Dinh Ly Fermat

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Vietnamese

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t nh c p Ton hio nh Ton a nm 1640,

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Copyright 2007 Vietnamese Kvant Group

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2 2 21 (5 2) 1 5(5 4 1)a k k k+ = + + = + + chia ht cho 5 . Tng t vi 3k = . Vy ta thu c iu phi chng minh. Ta c th phn tch 2 1 ( 2)( 2) 5a a a+ = + + v do

5 ( 2)( 1) ( 1)( 2) 5( 1) ( 1)a a a a a a a a a a = + + + + c cng s d vi ( 2)( 1) ( 1)( 2)a a a a a+ + khi chia cho 5. Tch 5 s t nhin lin tip ( 2)( 1) ( 1)( 2)a a a a a+ + chia ht cho 5, do hiu 5a a cng chia ht cho 5. Cng nhn c h qu l hiu ny chia ht cho 30.

Vo nm 1801 K. F. Gauss a ra k hiu ng d. Ta s dng chng n gin ho din t s chia ht.

Hai s nguyn ,a b gi l ng d modulo n nu chng c cng s d khi chia cho s nguyn n . K hiu l (mod )a b n . Gi s (mod ), (mod )a b n c d n , d dng chng minh:

i. (mod )a c b d n+ + ii. (mod )ac bd n iii. (mod )m ma b n .

vi , ,a b n nguyn v m khng m.

Bi tp

2. Gii phng trnh ng d 3 11(mod101).x 3. Gii phng trnh ng d 14 0(mod12).x 4. Vi 0k . Chng minh rng a. Nu (mod )ka kb kn th th (mod )a b n . b. Nu (mod )ka kb n v k nguyn t cng nhau vi n th th (mod ).a b n 4. 7p = . Xt hiu 7 2 2( 1) ( 1)( 1)( 1)a a a a a a a a a = + + + + . Hy th mt s gi tr ca a : 70 0 0 = , 71 1 0 = , 72 2 126 7.18 = = , 76 7 279930 7.39990 = = . By gi ta s chng minh hiu trn chia ht cho 7 vi mi s t nhin a . Ta c:

2 21 6 ( 2)( 3) (mod 7)a a a a a a+ + + + v 2 21 6 ( 2)( 3) (mod 7)a a a a a a + = +


5

Suy ra hiu trn ng d vi tch by s t nhin lin tip. S t nhin a chia cho 7 c s d l 0, 1, 2, 3, 4, 5, 6 nn tch ( 3)( 2)( 1) ( 1)( 2)( 3)a a a a a a a+ + + chia ht cho 7. Bi tp

5. a. Chng minh rng 7 (mod 42)a a . b. Chng minh rng 9 (mod30)a a . 5. 11p = . Xt hiu 11 4 3 2 4 3 2( 1) ( 1)( 1)( 1)a a a a a a a a a a a a a = + + + + + + + Ta c 2 2 2 2( 3)( 4)( 5)( 9) ( 7 12)( 14 45) ( 4 1)( 3 1)a a a a a a a a a a a a = + + + + +

= 4 3 2 4 3 210 1 1 (mod11)a a a a a a a a+ + + + + + + . Tng t, bn c th ch ra: 4 3 2( 2)( 6)( 7)( 8) 1a a a a a a a a = + + .

Nh vy hiu trn ng d vi tch 11 s t nhin lin tip, v tch ny chia ht cho 11.

Tip tc vi 13p = hoc nhng s nguyn t ln hn bn c th a ra tng li gii ring bit cho tng trng hp. Tuy nhin, n lc chng ta tip cn vi trng hp tng qut ca nh l Fermat nh i vi mi s nguyn t p .

Bi tp

6. Chng minh rng

a. Tch ca 4 s nguyn lin tip th chia ht cho 24.

b. Tch ca 5 s nguyn lin tip th chia ht cho 120.

c. 5 35 4a a a + chia ht cho 120 vi mi s nguyn a . 7. Chng minh rng 5a v a c ch s tn cng ging nhau.

8. Chng minh rng 5 5m n mn= chia ht cho 30 vi bt k s nguyn , .m n 9. Nu s k khng chia ht cho 2, 3, 5 th th 4 1k chia ht cho 240. 10. a. Chng minh rng 5555 22222222 5555+ chia ht cho 7.

b. Tm d s ca php chia 2014 16 19(13 15 ) 18+ + cho 7.

