Theory Numbers Khmer

PaBEckdac; nig viFIEckGWKøIt

- 1 - Prepared by Lim Phalkun

PaBEckdac;kñúgZ

niymn&y cMnYKt;rWuLaTIb a CaBhuKuNéncMnYnKt; b luHRtaEtman cMnYnKt;rWuLaTIb q mYyeTotEdl a b q . kñúgkrNIenH b ehAfatYEckén a . Knøw¼RsayPaBEckdac edIm,IRsayfacMnYKt;rWuLaTIb a Eckdac;nwgcMnYnKt;rWuLaTIbb eKRtUvRsay[eXIjfa a b q Edl q CacMnYnKt;rWuLaTIb .

lMhat´TI1 cUrRsaybBa¢ak;facMnYn 12 12A 2 3 Eckdac;nwg 97 . dMeNa¼Rsay eKman 12 12 4 3 4 3A 2 3 (2 ) (3 )

4 4 8 4 4 8

8 4 8

(2 3 )(2 2 .3 3 )

97(2 6 3 )

dUcenH 12 12A 2 3 Eckdac;nwg 97 .



lMhat´TI2 cUrRsaybBa¢ak;facMnYn ³

2010 2010 2010 670A 3 7 37 3 777 Eckdac;nwg 670 670 670B 3 7 37 . dMeNa¼Rsay Rsayfa A Eckdac;nwg B tamsmPaB

3 3 3 2 2 2a b c 3abc (a b c)(a b c ab bc ca) ebIeKyk 670 670 670a 3 , b 7 , c 37 ehIy 670 670 670 670abc 3 .7 .37 777 eK)an A B Q Edl 1340 1340 1340 670 670 670Q 3 7 37 21 111 259 dUcenH A Eckdac;nwg B . sMKal edIm,IRsaybBa¢ak;smPaB ³

3 3 3 2 2 2a b c 3abc (a b c)(a b c ab bc ca) eKRtUvtag

3 2

f (x) (x a)(x b)(x c)

x (a b c)x (ab bc ca)x abc



eday f (a) f (b) f (c) 0 eK)an ³ 3 2

3 2

3 2

a (a b c)a (ab bc ca)a abc 0 (1)

b (a b c)b (ab bc ca)b abc 0 (2)

c (a b c)c (ab bc ca)c abc 0 (3)

bUksmIkar (1) , (2) nig (3) rYcdak; a b c CaktþarYm enaHeK)ansmPaBdUcxageRkam ³

3 3 3 2 2 2a b c 3abc (a b c)(a b c ab bc ca) > lMhat´TI3 cUrRsaybBa¢ak;facMnYn ³ n n n nA 113 168 141 427 Eckdac;nwg 7 Canic©RKb;cMnYnKt;FmμCati n. dMeNa¼Rsay Rsayfa A Eckdac;nwg 7 tamrUbmnþ n n n 1 n 2 n 1a b (a b)(a a b ... b ) eK)an n n n nA (427 168 ) (141 113 ) 1 2

1 2 1 2

(427 168)m (141 113)m

259m 28m 7(37m 4m )

dUcenH A Eckdac;nwg 7 RKb; 1 2m , m IN * .



lMhat´TI4 cUrRsaybBa¢ak;facMnYn ³ n n n nA 90 118 168 287 Eckdac;nwg 91 Canic©RKb;cMnYnKt;FmμCati n. dMeNa¼Rsay Rsayfa A Eckdac;nwg 91 tamrUbmnþ n n n 1 n 2 n 1a b (a b)(a a b ... b ) eK)an n n n nA (287 168 ) (118 90 )

1 2

1 2

1 2 1 2

(287 168)m (118 90)m

119m 28m

7(17m 4m ) , m ,m IN

dUcenH A Eckdac;nwg 7 . mü:ageTot n n n nA (287 118 ) (168 90 )

1 2

1 2

1 2 1 2

(287 118)k (168 90)k

169k 78k

13(13k 6k ) , k ,k IN

dUcenH A Eckdac;nwg 13 . eday 7 nig 13 bfmrvagKña dUcenH A Eckdac;nwg 7 13 91 .



rebobRsayPaBEckdac´tamGnumanrYmKNitviTüa

edIm,IRsayfacMnYKt;rWuLaTIb na f (n) Eckdac;nwgcMnYnKt; rWuLaTIbb edayeFIVvicartamkMeNIneKRtUvGnuvtþn_dUcxageRkam ³ - cMeBaH n 1 RtUvbgðajfa 1a f (1) Eckdac;nwg b - ]bmafavaBitcMeBaH n k KW ka f (k) Eckdac;nwg b - RtUvRsayfavaBitcMeBaH n k 1 KW k 1a f (k 1) Eckdac;nwg b edayyk ka f (k) Casmμtikmμ . edIm,IRsayfa k 1a f (k 1) Eckdac;nwg b CadMbUgeKRtUv KNnaplsg k 1 ka a f (k 1) f (k) rYcsresr f (k 1) f (k) b g(k) rYceKTaj)an ³ f (k 1) f (k) b .g(k) eday f (k) Eckdac;nwg b enaH eKTaj)an f (k 1) f (k) b.g(k) Eckdac;nwg b . - snñidæan ³ na f (n) Eckdac;nwg b . -



lMhat´TI5 cUrRsaybBa¢ak;facMnYn nA n(n 1)(2n 1) Eckdac;nwg 6 Canic©cMeBaHRKb; n IN * . dMeNa¼Rsay Rsayfa nA Eckdac;nwg 6 -cMeBaH n 1 eK)an 1A 6 Eckdac;nwg 6 -]bmafavaBitcMeBaH n k KW kA k(k 1)(2k 1) Eckdac;nwg 6 . -eyIgnwgRsayfavaBitcMeBaH n k 1 KW ³ k 1A (k 1)(k 2)(2k 3) Eckdac;nwg 6 . eKman k 1 kA A (k 1)[(k 2)(2k 3) k(2k 1)] 2

k 1 kA A 6(k 1) eKTaj 2

k 1 kA A 6(k 1) eday kA Eckdac;nwg 6 dUcenH 2

k 1 kA A 6(k 1) Eckdac;nwg 6 . dUcenHcMeBaHRKb; n IN * cMnYn nA n(n 1)(2n 1) Eckdac;nwg 6 . sm:al; ³ 2 2 2n(n 1)(2n 1) 6(1 2 ... n ) .



lMhat´TI6 cUrRsaybBa¢ak;facMnYn n

nA 10 18n 1 Eckdac;nwg 27 Canic©cMeBaHRKb; n IN * . dMeNa¼Rsay Rsayfa nA Eckdac;nwg 27 -cMeBaH n 1 eK)an 1A 27 Eckdac;nwg 27 -]bmafavaBitcMeBaH n k KW k

kA 10 18k 1 Eckdac;nwg 27 . -eyIgnwgRsayfavaBitcMeBaH n k 1 KW ³ k 1

k 1A 10 18(k 1) 1 Eckdac;nwg 27 .

eKman k 1 kk 1 kA A 10 18(k 1) 1 10 18k 1

kk 1 kA A 9.10 18

k9(10 1) 27

9(10 1)g(k) 27

27[3g(k) 1] , g(k) IN *

eKTaj k 1 kA A 27[3g(k) 1] eday kA Eckdac;nwg 27 enaH 2

k 1 kA A 6(k 1) Eckdac;nwg 6 . dUcenH nA Eckdac;nwg 27 cRKb; n IN * .



lMhat´TI7 cUrRsaybBa¢ak;facMnYn n

nA 4 6n 1 Eckdac;nwg 9 Canic©cMeBaHRKb; n IN * . dMeNa¼Rsay Rsayfa nA Eckdac;nwg 9 -cMeBaH n 1 eK)an 1A 9 Eckdac;nwg 9 -]bmafavaBitcMeBaH n k KW k

kA 4 6k 1 Eckdac;nwg 9 . -eyIgnwgRsayfavaBitcMeBaH n k 1 KW ³ k 1

k 1A 4 6(k 1) 1 Eckdac;nwg 9 .

eKman k 1 kk 1 kA A 4 6(k 1) 1 4 6k 1

kk 1 kA A 3.4 6

k3(4 1) 9

3(4 1)g(k) 9

9[g(k) 1] , g(k) IN *

eKTaj k 1 kA A 9[g(k) 1] eday kA Eckdac;nwg 9enaH k 1 kA A 9[g(k) 1] Eckdac;nwg 9 . dUcenH nA Eckdac;nwg 9 cRKb; n IN * .



lMhat´TI8 edayeFIVvicartamkMeNIncUrRsaybBa¢ak´facMnYn ½

1n1n6n 92A Eckdacńwg11 Canic©cMeBa¼RKb´

cMnYnKt´Fm μCati n . dMeNa¼Rsay RsaybBa¢ak´facMnYn 1n1n6

n 92A Eckdacńwg11 -ebI 0n eKán 1192A0 Bit ( Eckdacnwg11 ) -ebI 1n eKán 191120992A 27

