Εισαγωγή Στη Θεωρία Αριθµών Για Το Λύκειο

. [email protected]

2012

1

pi pi pi pipi ( ) . pipi pi pipi pi, pi pi pi pi, . , - pipi pi pi pi pi . email [email protected], pi , pi - . pi , pi pi pi pi . , - pi pi pi pi.

. 2012

2

a|b : a b pi k Z, b = k a.pka : pk p pi a. pk

a ( pk|a pk+1 6 | a).a 6 | b : a b .min {a1, . . . , an} : a1, . . . an.max {a1, . . . , an} : a1, . . . an.(a1, . . . , an) : ... a1, . . . an.[a1, . . . , an] : ... a1, . . . an.n! : n pi n! = 1 2 n n 2

0!=1, 1!=1.a b (mod n) : a bmodulo n ( n)

n|(a b).ordn(a) : a modn (a, n) = 1, -

r pi ar 1 (mod n). pi (pi ) ordn(a)|(n).Z : {. . . ,2,1, 0, 1, 2, . . .}.N : {0, 1, 2, 3 . . .}. : pi pi. pi ( ).|a| : pi a |a| =

{a, a 0a, a < 0

. 3

1

1.1

pi () () 6= 0 pi pi(pi) (pipi)

= pi + , 0 <

pipi pipi .

1.1 6= 0, pi ,

= + , 0 < ||.

1.1 = 231 = 26 pi 231 26 231 = 8 26 + 23 pi

231 = 8 26 23= 8 26 26 + 26 23= 9 26 + 3

0 3 < 26 pi 231 26 9 pipi 3.

: pi 231 26 231 26.

2

: pi pi pi pipi, n pipi pi-pi n 0, 1, . . . , n 1. k n, k n + 1, . . . , k n + (n 1). n = 2 pipi 0, 1. = 0 = 2k , = 1 = 2k+1 pi.

1.2 a A =a(a2 + 2)

3 .

pi :pi pipi a 3 0,1,2, a pi

a = 3k a = 3k + 1 a = 3k + 2, k Z.

a = 3k A = 3k[(3k2)+2]3

= k(9k2 + 2) Z.

4

1

a = 3k + 1 A = (3k+1)[(3k+1)2+2]3

= (3k + 1)(3k2 + 2k + 1) Z.

a = 3k + 2 A = (3k+2)[(3k+2)2+2]3

= (3k + 2)(3k2 + 4k + 2) Z.

2

. 5

1.2

1.2

1.1 a b (b 6= 0) , pi-pi . pipi b () a a () pi b a pipi b, b|a a = pio.b.

b|a k Z a = k b.

: b a, b 6 | a a 6= pio.b. pi b|a a = kb pi k Z a = (k)(b) pi b a, b a. pi .

pi pipi :

(i) a|0 a Z,

(ii) 0|b, b = 0,

(iii) a|b a|b a| b |a| | |b|

(iv) 1|a a|a a Z.

(v) b|a, kb|ka, k Z.

pipi , pi .

( pi) -.

1.1 a, b, c, d Z. pi :

(i) a|b b|c, a|c.

(ii) a|b c|d, ac|bd.

(iii) a|b a|b Z.

(iv) a|b a|c, a|b+ c.

(v) a|b b 6= 0, |a| |b|.

(vi) a|b b|a, a = b a = b ( |a| = |b|).

6

1

: pi (iii), (iv) pipi pipi a|b a|c, a|kb + mc, k,m Z. kb + mc b c.

1.3 ( ) pi n pi n.

pi : k, k + 1, . . . , k + (n 1), n pi . A =

k(k + 1) (k + (n 1)). , pi , pi q, r ,

k = nq + r, 0 r n 1. r = 0, n|k, pi pi n|A. r 6= 0 1 n r n 1. pi

A = k(k + 1) (k + n r) (k + n 1)= (nq + r) (nq + r + n r) (nq + r + n 1).

nq + r + n r = n(q + 1), pi n|A.

2

1.4 pi m pi -pi m+ 1|m2 + 1.:pi m + 1|m + 1, pi 1.1 m +

1|m2 +m+ 2. m2 +m+ 2 = m(m+ 1) + 2 m+ 1|m(m+ 1), pi m+ 1|2 pi pi m+ 1 = 1,2 m = 3,2, 0, 1.

2

1.5 ( 1995) 6 - . a pi b . ab = 19951995;

: pi pipi 3,

n n 1, n, n+ 1 3n. pi a, b pipi 3 ab pipi 9. 19951995 pipi 9 9.

