oxford: Clarendon Press, 1938.

[14] Charles Haros. Tables pour valuer une fraction ordinaire avec autant de dcimalesquon voudra ; et pour trouver la fraction ordinaire la plus simple, et qui approchesensiblement dune fraction dcimale. Journal de lcole Polytechnique, 4:364368,1802.

[15] Brian Hayes. On the teeth of wheels. American Scientist, 88(4):296300,July-August 2000. [reprinted in [16]]

[16] Brian Hayes. Group theory in the bedroom, and other mathematical diversions.New York: Hill and Wang, 2008. [This collection of articles contains a reprint of [15]. [17] isa French translation of this book.]

[17] Brian Hayes. Lhorloge de lternit. Paris: Vuibert, 2010. [Translation of [16].]

[18] Walter James Langford, Thomas Arthur Alan Broadbent, and Reuben LouisGoodstein. Obituary: Professor Eric Harold Neville, M.A., B.Sc. The MathematicalGazette, 48(364):131145, May 1964.

[19] Roger Mansuy. Achille Brocot, mathmaticien ses heures, 2008. [available onhttp://www.math.ens.fr/culturemath/histoire%20des%20maths]

[20] Henry Edward Merritt. Gear trains, including a Brocot table of decimal equivalentsand a table of factors of all useful numbers up to 200,000. London: Sir IsaacPitman & Sons, Ltd., 1947. [The table of useful numbers was reconstructed in [25].]

[21] Eric Harold Neville. The structure of Farey series. Proceedings of the LondonMathematical Society, 51 (series 2):132144, 1949.

[22] Eric Harold Neville. The Farey series of order 1025, displaying solutions of theDiophantine equation bx ay = 1, volume 1 of Royal Society Mathematical Tables.Cambridge: University Press, 1950.

[23] Eric Harold Neville. Rectangular-polar conversion tables, volume 2 of Royal SocietyMathematical Tables. Cambridge: University Press, 1956. [not seen]

v

[24] Corina E. Patracu and Mihai Patracu. Computing order statistics in the Fareysequence. In Proceedings of the 6th Algorithmic Number Theory Symposium(ANTS 2004), volume 3076 of Lecture Notes in Computer Science, pages 358366.Springer, 2004.

[25] Denis Roegel. A reconstruction of Merritts table of useful numbers (1947).Technical report, LORIA, 2011. [This is a reconstruction of the table in [20].]

[26] Norman Routledge. Computing Farey series. The Mathematical Gazette,92(523):5562, March 2008.

[27] Moritz Abraham Stern. ber eine zahlentheoretische Funktion. Journal fr diereine und angewandte Mathematik, 55(3):193220, 1858.

vi

Nevilles Farey series of order 1025 (reconstruction, D. Roegel, 2011)

appendices

I

The Farey Series of Order with Decimal Equivalents

II

The Farey Series of Order

III

The Farey Integer-Series of Order

Nevilles Farey integer-series of order 100 (reconstruction, D. Roegel, 2011)

Nevilles auxiliary factor table (reconstruction, D. Roegel, 2011)

I. The Numbers 1549 factorized Completely

1 3 2 5 . 7 3 2

. 11 2. 13 . . 4 17 .2 19

2. . . 23 3. 2 . 3 2. 29

.. 31 5 . . . 2.2 37 . .

3. 41 .. 43 2. 2. . 47 4. 2

.2 . 2. 53 .3 . 3. . . 59

2.. 61 . 2. 6 . .. 67 2. .

.. 71 3.2 73 . .2 2. . .. 79

4. 4 . 83 2.. . . . 3. 89

.2. . 2. . . . 5. 97 .2 2.

2.2 101 .. 103 3. .. . 107 2.3 109

.. . 4. 113 .. . 2. 2. . .

3.. 2 . . 2. 3 .2. 127 7 .

.. 131 2.. . . 3. 3. 137 .. 139

2.. . . . 4.2 . . .2 2. 149

..2 151 3. 2. .. . 2.. 157 . .

5. . .4 163 2. .. . 167 3.. 2

.. 2. 2. 173 .. 2. 4. . . 179

2.2. 181 .. . 3. . .. . 2. 3.

.. 191 6. 193 . .. 2.2 197 .2. 199

3.2 . . . 2.. . . 2. 4. .

... 211 2. . . . 3.3 . . .

2.. . .. 223 5. 2.2 . 227 2.. 229

.. .. 3. 233 .2. . 2. . .. 239

4.. 241 .2 5 2. .2 .. . 3. .

.3 251 2.2. . . .. 8 257 .. .

2.. 2. . 263 3.. . .. . 2. 269

.3. 271 4. .. . 2. 2.. 277 . 2.

3.. 281 .. 283 2. .. .. . 5.2 2

.. . 2. 293 ..2 . 3. 3. . .

2..2 . . . 4. . .2. 307 2.. .