1 Bit ( Eckdacnwg11 ). eyIg«bmafavaBitdltYTI p KW 1p1p6

p 92A Eckdacńwg11 eyIgnwgRsayfavaBitdltYTI 1P KW

2p7p61p 92A

Eckdacńwg11 eyIgman 2p7p6

1p 92A

)9.29()92(2A 1p62p1p1p661p

1p

p1p

1pp1p

9.55A64A

)649(9A64A



edaytamkar«bma pA Eckdacńwg11 ehIy 1p9.55 Eckdacńwg11. eKTaján 1p

p1p 9.55A64A Eckdacńwg 11 Bit .

dUcen¼cMnYn 1n1n6n 92A Eckdacńwg11Canic©cMeBa¼

RKbćMnYnKt´Fm μCatin . lMhat´TI9 eK[ INn,610709743831E nnnn

n cUrbgHajfa nE Eckdacńwg 187 Canic©RKb´ INn . dMeNa¼Rsay bgHajfa nE Eckdacńwg 187 Canic©RKb´ INn ½ eday 1711187 Edl 11 nig 17 CacMnYnbfmrvagKña . dUcen¼edIm,IRsayfa nE Eckdacńwg 187 eyIgEtUvRsay [eXIjfa nE Eckdac´ 11 nig 17 . eKman N.)ba()b....baa)(ba(ba 1n2n1nnn Edl 1n2n1n b.......baaN eKán INn,610709743831E nnnn

n



*INq,p;)q9p8(11q99p88

q)610709(p)743831(

610709743831 nnnn

eKTaján nE Eckdacńwg 11 . müageToteKman½ INn,610709743831E nnnn

n

*IN;;)213(1734221

)743709()610831(

)743709()610831( nnnn

eKTaján nE Eckdacńwg 17 . dUcen¼ nE Eckdacńwg 187 Canic©RKb´ INn . lMhat´TI10 cUrRsaybBa¢ak;facMnYn nnnn

n 124122462447E Eckdac;nwg 221Canic©cMeBaHRKb;cMnYnKt;Fm μCati n . dMeNa¼Rsay RsaybBa¢ak;facMnYn nE Eckdac;nwg 221 eyIgeXIjfa 1713221 Edl 13 nig 17bzmrvagKña dUcenHedIm,IRsayfa cMnYn nE Eckdac;nwg 221eKRtUvRsay [eXIjfacMnYn nE vaEckdac;nwg !#pg nig Eckdac;nwg !& pg.



eKman nnnnn 124122462447E

)124462()122447(E nnnnn

tamrUbmnþ )b....baa)(ba(ba 1n2n1nnn eyIg)an 11n q).124462(p).122447(E )q26p25(13q338p325 1111 Edl *INq,p 11 . TMnak;TMngenHbBa¢ak;fa cMnYn nE Eckdac;nwg !# . müa:geToteKman )122462()124447(E nnnn

n

*INq,p,)q20p19(17

q340p323

q).122462(p).124447(

2222

22

22

TMnak;TMngenHbBa¢ak;fa cMnYn nE Eckdac;nwg !& . dUcenHsrubesckþIeTAeyIg)an cMnYn nE Eckdac;nwg 221Canic© cMeBaHRKb;cMnYnKt;FmμCati n .



viFIEckEbbGWKøIt niymn½y

eFIVviFIEckEbbGWKøIténcMnYnKt;rWuLaTIb a nigcMnYnKt;FmμCati b KWkMNt;cMnYnKt;rWLaTIb q nigcMnYnKt;FmμCati r Edl a b.q r ehIy 0 r b . a ehAfatMNagEck b ehAfatYEck q ehAfaplEck nig r ehAfasMNl; .

RTwsþIbT ebI a CacMnYnKt;rWuLaTIb nig b CacMnYnKt;FmμCati enaHman cMnYnKt;rWLaTIbEtmYyKt; q nigcMnYnKt;FmμCati r EtmYy Kt;Edl a bq r eday 0 r b .



lMhat´TI11 edaHRsaysmIkar 1y29x47 kñúgsMNMucMnYnKt;rWLaTIhV . dMeNa¼Rsay edaHRsaysmIkar 1y29x47 kñúgsMNMucMnYnKt;rWLaTIhV eyIgman 1812947 naM[ 294718 1111829 naM[ 47229)2947(29182911 711118 naM[ )47229()2947(11187 b¤ 3292477 41711 naM[ )329247()47229(7114 b¤ 5294734 1247 naM[ )329247(2).529473(7241 b¤ )13(29)8(471 eK)an )13(29)8(47y29x47



b¤ )13y(29)8x(47 naM[

Zq,q4713y

q298x

dUcenH Zq,13q47y,8q29x . lMhat´TI12 eK[cMnYnKt;viC¢man n . eKdwgfa n Ecknwg 7 [sMNl; % ehIy nEcknwg 8 [sMNl;# k> etIcMnYn n enaHEcknwg 56 [sMNl;b:un μan ? x> rkcMnYn n enaHedaydwgfa 5626n5616 . dMeNa¼Rsay k> etIcMnYn n enaHEcknwg 56 [sMNl;b:un μan ? tamsm μtikm μeKdwgfa n Ecknwg 7 [sMNl; % naM[man

INq1 Edl )1(5q7n 1 ehIymüa:geTot nEcknwg 8 [sMNl; # enaHnaM[man

INq2 Edl )2(3q8n 2 tam ¬!¦ nig ¬@¦ eK)anRbBn½æ

)2(3q8n

)1(5q7n

2

1

b¤

)4(21q56n7

)3(40q56n8

2

1



dksmIkar ¬# ¦ nig ¬$¦ eK)an 19)qq(56n 21 tag INq,qqq 21 eK)an 19q56n . TMnak;TMngenHmann½yfa cMnYn n enaHEcknwg 56 [sMNl; !( . x> rkcMnYn n enaHedaydwgfa 5626n5616 eKman 19q56n naM[ 562619q565616 b¤

56

5607q

56

5597 b¤

56

7100q

56

5399

naM[ 100q . cMeBaH 100q eK)an 5619195600n . dUcenHcMnYn n enaHKW 5619n .



lMhat´TI13 eK[cMnYnKt;FmμCati m nig n . eKdwgfa m Ecknwg 5 [sMNl; 3 ehIy n Ecknwg 5 [sMNl; 4 k> cUrbgðajfa 2 2m n Eckdac;nwg 5 . x> rksMNl;énviFIEck 3 3m n nwg 5 . dMeNa¼Rsay k> bgðajfa 2 2m n Eckdac;nwg 5 eday m Ecknwg 5 [sMNl; 3 ehIy n Ecknwg 5 [sMNl; 4 enaHeKmancMnYnKt;FmμCati 1q nig 2q Edl 1m 5q 3 nig 2n 5q 4 . eK)an 2 2 2 2

1 2m n (5q 3) (5q 4) 2 2 2 2

1 2 1 2m n 5(5q 5q 6q 8q 5) dUcenH 2 2m n Eckdac;nwg 5 . x> sMNl;énviFIEck 3 3m n nwg 5 eK)an 3 3 3 3

1 2m n (5q 3) (5q 4) 3 3 3 3m n 5q 3 4 5(q 18) 1 Edl q IN * dUcenH 3 3m n Ecknwg 5 [sMNl; 1 .



Knøw¼bMElgsV&yKuN eTVFajUtun n

n n k k

k 0(x y) C(n,k).x y

viFIbMElgsV&yKuN ebI a CacMnYnKt;rWuLaTIbEcknwg b CacMnYnKt;FmμCati [plEck CacMnYnKt;FmμCati q nigsMNl;CacMnYnKt;FmμCati r enaHeK)an a bq r eday 0 r b . eK)an n na (bq r) tameTVFajÚtuneK)an ³

nn n k k

k 0

n 1n k n

k 0

n 1n 1 k n k n

k 0

n

(bq r) C(n,k)(bq) r

C(n,k)(bq) C(n ,n)r

b C(n,k)b q r

b f (n) r , f (n) IN *

dUcenH n n na (bq r) b.f (n) r , f (n) IN *



lMhat´TI14 k> rksMNl;énviFIEckEbbGWKøIténcMnYn 20122 nwg 7 . x> kMNt;cMnYnKt;viC¢mantUcbMput k edIm,I[ 20122 k Eckdac;nwg 7 dMeNa¼Rsay k> sMNl;énviFIEckEbbGWKøIt eKman 32 8 7 1 eK)an 2010 6702 (7 1) 7q 1 , q IN * KuNGgÁTaMgBIrénsmIkarnwg 22 4 eK)an ³

20122 4 7q 4 . tamTMnak;TMngxagelIenHbBa¢ak;fa 20122 Ecknwg 7 [sMNl; 4 . x> kMNt;cMnYnKt;viC¢mantUcbMput k tamsMrayxagelIeyIg)an ³

20122 k 4 7q 4 k Edl q IN * ehtuenHedIm,I[ 20122 k Eckdac;nwg 7 luHRtaEt 4 k Eckdac; nwg 7 . dUcenH k 3 CacMnYnKt;viC¢mantUcbMputEdlRtUvrk .



lMhat´TI15 k> rksMNl;énviFIEckEbbGWKøIténcMnYn 20123 nwg 11 . x> kMNt;cMnYnKt;viC¢mantUcbMput k edIm,I[ 20123 k Eckdac;nwg 11 dMeNa¼Rsay k> sMNl;énviFIEckEbbGWKøIt eKman 53 243 22 11 1 eK)an 2010 4023 (22 11 1) 11q 1 , q IN * KuNGgÁTaMgBIrénsmIkarnwg 23 9 eK)an ³