2

1.6 ( 1995) pi - x, y pi pi x2 + 4y = 1995.

. 7

1.2

: x pi x = 2k+ 1, k Z x2 = 4k(k+ 1) + 1

x2 =pi.4+1. x x = 2k, k Z x2 = 4k2 x2 =pi.4.

pi 4y pi.4, x2 + 4y = pi.4 x2 + 4y =pi.4+1 1995=pi.4+3 1.

2

1.7 n

9|10n + 3 4n+2 + 5.

pi : pi.

P (n) = 10n + 3 4n+2 + 5. n = 0 P (0) = 54, pi pi 9. pi 9|P (k) 9|10k + 3 4k+2 + 5.

P (k + 1) = 10k+1 + 3 4k+3 + 5 = 10 10k + 3 4 4k+2 + 5= 10k + 3 4k+2 + 5 + 9 10k + 9 4k+2 = P (k) + 9(10k + 4k+2).

9|P (k), pi 1.1 9|P (k + 1). pi 9|P (n) n N.

2

1.8 n Z 4 6 | n2 + 2.pi : pi, , pi n 4|n2 + 2.

pipi n :

n = 2k, pi k Z, n2 + 2 = 4k2 + 2. 4|n2 + 2, pi 4|4k2 + 2, 4|2, pi.

n = 2k + 1, pi k Z, n2 + 2 = (2k + 1)2 + 2 = 4k2 + 4k + 3.pi 4|4k2 + 4k, pi 4|3, pi.

n Z 4 6 | n2 + 2.2

1 pi pi ( pi pi).

8

1

1.3 (...)

1.2 ( ) S pi N. S , , a S , a x, x S.

a1, . . . , an pi pi 6= 0. pi pi a1, . . . , an a1, . . . , an. S a1, . . . , an. S 1 S. ak 6= 0 d S d|ak pi d |ak|. S pipi. S pi (...) a1, . . . , an (a1, . . . , an). a Z, a pipi a. pi (a1, . . . , an) =(|a1|, . . . , |an|), ... pi. pi, 0, (0, a1, . . . , an) = (a1, . . . , an). pipi pi a1, . . . , an .

(a1, . . . , an) = 1, a1, . . . , an pi .pi (ai, aj) = 1 i, j {1, . . . , n} i 6= j, a1, . . . , an pi . pi a1, . . . , an pi , pi . .

1.2 ( Bezout) a1, . . . , an d =(a1, . . . , an). pi k1, . . . , kn ,

d = k1a1 + + knan.

1.1 a1, . . . , an . d ... a1, . . . , an , :

(i) d|a1, . . . , d|an,

(ii) |a1, . . . , |an, |d.

1.2 a1, . . . , an . d a1, . . . , an d = k1a1 + + knan, pi k1, . . . , kn Z, d = (a1, . . . , an).

1.3 a1, . . . , an . a1, . . . , an pi , , pi k1, . . . , kn Z 1 =k1a1 + + knan.

. 9

1.3 (...)

1.9 a, b pi .

(9a+ 7b, 4a+ 3b) = 1.

pi : d ... 9a+ 7b 4a+ 3b. d|9a+ 7b d|4a+ 3b.

pi d|4(9a+7b)9(4a+3b) d|3(9a+7b)7(4a+3b), pi pi pi d|b d|a . pi, 1.1 d|(a, b) pi pi d|1. pid = 1.

2

1.3 , a1, . . . , an . :

(i) (a1, . . . , an) = ||(a1, . . . , an),(ii) (a1, . . . , an) = d,

(a1d, . . . , an

d

)= 1,

(iii) (a1, . . . , an) = (a1 + k2a2 + + knan, a2, . . . , an), pi k2, . . . , kn Z.

1.10 a, b pi , (a+ b, a b) = 1 2.pi : d = (a + b, a b). d|a + b d|a b. pi

d|(a+ b) + (a b) d|(a+ b) (a b), d|2a d|2b pi d|(2a, 2b) 1.3(i) pi (2a, 2b) = 2(a, b) = 2 d|2 pi d = 1 2.

2

1.4 a1, . . . , an n > 2. k, 1 k n 2

(a1, . . . , an) = (a1, . . . , ak, (ak+1, . . . , an)) .

: pipi pi ... pipipi pi ... .

1.5 ( ) a, b, c . a|bc (a, b) = 1 a|c.