11. Chng minh tn cng ca 1011 1 c hai ch s tn cng l hai s 0. 12. a. Tm tt c cc s nguyn a sao cho 10 1a + c tn cng l s 0.


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b. Chng minh rng 100 1a + khng th tn cng l s khng vi bt k s nguyn a no. 13. Cho trc s chn n khc khng. Tm c chung ln nht ca cc s c dng na a , vi a thuc tp s nguyn.

14. Cho trc s t nhin 1n > . Chng minh rng c chung ln nht ca cc s dng na a , a thuc tp s nguyn trng vi c chung ln nht ca cc s dng na a , vi 1, 2,3,..., 2 .na =

Trng hp tng qut.

Xt s nguyn t p v s nguyn k khng chia ht cho p . Vi 19p = , 4k = . Lp bng xt cc s 1,2,...,18r = v cc d s ca 4r khi chia cho 19.

r 1 2 3 4 5 6 7 8 9 4r 4 8 12 16 20 24 28 32 36

4 mod19r 4 8 12 16 1 5 9 13 17 r 10 11 12 13 14 15 16 17 18

4r 40 44 48 52 56 60 64 68 72 4 mod19r 2 6 10 14 18 3 7 11 15

Ta nhn thy rng cc s d ca 4r khi chia cho 19 u i mt khc nhau v chnh l cc s r . Tng qut hn ta c khng nh

Nu s nguyn k khng chia ht cho s nguyn t p v 1 2,r r l hai s d phn bit trong php chia k

cho p th 1 2,kr kr c hai s d phn bit khi chia cho p .

Tht vy nu 1 2 1 2( ) 0(mod )kr kr k r r p = th do k khng chia ht cho p , hay ni cch khc nguyn t cng nhau vi p nn 1 2 (mod )r r p hay 1 2.r r= Bi tp

15. Tn ti hay khng s t nhin n sao cho 1999n c tn cng l 987654321?

16. Nu s nguyn k nguyn t cng nhau vi s t nhin n th tn ti s t nhin x sao cho 1kx chia ht cho n .

17. Nu ,a b nguyn t cng nhau, th bt k s nguyn no cng c th biu din di dng c ax by= + , vi ,x y nguyn.

By gi ta bn n li gii ca nh l Fermat nh. Ta c th vit 1( 1)p pk k k k = . Nh vy nu k chia ht cho p th nh l l hin nhin nn quan trng l trng hp k khng chia ht cho .p


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nh l 1.

Nu s nguyn k khng chia ht cho s nguyn t p th 1pk c s d l 1 khi chia cho p .

Chng minh

Cc s d ca cc s , 2 ,3 ,..., ( 1)k k k p k i mt khc nhau, v l 1,2,3,..., 1p . Nh vy .2 .3 ...( 1) 1.2.3...( 1) (mod )k k k p k p p

Hay 1( 1)! ( 1)! (mod )pk p p p , suy ra 1 1(mod )pk p (*).

y chng ta s dng kt qu Bi tp 4. C th bin i 1( 1)( 1)! 0 (mod )pk p p v do ( 1)!p nguyn t cng nhau vi p nn dn n (*). Bi tp

18. Tm phn d khi chia 20003 cho 43 .

19. Nu s nguyn a khng chia ht cho 17, th th 8 1a hoc 8 1a + chia ht cho 17. 20. Chng minh rng 61 61m n mn chia ht cho 56786730 vi mi s nguyn ,m n .

21. Tm tt c cc s nguyn t p sao cho 2

5 1p + chia ht cho p .

22. Chng minh rng 7 5 2p p chia ht cho 6p vi mi s nguyn t p l. 23. Vi p l s nguyn t th th tng 1 1 11 2 ... ( 1)p p pp + + + chia cho p d 1p . 24. Mt s c 6 ch s chia ht cho 7. Ta i ch ch s hng trm nghn lui v sau v hng n v. Chng minh rng s mi nhn c cng chia ht cho 7. Th d, 632387 v 200004 chia ht cho 7 sau khi bin i nhn c 323876 v 42 cng chia ht cho 7.