20123 9 11q 9 . tamTMnak;TMngxagelIenHbBa¢ak;fa 20123 Ecknwg 11 [sMNl; 9 . x> kMNt;cMnYnKt;viC¢mantUcbMput k tamsMrayxagelIeyIg)an ³

20123 k 9 11q 9 k Edl q IN * ehtuenHedIm,I[ 20123 k Eckdac;nwg 11 luHRtaEt 9 k Eckdac; nwg 11 . dUcenH k 2 CacMnYnKt;viC¢mantUcbMputEdlRtUvrk .



lMhat´TI16 cUrRsaybBa¢ak;facMnYn 5555 22222222 5555 Eckdac;nwg 7 . dMeNa¼Rsay eKman 2222 317 7 3 eK)an 5555 5555 5555

1 12222 (317 7 3) 7q 3 ,q IN * ehIy 5555 793 7 4 eK)an 2222 2222 2222

2 25555 (793 7 4) 7q 4 ,q IN * eKTaj 5555 2222 5555 2222

1 22222 5555 7(q q ) 3 4 eKman 63 104 7 1 enaH 5550 925

3 33 (104 7 1) 7q 1 , q IN * KuNGgÁTaMgBIrnwg 53 243 eK)an 5555

33 243 7q 243 ehIy 34 9 7 1 enaH 2220 740

4 44 (9 7 1) 7q 1 , q IN * KuNGgÁTaMgBIrnwg 24 16 eK)an 2222

44 16 7q 16 eK)an 5555 2222

3 4 3 43 4 7(q q ) 259 7(q q 37) eKTaj 5555 2222

1 2 3 42222 5555 7(q q q q 37) dUcenH 5555 22222222 5555 Eckdac;nwg 7 .



lMhat´TI17 cUrRsaybBa¢ak;facMnYn n 2 2n 12 3 Eckdac;nwg 7 RKb; n IN . dMeNa¼Rsay RsaybBa¢ak;facMnYn n 2 2n 12 3 Eckdac;nwg 7 eKman 23 7 2 eK)an 2n n n3 (7 2) 7q 2 , q IN * KuNGgÁTaMgBIrnwg 3 eK)an 2n 1 n3 3 7q 3 2 ehIy n 2 n 22 2 2 eK)an n 2 2n 1 n 22 3 2 (2 3) 3 7q n7(2 3q) dUcenH cMnYn n 2 2n 12 3 Eckdac;nwg 7 .



lMhat´TI18 eK[ m nig n CaBIrcMnYnKt;viC¢manEdl 1 m n . cUrkMNt;témøtUcbMputén m n edIm,I[cMnYn m22 nig n22 manelx BIrxÞg;cugeRkaydUcKña . dMeNa¼Rsay kMNt;témøtUcbMputén m n ³ edIm,I[cMnYn m22 nig n22 manelxBIrxÞg;cugeRkaydUcKñaluHRtaEt

n m22 22 Eckdac;nwg 100 . eKman n m m n m22 22 22 (22 1) eday m22 CacMnYnKU nig n m22 1 CacMnYness . ehtuenH m n m22 (22 1) Eckdac;nwg 100 4 25 luHRtaEt

m22 Eckdac;nwg 4 nig n m22 1 Eckdac;nwg 25 . cMnYn m22 Eckdac;nwg 4 luHRtaEt m 2 . müa:geToteKman 222 96 5 4 elIkCakaereK)an 4 222 (96 5 4)

4 2 2

2

2

22 (96 5) 2(96 5).4 4

(96 154) 25 6

25a 6 , a 96 154



eK)an 4p p p22 (25a 6) 25q 6 , q IN eday 6 5 1 enaHeK)an ³

pp p k

k 0

pk

k 2

6 (5 1) C(p,k)5

1 5p C(p,k)5

eKTaj p

4p k

k 222 1 5p 25q C(p,k)5

b¤ p

4p k

k 222 1 5p 25q C(p,k)5

eday p

k

k 225q C(p,k)5

Eckdac;nwg 25 ehtuenH 4p22 1

Eckdac;nwg 25 luHRtaEt 5p Eckdac;nwg 25 eBalKWeKRtUv[ p 5k , k IN * . dUcenH n m22 1 Eckdac;nwg 25 luHRtaEt n m 4p 20k eday m 2 enaH m n (n m) 2m 20k 4 24 dUcenHtémøGb,brmaén m n KW 24 .



lMhat´TI19 eK[ 2f (n) n 2n Edl n IN * cUrkMNt;RKb;témø n edIm,I[ f (n) Eckdac;nwg 7 . dMeNa¼Rsay kMNt;RKb;témø n ³ -krNITI 1 ³ ebI n Eckdac;nwg 7 enaHeK)an n 7k , k IN * eK)an 2f (7k) 49k 14k 7k(7k 2) Eckdac;nwg 7 . dUcenHeK)an n 7k , k IN * CacemøIy . -krNITI2 ³ ebI n Eckmindac;nwg 7 enaHeK)an n 7k r Edl k IN * nig r 1 , 2 , 3 , 4 , 5 , 6 . eK)an 2f (7k r) (7k r) 2(7k r)

2 2

2

49k 14kr r 14k 2r

7k(7k 2r 2) r 2r

edIm,I[ f (7k r) Eckdac;nwg 7 luHRtaEt 2r 2r Eckdac;nwg 7 eday r 1 , 2 , 3 , 4 , 5 , 6 enaHtémøEdleFIV[ 2r 2r Eckdac;nwg 7 manEt r 5 . dUcenH n 7k 5 , k IN * CacemøIy .



lMhat´TI20 eK[ 2f (n) n 7n 18 Edl n IN * cUrbgðajfamantémø n eRcInrab;minGs;EdleFIV[cMnYn f (n) Eckdac;nwg 121 . dMeNa¼Rsay kMNt;RKb;témø n ³ eKman 2f (n) n 7n 18

2

2

(n 4n 4) (11n 22)

(n 2) 11(n 2)

eyIgBinitüeXIjfaebI (n 2) Eckdac;nwg 11 enaH 2(n 2) nig 11(n 2) suTæEtEckdac;nwg 121 enaHeK)an f (n) Eckdac;nwg 121 Edr . tag n 2 11k n 11k 2 , k IN . dUcenHeKGacsnñidæanfamantémø n eRcInrab;minGs;EdleFIV[cMnYn f (n) Eckdac;nwg 121 .



lMhat´TI21 eK[ 2f (n) 4n 7n 34 Edl n IN * cUrbgðajfaKμan n EdleFIV[cMnYn f (n) Eckdac;nwg 121 eT. dMeNa¼Rsay karbgðaj eKman 2f (n) 4n 7n 34

2

2

(4n 4n 1) (11n 33)

(2n 1) 11(n 3)

eyIgeXIjfaedIm,I[ f (n) Eckdac;nwg 121 luHRtaEt n 3 nig 2n 1 Eckdac;nwg 11 naM[eKman 1 2q ,q IN * Edl

1n 3 11q nig 22n 1 11q . eK)an 1 22(n 3) (2n 1) 22q 11q b¤ 1 211(2q q ) 7 CasmIkarKμancemøIykñúgsMNMu IN * BIeRBaH GgÁTImYyénsmIkarEckdac;nwg 11 EtGgÁTIBIrminEckdac;nwg 11 . dUcenH Kμan n EdleFIV[cMnYn f (n) Eckdac;nwg 121 eT.



lMhat´TI22 eK[ 2f (a) 16a 50a 13 Edl aZ kMNt;RKb;témø a EdleFIV[cMnYn f (a) Eckdac;nwg 49 dMeNa¼Rsay kMNt;RKb;témø a eKman 2f (a) 16a 50a 13

2

2

(16a 8a 1) (42a 14)

(4a 1) 7(6a 2)

edIm,I[ f (a) Eckdac;nwg 49 luHRtaEteKman p , qZ Edl 4a 1 7p nig 6a 2 7q . eK)an 3(4a 1) 2(6a 2) 21p 14q 7 7(3p 2q) b¤ 3p 2q 1 eKGacsresr 3(p 1) 2(q 1) eKTaj p 1 2k nig q 1 3k RKb; kZ b¤ p 2k 1 , q 3k 1 eKTaj)an 4a 1 7(2k 1) naM[ 7k 3

a2



eday aZ enaH 7k 3

2

Z .

edIm,I[ 7k 3

2

Z luHRtaEt k CacMnYnKt;essKW ³

k 2 1 , Z enaHeK)an 7(2 1) 3a 7 2

2

dUcenH a 7 2 , Z . lMhat´TI23 cUrbgðajfaebI n Eckdac;nwg 3 enaH n2 1 Eckdac;nwg 7 . ¬ n CacMnYnKt;viC¢man ¦ dMeNa¼Rsay karbgðaj ebI n Eckdac;nwg 3 enaH n 3p , p IN * eK)an n 3p2 1 2 1

p 1 p 2

p 1 p 2

(8 1)(8 8 .... 8 1)

7( 8 8 .... 8 1)

dUcenH ebI n Eckdac;nwg 3 enaH n2 1 Eckdac;nwg 7 .