1.11 ( 2000) pi pi - x, y z pi

5x 3y + 10z = 0 y(x+ 2z) = 2004.

pi :

10

1

pi

5x 3y + 10z = 0 (1) y(x+ 2z) = 2004.

(1) : 5x + 10z = 3y 5(x + 2z) = 3y. pi 5|3y pi(5, 3) = 1 5|y. 5|y(x+ 2z) 5|2004 ( y(x+ 2z) = 2004). , pi pi .

2

. 11

1.4

1.4

pi ... .

pi a, b Z, , b > 0, b < 0, (a, b) = (a, |b|), b = 0, (a, b) = |a|. d := (a, b).

pi pi q r , a = q0b+ r0 pi 0 r0 < b.

(a, b) = (b, r0), pi d | a d | b, pi d | r0 = a q0b. b r0 d d, d a b, pi pi d . pi, d = (b, r0).

pi pi , b r0. - pi

a = q0b+ r0 pi 0 r0 < bb = q1r0 + r1 pi 0 r1 < r0r0 = q2r1 + r2 pi 0 r2 < r1r1 = q3r2 + r3 pi 0 r3 < r2

...

pi pi b > r0 >r1 > r2 > . . . , pi pipi pi 0. pi n- . rn = 0 .pi pi pi d = (rk1, rk) = (rk, rk+1) k = 0, 1, 2, . . ., pi r1 = b. pi, d =(rn1, rn) = (rn1, 0) = rn1.

, ... pipi pi-pi .

pipi pipi pi ... pi ... d, a b, Bezout. ( 1.2).

12

1

pi pi pi,

a q0b = r0b q1r0 = r1r0 q2r1 = r2r1 q3r2 = r3

...rn3 qn1rn2 = rn1

rn2 = qnrn1

pi, pipi rk pi pi

b q1(a q0b) = k1a+ l1b = r1(a q0b) q2(k1a+ l1b) = k2a+ l2b = r2

(k1a+ l1b) q3(k2a+ l2b) = k3a+ l3b = r3... ...

(kn3a+ ln3b) qn(kn2a+ ln2b) = kn1a+ ln1b = rn1

1.12 ... 391 323, - ... 391 323.

: 391 = 1 323 + 68323 = 4 68 + 5168 = 1 51 + 1751 = 3 17 + 0.

pi (391,323)=17. x, y , 17 =391x+ 323y.

17 = 68 51 = 68 (323 4 68) = 323 + 5 68= 323 + 5(391 323) = 5 391 6 323

pi 17 = 5 391 + (6) 323. 1.13 ... a = 756 b = 595. pipi, r pi pipi pi pi , q pi k, l - ki li pi pi pipi. pi,(756, 595) = 7 37 756 47 595 = 7.

. 13

1.5 pi (...)

r q k l756 1 0595 1 0 1161 3 1 1112 1 3 449 2 4 514 3 11 147 2 37 470

1.5 pi (...)

a1, . . . , an Z. pipi a1, . . . , an a1|, . . . , an|. pi a1, . . . , an 0, pipi 0. pi a1, . . . , an . |a1 an| pipi a1, . . . , an. pi, ( 1.2) pipi a1, . . . , an , pi . pi (...) a1, . . . , an [a1, . . . , an]. pi-pi a1, . . . , an |a1|, . . . , |an|, pi [a1, . . . , an] = [|a1|, . . . , |an|].

1.6 a1, . . . , an . m ... a1, . . . , an, ,

(i) a1|m, . . . , an|m,(ii) a1|, . . . , an|, m|.

1.7 , a1, . . . , an . :

(i) [a1, . . . , an] = ||[a1, . . . , an],

(ii) [a1, . . . , an] = m (ma1, . . . , m

an

)= 1.

1.8 a1, . . . , an n > 2. k, 1 k n 2

[a1, . . . , an] = [a1, . . . , ak, [ak+1, . . . , an]] .

: pipi pi ... pipipi pi ... .

14

1

1.6

1.2 p > 1 pi 1 p. pi pi m pi pi pi m. n > 1 pi pi, . pipipi d, e ,

n = d e 1 < d e < n.( 2 pi ).

1.9 a > 1 pi .

1.14 n pi 3n 4, 4n 5, 5n 3 pi .: 3 , pi pi

. pi 2. 3n4 5n3 pi . pi 3n 4 = 2 5n 3 = 2 pin = 2 n = 1 . n = 2 3 pipi pi .