.. 311 3.. 313 . 2.. 2. 317 .. .

6. . .. . 2.4 2. . . 3. .

... 331 2. 2. . . 4.. 337 .2 .

2.. . .2. 3 3. .. . 347 2.. 349

.2. 3. 5. 353 .. . 2. .. . 359

3.2. 2 . .2 2.. . .. 367 4. 2.

.. . 2.. 373 .. .3 3. . .3. 379

2.. . . 383 7. .. . 2. 2. 389

... . 3.2 . . . 2.2. 397 . ..

4.2 401 .. . 2. 4. .. . 3.. 409

.. . 2. . .2. . 5. . .. 419

2... 421 . 2. 3. 2. .. . 2. ..

.. 431 4.3 433 .. .. 2. . .. 439

3.. 2.2 .. 443 2.. . . . 6. 449

.2.2 . 2. . . .. 3.. 457 . 3.

2.. 461 ... 463 4. .. . 467 2.2. .

.. . 3. . .. 2. 2.. 2. . 479

5.. . . .. 2.2 . .5 487 3. .

..2 491 2.. . .. 2.. 4. . .. 499

2.3 . . 503 3.2. . .. .2 2. 509

... . 9 3. . . 2.. . .. .

3.. 521 .2. 523 2. .2. . . 4.. 2

.. 2. 2.. . .. . 3. . . 2.

2.3. 541 . . 5. . ... 547 2. 2.

Nevilles auxiliary factor table (reconstruction, D. Roegel, 2011)

I. The Numbers 5001049 factorized Completely

2.3 . . 503 3.2. . .. .2 2. 509

... . 9 3. . . 2.. . .. .

3.. 521 .2. 523 2. .2. . . 4.. 2

.. 2. 2.. . .. . 3. . . 2.

2.3. 541 . . 5. . ... 547 2. 2.

.2. . 3.. . . .. 2. 557 .2. .

4.. .. . 563 2.. . . 4. 3. 569

... 571 2.. . .. 2. 6.2 577 .2 .

2.. . .. . 3. 2.. . 587 2..2 .

.. . 4. 593 .3. .. 2. . .. 599

3..2 601 .. 2. 2. .2 .. 607 5. ..

.. . 2.2. 613 . .. 3.. 617 .. 619

2.. 3. . . 4.. 4 . .. 2. .

.2.. 631 3. . . . 2.. 2. .. 2.

7. 641 .. 643 2.. .. .. 647 3.4 .

.2. .. 2. 653 .. . 4. 2. .. 659

2... 661 . .. 3. .. .2. . 2. .

.. . 5.. 673 . 3.2 2.2 677 .. .

3.. . .. 683 2.2. . .3 . 4. .

... 691 2. 2.. . . 3.. . . .

2.2. 701 .3. . 6. .. . . 2.. 709

.. 2. 3. . ... .. 2. . . 719

4.2. . .2 . 2. 2. ..2 727 3.. 6

.. . 2.. 733 . ..2 5. . .2. 739

2.. .. .. 743 3.. . . 2. 2.. .

..3 751 4. . .. . 2.3. 757 . ..

3.. 761 .. . 2. 2.. . . 8. 769

... . 2. 773 .2. 2. 3. .. . .

2... . .. 3. 4.2 . .. 787 2. .

.. . 3.2. . . .. 2. 797 ... .

5.2 2. . . 2.. .. .. . 3. 809

.4. 811 2.. . .. . 4.. . . 2..

2.. 821 .. 823 3. .2. .. 827 2.2. 829

.. . 6. 2. .. . 2.. 3. . 839

3... 2 . . 2. .2 .2. .2 4. .

.2. . 2.. 853 .. 2.. 3. 857 ... 859

2.. .. . 863 5.3 . . .2 2.. .

... . 3. 2. .. 3. 2.. 877 . .

4.. 881 .2.2 883 2.. .. . 887 3.. .

.. 4. 2. . .. . 7. .. . .

2.2.2 . .. .. 3. . .. 907 2. 2.

... 911 4.. . . .. 2. . .3. 919

3.. . . . 2... 2. . 2. 5. 929

... 2. 2. . . .. 3.2. 937 .. .

2.. 941 .. . 4. 3.. .. 947 2.. .

.2. . 3.. 953 .2. . 2. .. . .

6.. 2 .. 2. 2. . ... 967 3.2 ..

.. 971 2.5 . . .2. 4. 977 .. .

2..2 2. . 983 3.. . .. .. 2.. .

.2.. 991 5. . .. . 2.. 997 . 3.

3.3 .. .. . 2. .. . . 4.2. 1009

.. . 2.. 1013 ..2 .. 3. 2. . 1019

2... 1021 .. .. 10 2. .3. . 2. .3

.. 1031 3.. 1033 .. 2.. 2.. . .. 1039

4.. . . . 2.2. .. . . 3. 1049