lMhat´TI24 cUrkMNt;RKb;cMnYnKt;viC¢man x edIm,I[ 5 3 2x 5x x 7x 13 Eckdac;nwg x 2 . dMeNa¼Rsay kMNt;cMnYnKt;viC¢man x ³ BinitüGnuKmn_ 5 3 2f (x) x 5x x 7x 13 tamrUbmnþBnøat Taylor GnuKmn_ f (x) GacsresrCaes‘rIsV½yKuN énktþa (x 2) dUcxageRkam ³

2 5(5)x 2 (x 2) (x 2)

f (x) f ( 2) f '( 2) f ''( 2) ... f ( 2)1! 2! 5!

eday f ( 2) 32 40 4 14 13 39 ehIy 4 2f '(x) 5x 15x 2x 7 enaH f '( 2) 19 3 2f ''(x) 20x 30x 2 enaH f ''( 2) 278 --------------------------------------------- nig (5)f (x) 120 enaH (5)f ( 2) 120 eK)an 2 5f (x) 39 19(x 2) 139(x 2) ... (x 2) dUcenHedIm,I[ f (x) Eckdac;nwg x 2 luHRtaEt 39 Eckdac;nwg x 2 . eday 39 3 13 enaHeK)an x 1 b¤ x 11 .



lMhat´TI25 kMnt;RKb;cMnYnKt;viC¢man x edIm,I[ 5 3 2x 3x 15x 26x 2012 Eckdac;nwg 2(x 1) . dMeNa¼Rsay kMnt;RKb;cMnYnKt;viC¢man x tag 5 3 2f (x) x 3x 15x 26x 2012 tamrUbmnþBnøat Taylor GnuKmn_ f (x) GacsresrCaes‘rIsV½yKuN énktþa (x 1) dUcxageRkam ³

2 5(5)x 1 (x 1) (x 1)

f (x) f ( 1) f '( 1) f ''( 1) ... f ( 1)1! 2! 5!

eKman f ( 1) 1 3 15 26 2012 2025 4 2f '(x) 5x 9x 30x 26 enaH f '( 1) 0

3f ''(x) 20x 18x 30 enaH f ''( 1) 32 --------------------------------------------- nig (5)f (x) 120 enaH (5)f ( 1) 120 GnuKmn_ f (x) Gacsresr ³

2 5f (x) 2025 16(x 1) ... (x 1) edIm,I[GnuKmn_ f (x) Eckdac;nwg 2(x 1) luHRtaEt 2025



Eckdac;nwg 2(x 1) . eday 2 2 2 22025 45 3 3 5 eK)an 2 2(x 1) 3 enaH x 2 b¤ 2 2(x 1) 5 enaH x 4 b¤ 2 2(x 1) 9 enaH x 8 b¤ 2 2(x 1) 15 enaH x 14 b¤ 2 2(x 1) 45 enaH x 44 dUcenH x {2 , 4 , 8 , 14 , 44 } .

lMhat´TI26 kMNt;RKb;cMnYnKt;viC¢man x edIm,I[ 3 2x 10x 115x 237 Eckdac;nwg 343 . dMeNa¼Rsay kMNt;RKb;cMnYnKt;viC¢man x tag 3 2f (x) x 10x 115x 237

3 2

2

(x 1) 7(x 1) 49(2x 5)

(x 1) (x 8) 49(2x 5)

edIm,I[ f (x) Eckdac;nwg 343 luHRtaEt ³ x 1 nig 2x 5 Eckdac;nwg 7 b¤ x 8 Eckdac;nwg 343 nig 2x 5 Eckdac;nwg 7 .



-krNI x 1 nig 2x 5 Eckdac;nwg 7 enaHeKman 1 2q , q IN * Edl 1

2

x 1 7q

2x 5 7q

eK)an 1 22(x 1) (2x 5) 14q 7q

1 2

1 2

1 2

7 7(2q q )

1 2q q

2(q 1) (q 1)

eKTaj 1

2

q 1 k

q 1 2k

b¤ 1

2

q k 1

q 2k 1 , k IN

eKTaj)an x 1 7(k 1) b¤ x 7k 6 , k IN dUcenH x 7k 6 , k IN CacemøIy . -krNI x 8 Eckdac;nwg 343 nig 2x 5 Eckdac;nwg 7 eK)an 1

2 1 2

x 8 343m

2x 5 7m , m ,m IN *

eKman 1 22(x 8) (2x 5) 686m 7m 1 2

1 2

21 686m 7m

3 98m m

eKGacsresr 1 298(m 1) m 95 naM[ 1

2

m 1 k

m 95 98k

b¤ 1

2

m k 1

m 98k 95 , k IN

dUcenH x 343(k 1) 8 343k 335 RKb; k IN .



lMhat´TI24 kMnt´rkelxGBaØat x nig y éncMnYn 5x2y edIm,I[cMnYnen¼ Eckdacnwg 7 ehIycMnYnen¼CakaerRákd . dMeNa¼Rsay kMnt´rkelxGBaØat x nig y ½ edaycMnYn 5x2y Eckdac´nwg 7 ehIynigCakaer®ákdena¼ naM[vaEckdacnwg 49 . eyIgman 5x2y 5000 100x 20 y

5x2y 5020 100x y

2x y 225x2y 49 (102 2x )

49

cMnYn 5x2y ekItmankalNa 2x y 22IN *

49

eday 0 x , y 9 ena¼ 22 2x y 22 49 naM[ 22 2x y 22

149 49

eKTaj 2x y 22

149

¦ y 27 2x (1) eyIgán 5x2y 49 (103 2x)



edIm,I[cMnYnen¼Cakaer®ákdluu¼RtaEt 2103 2x k , k IN * eKán

2k 103x

2

eday 0 x 9

eKTaj 2k 103

0 92

¦ 2103 k 121

naM[ k 11 cMeBa¼ k 11 ena¼ 121 103

x 92

ehIytam

(1) : y 27 2(9) 9 . dUcen¼ x 9 ; y 9 nig 25x2y 5929 77 . lMhat´TI25 kMnt´rkelxGBaØat a nig b éncMnYn aab edIm,I[cMnYnen¼ Cakaer®ákdehIycMnYnen¼Eckdac´nwg 7 . dMeNa¼Rsay kMnt´rkelxGBaØat a nig b eKman aab 110a b edaycMnYnen¼Cakaer®ákd ehIyEckdac´nwg 7 ena¼vaRtUvEt Eckdacnwg 49 .



eyIgman 12a baab 110a b 49 (2a )

49

cMnYnen¼ekItmankalNa 12a bIN *

49

eday 0 a 9 , 0 b 9 ena¼ 0 12a b 117 eKTaján 12a b 49 ¦ 12a b 98 -krNI 12a b 49 ena¼eKán b 49 12a eday 0 b 9 ena¼ 0 49 12a 9 ¦ 4 1

3 a 412 12

naM[ a 4 cMeBa¼ a 4 ena¼ b 1 . kñúgkrNIen¼eKán 2aab 441 21 . dUcen¼ a 4 , b 1 CacemøIy . -krNI 12a b 98 ena¼eKán b 98 12a eday 0 b 9 ena¼ 0 98 12a 9 ¦ 4 2

7 a 812 12

naM[ a 8 cMeBa¼ a 8 ena¼ b 2 . kñúgkrNIen¼eKán aab 882 (minEmnCakaerRákd ) .



lMhat´TI26 kMnt´rkelxGBaØat x nig y éncMnYn 4x87y edIm,I[ cMnYnen¼Eckdacnwg 7 ehIycMnYn 4x87y CaKUbéncMnYnKt´ dMeNa¼Rsay eKman 4x87y 40000 1000x 870 y 4x87y 40870 1000x y

4x87y 7 ( 5838 143x) (4 y x)

edIm,I[4x87y Eckdac´nwg 7 lu¼RtaEt 4 y x Eckdacnwg 7 . eday 0 x , y 9 ena¼ 5 4 y x 13 eKTaj 4 y x 0 ¦ 4 y x 7 . -krNI 4 y x 0 eKán 4x87y 7(5838 143x) eday 4x87y CaKUbéncMnYnKtena¼

35838 143x 49k , k IN eKTaj

349k 5838x

143



eday 0 x 9 ena¼ 5838 5838 143x 7125 eKTaj 35838 49k 7125 ¦ 37 20

119 k 14549 49

naM[ k 5

cMeBa¼ k 5 ena¼ 349(5) 5838 287

x IN *143 143

-krNI 4 y x 7 eKán 4x87y 7(5838 143x) 7 7(5839 143x) eday 4x87y CaKUbéncMnYnKtena¼

35838 143x 49k , k IN eKTaj

349k 5839x

143

eday 0 x 9 ena¼ 5839 5839 143x 7126 eKTaj 35839 49k 7126 ¦ 38 21

119 k 14549 49

naM[ k 5

cMeBa¼ k 5 ena¼ 349(5) 5839

x 2143

.

ebI x 2 ena¼ 4 y 2 7 ¦ y 5 . dUcen¼ x 2 , y 5 ehIy 342875 (35) .



lMhat´TI27 kMnt´rkelxGBaØat x nig y éncMnYn 7x4y edIm,I[ cMnYnen¼Eckdacnwg 11 ehIycMnYnen¼Cakaer®ákd . dMeNa¼Rsay rkelxGBaØat x nig y eyIgman 7x4y 7000 100x 40 y 7x4y 100x y 7040

7x4y 11 (9x 640) x y

edIm,I[cMnYnen¼Eckdacnwg 11 lu¼RtaEt x y Eckdac´nwg 11 . eday 0 x , y 9 ena¼eKán x y 11 eKán 7x4y 11(9x 640) 11 7x4y 11(9x 640 1) 11(9x 641) eday 7x4y Cakaer®ákdena¼

29x 641 11 k , k IN * eKTaj

211k 641x

9

ehIy x 0

ena¼ 211k 641 0 ¦ k 7 eKán k { 8 , 9 } .