2

1.15 (AHSME 1976) p q pi x2 px+ q = 0 , p q.: x1 x2 x1 < x2, .

x2 px+ q = (x x1)(x x2), pi p = x1 + x2 q = x1x2. q pi, x1 = 1. pi q = x2 p = x2 + 1 pi , q = 2 p = 3.

2

1.16 (ARML 2003) 1001001001 pi pi 10000.

:

1001001001 = 1001 106 + 1001 = 1001 (106 + 1) = 7 11 13 (106 + 1).

x6 + 1 = (x2)3 + 1 = (x2 + 1)(x4 x2 + 1).

. 15

1.6

106 + 1 = 101 9901, 1001001001 = 7 11 13 101 9901. 7,11,13 101 pi pi 9901 1000, pi 9901.

2

1.17 n . pi 32n

+ 1 pi 2 pi 4 2.

pi :, 32

n pi 32

n+ 1 . pi,

32n

= (32)2n1

= 92n1

= (8 + 1)2n

.

pi

(x+ y)m = xm +

(m

1

)xm1y +

(m

2

)xm2y2 + +

(m

m 1)xym1 + ym,

x = 8, y = 1 m = 2n1, pi (pi ym = 1), pipi 8 ( pi pipi 4).pi pipi 32

n 4 1, pipi

32n

+ 1 4 2.

: pipi pi pipi modulo 4 (pi pi).

2

1.18 n 2n31024 1.: pi 12. 1024 = 210 x2y2 = (xy)(x+y).

,

3210 1 =

(32

9

+ 1)(

329 1

)=(

329

+ 1)(

328

+ 1)(

328 1

)= =

(32

9

+ 1)(

328

+ 1) (

321

+ 1)(

320

+ 1)

(3 1)

pi pi (1.17), 232k + 1 k. pi pi 9+2+1=12.

2

2 232n + 1.

16

1

pi pi pi pi .

1.10 pi 6k + 1 6k + 5.

1.3 pi pi pi.

pi: 3

pi p1, . . . , pn pi .

A = p1 pn + 1. 1.9, pi pi p , p|A. p1, . . . , pn pi, p = pj pi j 1 j n. pip|A p|p1 pn pi pi pi p|1 pi pi. pi, pi pi pi.

2

1.11 pn n- pi , (pi pi)

pn 22n1 . 1.19 n > 1 pi n , pi pi pi .

pi : n

(n+ 1)! + 2, (n+ 1)! + 3, , (n+ 1)! + (n+ 1). pi pi, m = 2, 3, , n+1, (n+ 1)! +m m 4.

: pi pipi pi pipi pi pi , .

, pi pi pi pi (pi ) pi pi . pipi pi . pi.. pipipi pi 5 pi pi- 6!+2, 6!+3, 6!+4, 6!+4, 6!+6 722, 723,724, 725, 726. 5 pi pi 24,25,26,27,28 ( pi pi pi pi pipi pipi ).

3 pi pi pi pipi .

4pi (n+ 1)! (n+ 1), . . . , (n+ 1)! 3, (n+ 1)! 2 pi.

. 17

1.6

2

pi pi - pi.

1.12 a > 1, pi p, p a. 1.4 a > 1 pi pi p, p n, a pi. 1.20 383 pi. 19 1 - pi pi, pi pi .

1.3 pipi , a > 1, pi pi p1, . . . , pk a1, . . . , ak > 0 ,

a = pa11 pakk . pipi a pi a.

1.14 a > 1 a = pa11 pakk pi . d a, , d = p11 pkk 0 i ai (i = 1, . . . , k). 1.21 ( 1995) pi x, y pi pi x2 = y2 + 2y + 9.

:x2 = y2 + 2y + 9 x2 (y + 1)2 = 8 (x y 1)(x+ y + 1) = 8.

x y, x + y pi, x y1, x+ y+ 1 pi pi, pi. x y pi x + y pi. x + y + 1 pipix y 1 . (x y 1, x+ y + 1) = (2, 4) (4, 2) pi pi pi (x, y) = (3, 0).