25. Xt s nguyn t p khc 2, 3, 5. Chng rng s c lp bi 1p s 1 s chia ht cho p . Th d, 111111 chia ht cho 7.

26*. Chng minh rng vi bt k s nguyn t p th s 9p ch s 111122229999 (trong c p ch s 1, p ch s 2,, p ch s 9) ng d vi s 123456789 khi chia cho p .

Cc bng nhn modulo.

Hy xem xt 1n s d khc khng trong php chia mt s cho n . Lp cc bng m ca hng th a v ct th b l s d ca php chia ca tch ab cho n , trong 0 ,a b n< < .


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Th d vi 5n = ta c Bng 1, 11n = ta c Bng 2.

Bng 2.

Bng 1. Bng 3.

Bng 5.

Bng 4. Bng 6.

Ta thy rng trong cc bng ny s d trong cc cng khc khng. i vi n l cc s nguyn t th s d tch hai s d khc khng cng c s d khc khng trong php chia cho .n

i vi cc hp s n th tch hai s d ca n c th c d bng 0 trong php chia cho n . Th d 2.2 0(mod 4) (xem Bng 3.), v i vi 12n = th xy ra nhiu trng hp hn na (xem Bng 4.) By gi t cc bng c xo i cc ct v hng c cha s d bng 0. Th d Bng 2 ta xo i hng


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th 2 v ct th 2 th thu c bng 5, Bng 4 xo i cc ct v hng th 2, 6, 8, 9, 10 th thu c Bng 6. cc bng m n l s nguyn t th khng cn phi xo i hng hay ct no c.

Nhn thy rng cc hng v ct c li l nhng hng v ct c s c nh l nguyn t cng nhau vi n . Ta c khng nh (hy chng minh)

Hai s nguyn t cng nhau vi n th tch ca chng s c s d khc khng modulo n .

Bi tp

27. Lm sng t tnh i xng ca cc cp s d qua cc ng cho cc bng 1-6.

nh l Euler.

By gi ta s c mt s tng qut ca nh l Fermat nh cho trng hp php chia i vi mt s t nhin n bt k. Ta xem xt cc bng nhn modulo phn trc v nhn thy rng ca hng v ct c nh s l nguyn t cng nhau vi n th s c d s khc khng molulo n . Hn na trong mi hng hoc mi ct trong cc bng mi nhn c u c cha s d i mt khc nhau modulo n . C th khng nh nu cc s d 1 2 3, , ,..., ra a a a modulo n i mt khc nhau v nguyn

t cng nhau vi n th cc s 1 2, ,..., rka ka ka c s d cng chnh l cc s 1 2 3, , ,..., ra a a a (hy chng minh). Ta c

1 2 1 2. ... . ... (mod )n nka ka ka a a a n

T 1 2 1 2( 1) ... ... (mod )r

n nk a a a a a a n do 1 2, ,..., ra a a u nguyn t cng nhau vi n nn 1(mod )rk n . Nu n l s nguyn t th 1r n= , ta thu c khng nh ca nh l Fermat nh.

Khng nh tng qut c mang tn nh l Euler.

nh l 2.

Nu s nguyn k nguyn t cng nhau vi s t nhin n th 1rk chia ht cho n , vi r l s cc s t nhin nguyn t cng nhau vi n m khng vt qu n .

Bi tp

28. Chng minh rng nu k chia ht cho 3, th th

a. 3k chia cho 9 c d s l 1 hoc 8.

b. 81k chia cho 243 c d s l 1 hoc 242.

29. Chng minh rng

a. Nu 3 3 3a b c+ + chia ht cho 9, th th mt trong cc s , ,a b c chia ht cho 3.


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b. Tng cc bnh phng ca 3 s nguyn chia ht cho 7 khi v ch khi tng cc lu tha bc 4 ca nhng s nguyn chia ht cho 7.

30. Chng minh rng 777 77 77 77 7 chia ht cho 10.

31. Tm 3 ch s cui cng ca 99997 .

32. Nu s nguyn a nguyn t cng nhau vi s t nhin 1n > , chng minh rng phng trnh ng d(mod )ax b n tng ng vi 1 (mod )rx a b n . Trong r l s cc s t nhin khng b hn n

nguyn t cng nhau vi n .