-cMeBa¼ k 8 ena¼ 211(8) 641

x 79

ehIy y 11 x 4 -cMeBa¼ k 9 ena¼

211(9) 641 250x

9 9

( minyk )

dUcen¼ x 7 , y 4 . epÞógpÞat´ ½ cMeBa¼ x 7 , y 4 eKán 27x4y 7744 88 Bit . lMhat´TI28 eK[sVúItnBVnþ (U) mantY 5 , 9 , 13 ,17 ...... nigsIVútnBVnþ (V) mantY 7 , 13 , 19 ,25, .... k> etIkñúgcMeNam $( tYénsIVút U manb:unaμntYEdlCakaerR)akd ? cUrkMnt;rktémøéntYTaMgenaH. x> etIkñúgcMeNam $( tYénsIVút V manb:unaμntYEdCakaerR)akd ? cUrkMnt;rktémøéntYTaMgenaH. K> etIkñúgcMeNam $(tYénsIVútTaMgBIrenHmanb:unμantYEdlmantémøesμIKña ?



dMeNa¼Rsay k> etIkñúgcMeNam $( tYénsIVút U manb:unaμntYEdlCakaerR)akd ? tYTI p énsIVútnBVnþ (U) EdlmantYTImYy 1U 5 nigplsgrYm d 9 5 4 kMnt;eday pU 5 4(p 1) 4p 1 cMeBaHRKb; p IN * . cMeBaHRKb; 1 p 49 ]bmafa 2

pU N , N IN *

eK)an 24p 1 N naM[ 2N 1

p (1)4

tamTMnak;TMng (1) edIm,I[ p CacMnYnKt;luHRtaEt k CacMnYness . ebI N 2r 1 , r IN * ¬cMnYness ¦

enaH 2

2(2r 1) 1p r r

4

Et 1 p 49 naM[ 21 r r 49 eday r IN * eK)an 1 r 6 b¤ r { 1 ,2 , 3 ,4 ,5 ,6 } ehIy p {2,6,12,20,30,42} dUcenHkñúgcMeNam $( tYénsIVút U man ^ tYEdlCakaerR)akd . tYTaMgenaHKW 2

2U 4(2) 1 9 3



26

212

220

230

242

U 4(6) 1 25 5

U 4(12) 1 49 7

U 4(20) 1 81 9

U 4(30) 1 121 11

U 4(42) 2 169 13

x> etIkñúgcMeNam $( tYénsIVút V manb:unaμntYEdCakaerR)akd ? tYTI k énsIVútnBVnþ (V) EdlmantYTImYy 1V 7 nigplsgrYm d 13 7 6 kMnt;eday kV 7 6(k 1) 6k 1 cMeBaHRKb; k IN * . cMeBaHRKb; 1 k 49 ]bmafa 2

kV t , t IN *

eKTaj)an 26k 1 t naM[ 2t 1

k (2)6

tam (2) edIm,I[ k CacMnYnKt;viC©manluHRtaEt 2t 1 Eckdac;nwg ^ eyIgyk t 6m r Edl m IN * , r , 5 r 5 eK)an 2 2 2 2t 1 (6m r) 1 6(6mr 2mr) r 1 edIm,I[ 2t 1 Eckdac;nwg ^ luHRta 2r 1 Eckdac;nwg ^ . eday 5 r 5 enaHeK)an r { 1 , 2 , 3 , 4 , 5} -cMeBaH r 1 enaH 2 2r 1 1 1 0 yk



-cMeBaH r 2 enaH 2 2r 1 2 1 3 ¬minyk¦ -cMeBaH r 3 enaH 2 2r 1 3 1 8 ¬minyk¦ -cMeBaH r 4 enaH 2 2r 1 4 1 15 ¬minyk¦ -cMeBaH r 5 enaH 2 2r 1 5 1 24 yk dUcenHeK)an t 6m 5, t 6m 1 , t 6m 1, t 6m 5 Eday 6m 1 6(m 1) 5 6m' 5

6m 5 6(m 1) 1 6m' 1

dUcenHtémø t EdlRtUvykmanEt t 6m 5 , t 6m 1 -ebI t 6m 5

eK)an 2

2(6m 5) 1k 6m 10m 4

6

eday 1 k 49 naM[ 21 6m 10m 4 49 b¤ eKTaj)an 2 m 3 kñúgkrNIenHeKTaj)antémø k { 8 , 28 }

-ebI t 6m 1 eK)an 2

2(6m 1) 1k 6m 2m

6

eday 1 k 49 naM[ 21 6m 2m 49 b¤ 1 m 3



kñúgkrNIenHeKTaj)antémø k { 4, 20 , 48} dUcenHeKTTYl)antémø k { 4,8,20,28,48} dUcenHkñúgcMeNam $( tYénsIVút V man % tYEdlCakaerR)akd . tYTaMgenaHKW 2

4V 6(4) 1 25 5

28

220

228

248

V 6(8) 1 49 7

V 6(20) 1 121 11

V 6(28) 1 169 13

V 6(48) 1 289 17

K> etIkñúgcMeNam $( tYénsIVútTaMgBIrenHmanb:unμantYEdlmantémøesμIKña ? -tYTI p énsIVútnBVnþ (U) EdlmantYTImYy 1U 5 nigplsgrYm d 9 5 4 kMnt;eday pU 5 4(p 1) 4p 1 cMeBaHRKb; p IN * . -tYTI k énsIVútnBVnþ (V) EdlmantYTImYy 1V 7 nigplsgrYm d 13 7 6 kMnt;eday kV 7 6(k 1) 6k 1 cMeBaHRKb; k IN * . cMeBaH 1 p 49 nig 1 k 49 ebI p kU V enaHeK)an ³ 4p 1 6k 1 b¤ 2p 3k naM[ p 3r

k 2r , r IN *



eday 1 p 49 nig 1 k 49 eK)an 1 3r 49

1 2r 49

eKTaj 1 r 16 b¤ r { 1,2,3, .....,16 } . dUcenHkñúgcMeNam $(tYénsIVútTaMgBIrenHman16 tYEdlmantémøesμIKña . lMhat´TI29 cMnYnKt;viC¢man n Ecknwg * [sMNl; ! . cMnYn n enaHEcknwg % [sMNl; @ . k-ebIcMnYn n enaHEcknwg $0 [sMNl;b:unμan ? x-rkcMnYn n enaHedaydwgfa 3940 n 4000 . ( RbLgGaharUbkrN_eTACb¨un éf¶TI 02 Ex sIha qñaM 2001 ) dMeNa¼Rsay k> ebIcMnYn n enaHEcknwg 40 [sMNl;b:un μan ? ]bmafa n Ecknwg * [plEck 1q IN nigsMNl; ! nig cMnYn n enaHEcknwg % [plEck 2q IN nigsMNl; @

tamGWKøIt eyIg)an 1

2

n 8q 1 ( 15)

n 5q 2 (16)

b¤ 1

2

15n 120q 15 (1)

16n 80q 32 (2)



bUksmIkar ¬!¦ nig ¬@¦ eyIg)an 2 1n 80q 120q 17 40q 17

Edl 2 1q 2q 3q . tamTMnak;TMng n 40q 17 bBa¢ak;fa ebIcMnYn n enaHEcknwg40 [sMNl; r 17 . x> rkcMnYn n enaHedaydwgfa 3940 n 4000 eyIgman n 40q 17 eday 3940 n 4000 eKTaj 3940 40q 17 4000 b¤ 3 17

98 n 10040 40

eday q IN naM[eKTaj)an q { 99 , 100} ehIy n { 3977 , 4017 } . dUcenH n { 3977 , 4017 } .

lMhat´TI30 cUrkMnt´RKb´KU )n;m( éncMnYnKt´viC¢manebIeKdwgfa ½ )nm(13nm 22 .



dMeNa¼Rsay kMnt´RKb´KU )n;m( ½ eKman )1()nm(13nm 22 -krNITI1 nm eKán n26n2 2 naM[ 13n dUcen¼ 13nm . -krNITII2 nm eyIgBinitüeXIjfaebI )n;m( CaKUcemøIyrbs )1( ena¼eKán

)m;n( k¾CaKUcemøIyrbs´ )1( Edr . sn μtfa nm . tamvismPaB Cauchy Schwarz

eyIgman )nm(2)nm( 222

26nm

)nm(26)nm( 2

eday nm ena¼eKán 26nmm2 ¦ 13m eday *INm ena¼eKTaj 12m1 . müageTotsmIkar )1( Gacsresr ½

)2(0m13mn13n 22



DIsRKImINg´ )m13m(4169 2 smIkar )2( mancemøIykñúg *IN kalNa CakaerRákd éncMnYnKt´viC©maness . eKyk INk)1k2()m13m(4169 22 eKán )1k(k41)1k2()m13m(4168 22 eKTaj )m13(m42)1k(k eday 12m1 ena¼témøEdlGacrbsplKuN )1k(k KW ½

}84,82,78,72,64,54{)1k(k . kñúgtémøTaMgRáMmYyen¼témøEdlCaplKuNcMnYnKt´tKñamanEt témø 9872 mYyKt´EdlRtUvnwg }10;3{m . -cMeBa¼ 3m eKán 0)2n)(15n(30n13n2 naM[ 15n . -cMeBa¼ 10m eKán 0)2n)(15n(30n13n2 naM[ 15n . srubmkeKTTYlánKUcemøIy®áMKUKW ½

})10;15(;)3;15(;)13;13(;)15;10();15;3({)n;m( .



lMhat´TI31 k¿cUrkMnt´elxénGBaØat d,c,b,a éncMnYn abcd kñúgráb´ eKal 10 ebIeKdwgfa ½

dcba9abcd x¿cMeBa¼témø d,c,b,a EdlánrkeXIjxagelIcUrbBa¢ak´fa cMnYn abcd nig dcba suTæEtCakaer®ákd . dMeNa¼Rsay k¿kMnt´elxénGBaØat d,c,b,a ½ eKman )1(dcba9abcd tamTMnak´TMng )1( eKTajántémø a EtmYyKt´KW 1a . cMeBa¼ 1a eKán )2(1dcb9bcd1 tamTMnak´TMng )2( eKTaján 9d eRBa¼ 819d manelxxagcugesμI 1 . cMeBa¼ 9d eKán )3(1cb999bc1 tamTMnak´TMng )3( eKTaján 0b ( eRBa¼ 9b minGacmanRtaTukeT ) cMeBa¼ 0b eKán )3(01c999c10



tamTMnak´TMng )3( eKTaján 8c eRBa¼ 72989c Efm 8 [elxxagcuges μI 0 . cMeBa¼ 8c eKán 980191089 . dUcen¼ 9d,8c,0b,1a . x¿bBa¢ak´facMnYn abcd nig dcba suTæEtCakaer®ákd ½ cMeBa¼ 9d,8c,0b,1a eKán ½

2331089abcd nig 2999801dcba suTæEtCakaer®ákd .



lMhat´TI32 sn μtfa HUNSEN CacMnYnmYymanelxRáMmYyxÞg´Edl

N,E,S,N,U,H Caelx . eBlEdleKKuNcMnYnen¼nwg 3 eKTTYlTæplCacMnYnmanelx RáMmYyxÞg´KW STRONG Edl G,N,O,R,T,S Caelx cUrrkcMnYn HUNSEN nig STRONG cMeBa¼ 3S,6N nigcMeBa¼ 9S,8N . (GkßrxusKñatagedayelxxusKña ).

STRONG

3

HUNSEN

dMeNa¼Rsay rkcMnYn HUNSEN nig STRONG eKman ½

STRONG

3

HUNSEN



k¿cMeBa¼ 3S,6N

G6TRO3

3

6E63HU

tamtaragRbmaNviFIKuNen¼bBa¢ak´fa 1H cMeBa¼ 1H taragviFIKuNeTACa ½

G6TRO3

3

6E63U1

tamtaragviFIKuNen¼eKTaján 8G eRBa¼ 36 manelxagcuges μI 8 . cMeBa¼ 8G taragviFIKuNeTACa ½

68TRO3

3

6E63U1

tamtaragviFIKuNen¼eKTaján 5E eRBa¼ 3E Efm 1 manelxcugeRkayesμInwg 6 . cMeBa¼ 5E taragviFIKuNeTACa ½



68TRO3

3

6356U1

tamtaragviFIKuNen¼eKTaján 0O eRBa¼ 33 Efm 1 manelxcugeRkayesμInwg 0 . cMeBa¼ 0O taragviFIKuNeTACa ½

068TR3

3

6356U1

tamtaragviFIKuNen¼eKTaján 9R eRBa¼ 36 Efm 1 manelxcugeRkay es μInwg 9 . cMeBa¼ 9R taragviFIKuNeTACa ½

9068T3

3

6356U1

tamtaragen¼bBa¢ak´fa U minGacFMCag 2 eTehIyedayGkßr xusKñatagedayelxxusKñaena¼eKTaján 2U EtmYyKt´ BIeRBa¼ 1H,0O ehIy U minGacyktémø 0 ¦ 1



dEdleTotáneT . cMeBa¼ 2U taragviFIKuNeTACa ½

9068T3

3

126356

tamtaragviFIKuNen¼eKTaján 7T BIeRBa¼fa ½

379068

3

126356

edayeRbobeFobtaragviFIKuN ½

STRONG

3

HUNSEN

nwg 379068

3

126356

eKTaján ½

126356HUNSEN nig 379068STRONG CacMnYnEdlRtUvrk .



x¿cMeBa¼ 9S,8N eyIgeda¼RsaytamrebobdUcxagelIeKán ½

956784

3

318928

edayeRbobeFobtaragviFIKuN ½

STRONG

3

HUNSEN

nwg 956784

3

318928

eKTaján ½ 318928HUNSEN nig 956784STRONG

CacMnYnEdlRtUvrk .



lMhat´TI33 cMnYnmYymanelxbYnxÞg´EdlelxxÞg´vaerobtamlMdab´

b;b;a;a . rkcMnYnena¼ebIeKdwgfavaCakaerRákd . dMeNa¼Rsay rkcMnYnEdlCakaerRákd ½ tag N CacMnYnEdlRtUvrk eyIgán bb10a100a1000aabbN )1()ba(a9911)ba100(11N

b11a1100N

eday 9a0 nig 9b0 ena¼ 18ba0 tamTMnak´TMng )1( edIm,I[ N GacCakaerRákdlu¼RtaEt

ba CaBhuKuNén 11 ehIy 18ba0 ena¼eKRtUv[ 11ba EtmYyKt´ .

TMnak´TMng )1( Gacsresr )1a9(1111a9911N 2 ehtuen¼ N Cakaer®ákdkalNa 1a9 CakaerRákd . eday 9a1 naM[ 811a910 eKTaj ½



161a9 (Kμan¦skñúg IN ) 251a9 (Kμan¦skñúg IN ) 361a9 (Kμan¦skñúg IN ) 491a9 (Kμan¦skñúg IN ) 641a9 naM[ 7a ehIy 4711b 811a9 (Kμan¦skñúg IN ) . dUcen¼cMnYnEdlRtUvkMnténa¼KW 2887744 . lMhat´TI34 cUrkMntélx a nig b edIm,I[cMnYn abba CaKUbéncMnYnKt´ . dMeNa¼Rsay kMntélx a nig b tag ab10b100a1000abbaN

)b10a91(11N

b110a1001N

edIm,I[ N CaKUbéncMnYnKt´lu¼RtaEt *INk,k121k11b10a91 332

eday 9b0,9a0 ena¼ 909b10a910



eKán 909k1210 3 smmUl

121

627

121

909k0 3 naM[ 1k

eKTaján 121b10a91 naM[ 10

a91121b

eday 0b ena¼ 010

a91121

¦

91

301

91

121a naM[ 1a ehIy 3

10

91121b

.

dUcen¼ 3b,1a nig 3111331abba . lMhat´TI35 cUrkMnt´RKb´KUtémøKt´viC¢man )b,a( ebIeKdwgfacMnYn ½

1bab2

a32

2

CacMnYnKt´viC¢manEdr .

dMeNa¼Rsay kMnt´RKb´KUtémøKt´viC¢man )b,a( ½ yk k

1bab2

a32

2

Edl *INk eKán )1(0)1b(kakb2a 322 DIsRKImINg´énsmIkar )1b(k4bk4 342 222 bk4)bkb2(



smIkar )1( mancemøIykñúg *IN lu¼RtaEt CakaerRákd mann&yfa 2222 dbk4)bkb2( Edl d CacMnYnKt´. -ebI 0bk4 2 ¦

4

bk

2

eyIgTTYlán

2

bb

2

bkb2a

32

¦ 2

ba

eday b,a CacMnYnKt´viC¢man ehtuen¼eKRtUv[ ½ INp,p2b

eKTaj pp8p4

)p2()p2(2a 4

22

ehIy p2

p2a .

dUcen¼ *INp,)p2,p(;)p2,pp8()b,a( 4

-ebI 0bk4 2 eKán *INk,)1bkb2(dbk4)bkb2( 222222 ¦ 0)1b()1b(k4 22 eKTaján 1b kñúgkrNIsmIkar )1( køayCa 0ak2a2 naM[ k2a dUcen¼ )1,k2()b,a( cMeBa¼RKb´ *INk .