33. Chng minh rng nu n l s t nhin l th th !2 1n chia ht cho n . 34*. Tm tt c cc s t nhin 1n > sao cho tng 1 2 ... ( 1)n n nn+ + + chia ht cho n . 35*. Chng minh rng vi mi s t nhin s th tn ti mt bi s n ca n sao cho tng cc ch s ca n chia ht cho s .

Hm Euler.

Nm 1763, Leonard Euler (1707-1783) a ra k hiu ( )n ch s lng cc s d modulo n m nguyn t cng nhau vi n . Th d : (1) 1 = , (4) 2 = , (12) 4 = . Nu p l s nguyn t th th ( ) 1p p = . Hy xt ( )mp vi m l s t nhin. Cc s d modulo

mp l 0,1,2,..., 1mp . Trong c 1mp s chia ht cho p l 0, , 2 ,..., mp p p p . Suy ra :

1 1( ) (1 )m m m mp p p pp

= =

By gi th tnh (1000) , l s tt c cc s t nhin b hn 1000 v khng chia ht cho 2 v 5. C 500 s chn trong s cc s t nhin b hn 1000. C 200 s chia ht cho 5 trong s cc s t nhin b hn 1000. C 100 s chia ht cho 2 v 5 trong s cc s t nhin b hn 1000. Nh vy thu c :

(1000) 1000 (500 200 100) 400. = + = Bi tp

36. Tnh (2 5 )a b vi a v b l cc s t nhin. 37. Vi ,p q l hai s nguyn t khc nhau. Tnh ( )pq v ( )a bp q vi ,a b l cc s t nhin. 38. Gii phng trnh :


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a. (7 ) 294x = b. (3 5 ) 360x y = . p dng phng php tng t ta c th tnh ( )n vi mi s nguyn dng n . Mt th d phc tp hn lm r hn phng php ny, hy tnh (300) . Cc s t nhin b hn 300 c 150 s chn, 100 s chia ht cho 3, 60 s chia ht cho 5. Trong s nhng bi ny li c 50 s chia ht cho 2.3=6, 30 s chia ht cho 2.5=10 v 20 s chia ht cho 3.5=15. Trong s cc bi ca 6, 10 v 15 ny li c 10 s chia ht cho 2.3.5=30. Nh vy ta c :

(300) 300 [150 100 60 (50 30 20 10)] 80 = + + + + = c php chng minh tng qut s dng phng php ny bn c th c bi bo S hc v nhng nguyn l m ca N. Basileva v V. Gytenmakhera ng trong s Kvant No2, nm 1994. V di y l mt cch chng minh khc.

nh l 3.

Hm Euler c tnh cht nhn, tc l ( ) ( ) ( )mn m n = vi ,m n nguyn t cng nhau. H qu.

Nu n c dng phn tch chnh tc 1 21 2 ... saa asn p p p= vi 1 2, ,..., sp p p l cc c nguyn t phn bit ca

n v 1 2, ,..., sa a a l cc s t nhin. Th th

1 2 1 1 2 2 11 11 2 1 1 2 2( ) ( ) ( ).... ( ) ( )( )...( )s s s

a a aa a a a a as s sn p p p p p p p p p = =

Chng minh nh l 3

Xt cc s c dng mx ny+ vi 0 , 0x n y m < < . Vit chng thnh mt bng m n . Th d vi 5, 8n m= = , ta c bng sau :

x \ y 0 1 2 3 4 5 6 7 0 0 5 10 15 20 25 30 351 8 13 18 23 28 33 38 43 2 16 21 26 31 36 41 46 51 3 24 29 34 39 44 49 54 59 4 32 37 42 47 52 57 62 67

Cc s trong bng trn phi c d s i mt khc nhau khi chia cho mn . Tht vy nu c

1 1 2 2 (mod ).mx ny mx ny mn+ +

vi 1 2 1 20 , ,0 ,x x n y y m < < . Th th ta c :


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1 1 2 2 (mod )mx ny mx ny n+ + (1) v 1 1 2 2 (mod )mx ny mx ny m+ + (2)

T (1) suy ra 1 2 (mod )ny ny m . Do ,m n nguyn t cng nhau nn 1 2 (mod )y y m . Hn na 1 2,y y m< nn 1 2y y= . Tng t vi (2) ta c 1 2x x= .