-ebI 0bk4 2 eKán 222222 )1bkb2(dbk4)bkb2( smmUl 0)1bkb2(bk4)bkb2( 22222 ¦ 0)1k4()1b(b2)3k4(b2 ( minBitkñúg *IN ) srubmkeKánKUcemøIybImanragdUcxageRkam ½

)k2,kk8(;)k2,k(;)1,k2()b,a( 4 Edl *INk . lMhat´TI36 cUrkMnt´RKb´KUtémøKt´ 3n,m ebIeKdwgfacMeBa¼RKb´ cMnYnKt´viC¢man a eKman

1aa

1aa2n

m

CacMnYnKt´ .

dMeNa¼Rsay kMnt´RKb´KUtémøKt´viC¢man )n,m( ½ edIm,I[

1aa

1aa2n

m

CacMnYnKt´lu¼RtaEt 1aa 2n

CaktþarYmén 1aam ehIy nm . eyIgyk *INk,knm



eyIgán ½

)1aa)(a1()1aa(a

1aa1aak1k2nk

knm

tamTMnak´TMngen¼edIm,I[ 1aa 2n CaktþarYmén 1aam lu¼RtaEt 1kn nig 2k . dUcen¼ )3,5()n,m( . lMhat´TI37 eK[bIcMnYnKt´viC¢man c,b,a Edl 10cba . cUrrktémøFMbMputén cbaP . dMeNa¼Rsay rktémøFMbMputén cbaP eyIgman 10cba naM[ ba10c eyIgán a)b10(bab)ba10(abP 2 tag a)bb10(ba)a(P 22 eKán 2bb10ab2)a('P ebI 0)a('P eKTaján

2

b5a

eday *INb,0b2)a(''P



ehtuen¼ )a(P mantémøGtibrmacMeBa¼témø 2

b5a .

eday *INa ena¼ b RtUvEtCacMnYnKU ebI k2b ena¼ k5a ehIy k5k2)k5(10c

eday *INc,b,a ena¼

1k5c

1k2b

1k5a

¦ 4k1

-cMeBa¼ 1k eKán 4c,2b,4a naM[ 32424P -cMeBa¼ 2k eKán 3c,4b,3a naM[ 36343P -cMeBa¼ 3k eKán 2c,6b,2a naM[ 24262P -cMeBa¼ 4k eKán 1c,8b,1a naM[ 32181P dUcen¼témøGtibrmaén c.b.aP esμInwg 36 .



lMhat´TI38 bgðajfaKμancMnYnKt; z;y;x NaEdlepÞógpÞat;smIkar

6z8yx 22 eT. dMeNa¼Rsay bgðajfaKμancMnYnKt; z;y;x EdlepÞógpÞat; 6z8yx 22 eT eyIg)an )1()3z4(2yx 22 -ebI x CacMnYnKU nig y CacMnYnessenaH 22 yx CacMnYness dUcenHsmIkar )1( KμancemøIy ¬eRBaH )3z4(2 CacMnYnKU¦ . -ebI x CacMnYnessnig y CacMnYnKUenaH 22 yx CacMnYness dUcenHsmIkar )1( KμancemøIy¬ eRBaH )3z4(2 CacMnYnKU ¦. -ebI x CacMnYnKU y CacMnYnKU enaHeKGactag )Zn,m(n2y;m2x smIkar )1( Gacsresr )3z4(2)n2()m2( 22 b¤ 3z4)nm(2 22 CasmIkarKμanb¤skñúgsMNMucMnYnKt; . ¬ eRBaH )nm(2 22 CacMnYnKU ehIy 3z4 CacMnYness ¦ -ebI x CacMnYness y CacMnYnessenaHeKGactag 1n2y;1m2x cMeBaHRKb;cMnYnKt; m nig n .



smIkar )1( Gacsresr )3z4(2)1n2()1m2( 22 b¤ 6z82n4n4m4m4 22 b¤ 1z2)1n(n)1m(m CasmIkarKμanb¤skñúgsMNMucMnYnKt; . ¬ eRBaH )1n(n;)1m(m CacMnYnKU ehIy 1z2 CacMnYness ¦ srubmksmIkar 6z8yx 22 KμancemøIykñúgsMNMucMnYnKt; . dUcenH KμancMnYnKt; z;y;x EdlepÞógpÞat; 6z8yx 22 eT . lMhat´TI39 cUrbgðajfa

13n317n4

F

cMeBaHRKb; INn CaRbPaKsRmYlmin)an dMeNa¼Rsay sn μtfa 17n4 nig 13n3 mantYEckFMbMputesμI d . eyIg)an ad17n4 nig bd13n3 Edl a nig b bfmrvagKña eyIgman d)a3b4()17n4(3)13n3(4

d)a3b4(1

d)a3b4(51n1252n12

eday a3b4 nig d suTæEtCacMnYnKt;enaHeKTaj)an 1d naM[ 17n4 nig 13n3 CacMnYnbfmrvagKña . dUcenH

13n317n4

F

CaRbPaKsRmYlmin)an .



lMhat´TI40 eK[ x nig y CacMnYnBit . eKdwgfa 3322 yx,yx nig 44 yx CacMnYnsniTan . cUrRsayfa xy nig yx CacMnYnsniTan . dMeNa¼Rsay Rsayfa xy nig yx CacMnYnsniTan eyIgman )yx()yx(

21

yx 4422222 CacMnYnsniTan )yx(yx3)yx(yx 222232266 CacMnYnsniTan ehIy )yx()yx(

21

yx 6623333 CacMnYnsniTan

eyIgTaj)an 22

33

yxyx

xy CacMnYnsniTan .

müa:geTot xyyx

yxyx 22

33

CacMnYnsniTan .

dUcenH xy nig yx CacMnYnsniTan .



lMhat´TI41 cUrbgðajfa nP n(n 1)(n 2)(n 3)(n 4) Eckdac;nwg 120 cMeBaHRKb; n IN * . dMeNa¼Rsay karbgðaj eKman nP n(n 1)(n 2)(n 3)(n 4) cMeBaH 1n 1 : P 1 2 3 4 5 120 Eckdac;nwg 120 ]bmafavaBitcMeBaH n k KW kP Eckdac;nwg 120 eK)an kP k(k 1)(k 2)(k 3)(k 4) 120q ,q IN * eyIgnwgRsayfavaBitcMeBaH n k 1 KW k 1P Eckdac;nwg 120 . eKman k 1P (k 1)(k 2)(k 3)(k 4)(k 5)

k 1 kP P (k 1)(k 2)(k 3)(k 4)[(k 5) k] k 1 kP P 5(k 1)(k 2)(k 3)(k 4)

eday (k 4)! (k 1)(k 2)(k 3)(k 4)C(k 4,4)

4!.k! 24

eK)an k 1P 120q 120C(k 4,4) Eckdac;nwg 120 . dUcenH nP n(n 1)(n 2)(n 3)(n 4) Eckdac;nwg 120 cMeBaHRKb; n IN* .



lMhat´TI42 cUrbgðajfa nP n(n 1)(n 2)....(n k) Eckdac;nwg (k 1)! cMeBaHRKb; n IN * nig k IN * . dMeNa¼Rsay karbgðaj eKman nP n(n 1)(n 2)....(n k) cMeBaH 1n 1 : P 1 2 3.... (k 1) (k 1)! Eckdac;nwg (k 1)! Bit . ]bmafavaBitcMeBaH n p KW pP Eckdac;nwg (k 1)! eK)an pP p(p 1)(p 2)...(p k) (k 1)!q ,q IN * eyIgnwgRsayfavaBitcMeBaH n p 1 KW p 1P Eckdac;nwg (k 1)! eKman p 1P (p 1)(p 2)(p 3)...(p 1 k)

p 1 pP P (p 1)(p 2)...(p k)[(p 1 k) p] p 1 pP P (k 1)(p 1)(p 2)...(p k)

eday (p k)! (p 1)(p 2)...(p k)C(p k ,k)

k!p! k!

eK)an p 1P (k 1)!q (k 1)!C(p k ,p) Bit dUcenH nP n(n 1)(n 2)...(n k) Eckdac;nwg (k 1)! .



lMhat´TI43 edaHRsaysmIkarkñúgsMNMu IN * ³ k> x y 2012 x> 3 33x y 999 dMeNa¼Rsay k> x y 2012 eKman 2012 503 4 eK)an x y 2 503 eKTaj)an x 503 nig y 503 dUcenH x 503 , y 503 . x> 3 33x y 999

eKman 999 27 37 eK)an 3 33x y 3 37 eKTaj 3 3 33x 37 , y 2 37 naM[ x 37 , y 296 b¤ 3 3 33x 2 37 , y 37 naM[ dUcenH x 37 , y 296 b¤ x 296 , y 37 .



lMhat´TI44 edaHRsaysmIkarkñúgsMNMu IN * ³ k> 2 2x y 111 x> 3 3x y 61 dMeNa¼Rsay k> 2 2x y 111 eK)an (x y)(x y) 111 3 37 eday x y x y Canic©RKb; x , y IN * eK)an x y 1

x y 111

naM[ x 56 , y 55

b¤ x y 3

x y 37

naM[ x 20 , y 17

dUcenH x 56 , y 55 b¤ x 20 , y 17 . x> 3 3x y 61 eK)an 2 2(x y)(x xy y ) 1 61 eday 2 2x xy y x y Canic©RKb; x , y IN *

eK)an 2 2

x y 1

x xy y 61

naM[ x 5 , y 4

dUcenH x 5 , y 4 .



lMhat´TI45 edaHRsaysmIkarkñúg Z k> xy 7x 3y 194 x> 2x (y 1)x y 405 dMeNa¼Rsay k> xy 7x 3y 326 eKGacsresr (xy 7x) (3y 21) 21 194 b¤ (x 3)(y 7) 173 eyIgeXIjfa 173 Eckmindac;nwgcMnYnbfm 2 , 3 , 5 ,7 ,11 , 13 eT ehIy 213 169 173 enaH 173 CacMnYnbfm . eKTaj)an x 3 1

y 7 173

naM[ x 4 , y 166

b¤ x 3 173

y 7 1

naM[ x 176 , y 6

b¤ x 3 1

y 7 173

naM[ x 2 , y 180

b¤ x 3 173

y 7 1

naM[ x 170 , y 8



x> 2x (y 1)x y 405 smIkarenHGacsresrdUcxageRkam ³

2x xy x y 405

x(x y) (x y) 405

(x 1)(x y) 405

eday 405 CacMnYnbfmenaHeKTaj)an ³ x 1 1

x y 405

naM[ x 0 , y 405 x 1 405

x y 1

naM[ x 404 , y 403 x 1 1

x y 405

naM[ x 2 , y 403 x 1 405

x y 1

naM[ x 406 , y 405



lMhat´TI46 eK[cMnYn m nE(m,n ) 2 2 Edl m n 1 nig m,n IN * cUrkMNt;témøtUcbMputén m n edIm,I[ E(m,n ) Eckdac;nwg 448 dMeNa¼Rsay kMNt;témøtUcbMputén m n eKman m n 1 enaH m n2 2 RKb; m,n IN * eK)an n m nE(m,n) 2 (2 1) eday 6448 7 2 ehtuenHedIm,I[ n m nE(m,n) 2 (2 1) Eckdac;nwg 448 luHRtaEt n2 Eckdac;nwg 62 eBaHKW n 6 nig m n2 1 Eckdac;nwg 7 . eKman 32 1 7 enaHedIm,I[ m n2 1 Eckdac;nwg 7 luHRtaEt m n CaBhuKuNén 3 eBalKW m n 3k , k IN * eKman m n (m n) 2n 3k 12 dUcenHedIm,I[ m n mantémøGb,brmaluHRtaEt k 1 dUcenH témøtUcbMputén m n KW m n 15 .



lMhat´TI47 eKmancMnYn m nE(m,n ) 3 3 Edl m n 1 nig m,n IN * . cUrkMNt;témøtUcbMputén m n edIm,I[ E(m ,n )Eckdac;nwg 9801 dMeNa¼Rsay kMNt;témøtUcbMputén m n eKman 4 29801 3 11 eK)an m n n m nE(m,n ) 3 3 3 (3 1) edIm,I[E(m ,n )Eckdac;nwg 9801 luHRtaEt n3 Eckdac;nwg 43 nig m n3 1 Eckdac;nwg 211 . edIm,I[ n3 Eckdac;nwg 43 luHRtaEt n 4 . mü:ageToteKman 5 23 1 242 2 11 ehtuenH m n3 1 Eckdac;nwg 211 luHRtaEt m n Eckdac;nwg 5 enaHeK)an m n 5q , q IN * eday m n (m n) 2n 5q 8 13 RKb; q IN * dUcenH témøtUcbMputén m n KW m n 13 .



lMhat´TI48 cUrkMNt;RKb;cMnYnKt;viC¢man z,y,x ebIeKdwgfa

1997z

y

1996

x

1322 .

dMeNa¼Rsay kMNt;RKb;cMnYnKt;viC¢man z,y,x ³ eKman (*)

1997z

y

1996

x

1322

tag d CatYEckrYmFMbMputén x nig y enaH mdx nig ndy Edl m nig n CacMnYnKt;viC¢manbfmrvagKña . tamsmIkar (*) eKGacsresr

1997z

dn

1996

dm

132222

b¤ zdnmm19971996n199713 22222 eday m nig n CacMnYnKt;viC¢manbfmrvagKñaenaHeyIgRtUvman ³

199713 Eckdac;nwg 2m nig 19971996 Eckdac;nwg 2n . edayeKman 44921966 2 ehIy 1997,449,13 CacMnYnbfm dUcenHtémø m nig n bMeBjlkçxNÐxagelIenHKW 1n,1m b¤ 2n,1m .



-krNI 1n,1m eK)an 1997.41.71997).199613(zd 22 eday 1997 bfmCamYy 41 nig 7 enaHeKTaj)an 1d b¤ 7d dUcenHeK)ancemøIy 4011973z,1y,1x b¤ 81877z,7y,7x . -krNI 2n,1m eK)an 1997.21997).44913(zd 92 eKTaj)an 16,8,4,2,1d . dUcenHeKTaj)ancemøIy ³

)3994,32,16(;)16976,16,8(

)63904,8,4(;)255616,4,2(;)1022464,2,1()z,y,x(



lMhat´TI49 eK[

1998.1997

1....

5.4

1

4.3

1

2.1

1A

nig 1000.1998

1....

1996.10021

1997.10011

1998.10001

B cUrbgðajfa

B

A CacMnYnKt; ?

dMeNa¼Rsay bgðajfa

B

A CacMnYnKt; ³ eKman

1998.19971

....5.4

14.3

12.1

1A

19981

.....1002

11001

11000

1

)9991

...31

21

1()1998

1...

31

21

1(

)1998

1....

61

41

21

(2)1998

1....

31

21

1(

19981

19971

...61

51

41

31

21

1

eday )1(1998

1....

10011

10001

A nig )2(

1000

1....

1997

1

1998

1A

bUkTMnak;TMng )2(&)1( eK)an ³ B2998)

1000.19981

...1997.1001

11998.1000

1(2998A2

eKTaj)an 1499B

A CacMnYnKt; .



lMhat´TI50 ¬RbLgKNitviTüaGULaMBicGnþrCatiqñaM 1977 ¦ eK[BIcMnYnKt;viC¢man a nig b . eBlEdl 2 2a b Ecknwg a b enaHeK)anplEck q nigsMNl; r . cUrkMNt;RKb;KU (a , b ) edaydwgfa 2q r 1977 . dMeNa¼Rsay kMNt;RKb;KU (a , b ) tamviFIEckEbbGWKøIteKGacsresr 2 2a b (a b)q r (*) Edl 0 r a b 1 . eKman r a b 1 enaH 2 2q r q a b 1 b¤ 2q a b 1 1977 b¤ 2q a b 1978 (**) eKman r 0 enaH 2 2a b (a b)q r (a b)q

tamvismPaB 2

2 2 (a b)a b

2

eK)an 2(a b)

(a b)q2

b¤ a b 2q

tamvismPaB (**) eKGacsresr 2q 2q 1978 b¤ 2 2(q 1) 1979 44 43 eKTaj q 1 45 b¤ q 44



mü:ageTot 2 2 2q q r 1977 44 43 eKTaj)an q 44 . BIlTæplxagelIenHeKTaj)an ³ q 44 ehIy 2r 1977 44 41 smIkar (*) Gacsresr 2 2a b 44(a b) 41 b¤ 2 2(a 22) (b 22) 1009 tag u | a 22 | nig v | b 22 | Edl u , v IN eK)ansmIkar 2 2u v 1009 . eyIgdwgfaRKb;kaeréncMnYnKt;manelxcugeRkay {0,1,4,9,5,6 } ehtuenHplbUkkaerénBIcMnYnKt;EdlmanelxcugeRkayesμI 9luHRtaEt elxcugeRkayénkaercMnYnnImYyesμ IerogKña 4 nig 5 b¤ 5 nig 4 b¤ 0 nig 9 b¤ 9 nig 0 . BIsmIkar 2 2 2u v 1009 31 48 eKTaj)an 0 u 31 ]bmafa u v enaH 2 2 22u u v 1009 eKTaj 2 21009 41

u 222 2

b¤ u 23 ehtuenH 23 u 31 . eday 2u RtUvmanelxcugeRkay {0 , 4 , 5 ,9}



enaHeKGaceRCIserIstémø u CadMbUgdUcxageRkam ³ u {30,28,25,23,27} eday 2 2u v 1009 enaH 2 2v 1009 u -ebI u 30 enaH v 1009 900 109 minyk -ebI u 28 enaH v 1009 784 15 yk -ebI u 25 enaH v 1009 6225 384 minyk -ebI u 23 enaH v 1009 529 480 minyk -ebI u 27 enaH v 1009 729 280 minyk eday 2 2u v 1009 CasmIkarqøúHehtuenHebI (a ,b )CacemøIy rbs;smIkarenaH (b ,a) k¾CacemøIyrbs;smIkarEdr . eKTaj)ancemøIy u 28 , v 15 b¤ u 15 , v 28 -krNI u 28 , v 15 eK)an | a 22 | 28

| b 22 | 15

eKTaj)an a 50,b 37 b¤ a 50 , b 7 . -krNI u 15 , v 28 eK)an | a 22 | 15

| b 22 | 28

a 37,b 50 b¤ a 7 , b 50 . dUcenH (a,b) {(50,37);(37,50);(7,50);(50,7)} .



lMhat´TI51¬RbLgKNitviTüaGULaMBicGnþrCatiqñaM 1967 ¦ eK[ k , m , n CacMnYnKt;FmμCatiedaydwgfa m k 1 CacMnYn bfmFMCag n 1 . eKyk sc s(s 1) . cUrRsayfaplKuN ³

m 1 k m 2 k m 3 k m n k(c c )(c c )(c c )...(c c ) Eckdac;nwgplKuN 1 2 3 nc .c .c ...c . dMeNa¼Rsay

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