Nh vy cc s trn c s d i mt khc nhau khi chia chomn . Hn na tp cc s d ny chnh l tp 0,1,2,..., 1.mn Ni cch khc vi mi 0,1,..., 1d mn= u tn ti cp s ,x y sao cho 0 , 0x n y m < < v (mod )d mx ny mn + . Ta c ( , ) ( , ) ( , ).UCLN mx ny m UCLN ny m UCLN y m+ = = Tng t ( , ) ( , ).UCLN mx ny n UCLN x n+ = C ngha l nhng s trong bng trn nguyn t cng nhau vi m s nm ct m y nguyn t cng nhau vi m , v nhng s nguyn t cng nhau vi n s nm dng m x nguyn t cng nhau vi n . Cc s nguyn t cng nhau vi mn s l giao ca cc dng v cc ct .

iu ny chng t h thc ( ) ( ) ( )m n mn = . Bi tp

39. Vit cc s t 0 n 1mn vo bng sau 0 1 2 1n n 1n + 2n + 2 1n

2n 2 1n + 2 2n + 3 1n

( 1)m n ( 1) 1m n + ( 1) 2m n + 1mn

Chng minh nh l Euler thng qua nhng mnh sau

i. Nhng s nguyn t cng nhau vi n lp y ( )n ct ca bng trn. ii. m s t mt ct bt k ca bng trn c d s i mt khc nhau khi chia cho m . iii. T mi ct c ( )m s nguyn t cng nhau vi m . iv. Mt s nguyn t cng nhau vi mn khi v ch khi s nguyn t cng nhau vi m v n

40. Mt ng trn c phn chia thnh n phn bng nhau bi n im. C bao nhiu ng gp khc kn m cc on ca n u bng nhau v ly cc nh thuc vo tp n im trn.


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Quy c hai ng gp khc c coi l trng nhau nu ng ny s ng nht vi ng kia qua mt php quay.

41. Chng minh rng vi bt k s t nhin ,m n th

a. ( ) ( ) ( [ , ]) ( ( , ))m n BCNN m n UCLN m n = b. ( ) ( [ , ]) ( , )mn BCNN m n UCLN m n = c. ( ) ( ) ( , ) ( ) ( ( , )).m n UCLN m n mn UCLN m n = 42. Gii cc phng trnh

a. ( ) 18x = . b. ( ) 12x = . c. ( ) 12x x = . d. 2 2( )x x x = . e. ( )

2xx = .

f. ( )3xx = .

g*. ( ) , 3xx nn

= > v l s t nhin. h. ( ) ( )nx x = vi s t nhin 1n > .

Mt m vi cha kho m.

Hy tng tng rng bn nhn c mt thng ip m ho t mt ngi bn, nhng anh ta khng th gp bn trc , vy loi m ho g c th s dng c trong trng hp ny. C tn ti phng php m ho no m c th truyn tin khp th gii, thm ch c ngi bn ln k th u nhn c nhng k th hon ton khng th gii m c thng ip ca bn ?

tht s l mt loi m ho tuyt vi, n khc hon ton vi cc loi m ho thng s dng b mt ch yu l cha kho, khi nm m c cha kho th c th m ho hay gii m thng tin d dng. Loi mt m mi cp ti gi l Mt m vi cha kho m , khi m m ho thng ip th ch c tc gi mi c th gii m thng tin nhn c.

Mt m RSA.

Nm 1978, ba nh Ton hc Rivest, Shamir v Adleman m ho mt cu Anh ng v ha s trao gii 100 USD cho ai gii m c thng ip :


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96869613754622061477140922254355882905759991124574319874695120930816298225145708356931476622883989628013391990551829945157815154.y =

H gii thch chi tit phng php m ho. Cc ch ci c quy c 01, 02,..., 26a b z= = = v du cch l 00 . Sau h vit cu thng thng ip nh cc ch s trn thay th cho cc ch ci c sp lin tc thnh mt s x c 78 ch s. Tip theo h s dng mt s nguyn t p c 64 ch s, mt s nguyn t q c 65 ch s . V tch ca chng l :

114381625757888867669325779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541. pq =

V h chn s y l dng m ho ca thng ip nh cng thc :

9007 (mod )y x pq H cng b tch pq , s y v s nguyn t 9007 v chnh phng php m ho v cho bit s nguyn t p c 64 ch s, s nguyn t q c 65 ch s v x c 78 ch s. B mt ch nm hai s

,p q c gi tr bng bao nhiu. iu i hi l tm x tho mn phng trnh ng d trn.

Cu chuyn trn kt thc vo nm 1994, khi m Atkins, Kpaft, Lenstra v Leilang gii m c cu thng ip . V hai s nguyn t h tm c l :

Trong cun M u v L thuyt Mt m xut bn nm 1998 ca cc nh Ton hc ny vit rng : Kt qu k diu ny (s phn tch mt s c 129 ch s thnh nhn t) t c nh mt thut ton phn tch mt s thnh nhn t, c tn gi l phng php Sn bnh phng. Qu trnh thc hin tnh ton l nh vo s cng tc ca c mt i ng ng o. iu hnh d n l bn tc gi ca li gii vi s chun b bc u v l thuyt s khong 220 ngy cng vi s tham gia ca gn 600 ngi v khong 1600 my tnh lin kt vi nhau qua Internet.

ng tic l vic i su vo phng php phn tch ca h vt qu khun kh ca bi vit. Ta chp nhn b qua phn ny v tip tc bn lun v tng h thng mt m RSA ( chnh l cc ch ci u ca cc nh Ton hc pht minh ra loi mt m ny).

tng ny nh sau :

Cho cc s nguyn t ,p q , tnh c ( ) ( 1)( 1)pq p q = . Gi s

1 ( )ef k pq= +

349052951084765094914784961990389813341776463849338784399082057732769132993266709549961988190834461413177642967992942539798288533

pq==


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, ,e f k l cc s t nhin. Vi bt k s t nhin x nguyn t cng nhau vi pq th theo nh l Euler

( )( ) .1 (mod )ef k pqx x x x x pq= = Trong th d ca chng ta th 9007e = , v f tho mn phng trnh ng d 1(mod ( ))ef pq . y s e c chn sao cho phi nguyn t cng nhau vi ( 1)( 1)p q , c th ly 1e = hoc

( 1)( 1) 1e p q= nhng s khng hp l nu mun gi b mt. Khi f tn ti do thut ton Euclid. Do iu kin (mod )ey x pq nn

(mod )f efy x x pq Nh vy s x cn tm l phn d ca fy cho pq . Ti sao mt m RSA li gi l loi mt m vi cha kho m ? l ti v s e v tch pq c ngi m ho thng ip cng khai. Khi m m ho bt k thng ip no th ch cn c mt my tnh c nhn vi mt chng trnh tnh ton no l . Qu trnh gii m s d dng nu bit c s f . Nhng cch duy nht tnh c f th phi bit c gi tr ca p v q , tc l cn phn tch pq thnh nhn t. Thut ton phn tch mt s thnh tha s nguyn t l thut ton c phc tp m nn hi vng c c li gii l khng c hin thc. Ngay c s thnh cng nm 1994 ca bn nh Ton hc vi phng php phn tch ca h ch c hiu lc khi bit s ch s ca p v q , cn nu khng th c h thng lin kt ca 600 con ngi v 1600 my mc qua Internet phi u hng. Bi tp

43*. (Dnh cho cc bn yu lp trnh trn my tnh)

a. Tm s f m nm 1994 bn nh Ton hc Atkins, Kpaft, Lenstra v Leilang tnh ton c. b. Gii m cu Anh ng m nm 1978 c m ho bi Rivest, Shamir, Adleman.

Mt s bi ton ngh.

1. Ch ra s tn ti ca cc hp s n sao cho vi bt k s nguyn a th na a chia ht cho n (gi l cc s Carmichael)

2. Khng tn ti s t nhin n no 2 1n + chia ht cho 1n + . 3. Nu 2 1n + chia ht cho n th 1n = hoc 3n = . Hy chng mnh

iu ny. 4. Cc im c nh s 1,2,..., 1n c th c xp trn mt

ng trn sao cho bt k 3 s , ,a b c lin tip nhau th 2b ac chia ht cho n . Tm cc s n nh vy. (Hnh bn minh ho mt trng hp khi 7n = )


16

5. Vi nhng s nguyn t p no th tn ti s nguyn a sao cho 4 3 2 1a a a a+ + + + chia ht cho p .

(Cn tip k sau)

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Dinh Ly Fermat