18

1

2

1.22 ( 1997) a, b (a, b N) -,

a3 + 1

b+ 1+b3 + 1

a+ 1 N.

pi pi a3 + 1

b+ 1,b3 + 1

a+ 1 .

pi : (a+ 1, b+ 1) = d. a+ 1 = d p1p2 pk b+ 1 = d q1q2 ql, pi

pi, qj pi pi 6= qj, i = 1, 2 . . . , k j = 1, 2, . . . , l pi qj .

a3 + 1

b+ 1+b3 + 1

a+ 1=

(a+ 1)(a2 a+ 1)b+ 1

+(b+ 1)(b2 b+ 1)

a+ 1

=d p1p2 pk(a2 a+ 1)

d q1q2 ql +d q1q2 ql(b2 b+ 1)

d p1p2 pk=

p21p22 p2k(a2 a+ 1) + q21q22 q2l (b2 b+ 1)

q1q2 qlp1p2 pk N

qj q21q22 q2l (b2 b+ 1)

p21p22 p2k(a2 a+ 1) + q21q22 q2l (b2 b+ 1),

p21p22 p2k(a2 a+ 1).

qj p21p22 p2k qj a2 a+ 1

a2 a+ 1q1q2 ql N

d(a2 a+ 1)d q1q2 ql N

d(a2 a+ 1)b+ 1

N

p1p2 pk d(a2 a+ 1)b+ 1

N

(a+ 1)(a2 a+ 1)

b+ 1 N

a3 + 1

b+ 1 N

a3 + 1

b+ 1+b3 + 1

a+ 1 N b3+1

a+1 N.

2

. 19

1.7 ... ..

1.7 ... ..

1.15 a1, . . . , an

|a1| = pa111 pa1kk , . . . , |an| = pan11 pankkpi p1, . . . , pk pi aij (i = 1, . . . , n, j = 1, . . . , k).

(a1, . . . , an) = pd11 pdkk ,

pi dj = min {a1j, . . . , anj} (j = 1, . . . , k).

1.5 a1, . . . , an m N. (am1 , . . . , a

mn ) = (a1, . . . , an)

m.

1.16 a, b1, , bn (n 2) b1, , bnpi .

(a, b1, . . . , bn) = (a, b1) (a, bn).

1.6 a, b1, . . . , bn (n 2) b1, . . . , bnpi . b1|a, . . . , bn|a b1 bn|a.

1.23 n pi > 1.

24|n(n2 1).

pi : A = n(n2 1) = (n 1)n(n + 1)

pi pipi 3 3|A. pi n pi > 1, pi k N k 6= 0 , n = 2k + 1. pi

A = 4k(k + 1)(2k + 1).

pi k, k + 1 8|A. , (3, 8) = 1, pi (3.1) 24|A.

2

1.17 a1, . . . , an

|a1| = pa111 pa1kk , . . . , |an| = pan11 pankkpi p1, . . . , pk pi aij (i = 1, . . . , n, j = 1, . . . , k).

[a1, . . . , an] = pc11 pckk ,

pi cj = max {a1j, . . . , anj} (j = 1, . . . , k).

20

1

1.7 a1, . . . , an m N.

[am1 , . . . , amn ] = [a1, . . . , an]

m.

1.24 pi (1.15), (1.17) pi ... ... pi pipi pi pi . ... pi pi pi pi ( pi pi pi , 0, pi pi pi). ... pi pi pi pi . , ... 49000 = 23 53 72, 36400 = 24 52 7 13, 27500 = 22 54 11 22 52 70 110 130 = 100. ... 24 54 72 11 13 = 70070000.

1.18 a, b .

(a, b) [a, b] = |ab|.

: pi pi , pipi. ,

(a1, . . . , an) [a1, . . . , an] 6= |a1 an| n > 2.

pi, (6, 8, 10) [6, 8, 10] = 2 120 = 240 6= 480 = 6 8 10.

. 21

2

2.1

2.1 n . a, b pipi n, n pipi.

a b (mod n) a pipi b n. a pipi b n,

a 6 b (mod n)

.

pi 10 2 (mod 4), 11 15 (mod 13) 7 6 11 (mod 5).

2.1a b (mod n) n|a b

pi pi .

2.1 (i) a a (mod n) ( ).(ii) a b (mod n), b a (mod n) ( ).

(iii) a b (mod n) b c (mod n), a c (mod n)( ).

2.2 (i) a 0 (mod n) n|a,(ii) a , , a 0 (mod 2),

(iii) a pi, , a 1 (mod 2),(iv) a b (mod n) m|n a b (mod m),(v) a, b a b (mod 1),

(vi) pipi a n , a (mod n).

2.1 a, b, c, d Z f(x) pi -. a b (mod n) c d (mod n),

(i) a+ c b+ d (mod n) ac bd (mod n),(ii) am bm (mod n), m N,

22

2

(iii) f(a) f(b) (mod n). 2.2 a, b, k Z k 6= 0 d = (n, k).

ka kb (mod n) a b (mod nd

).

pi .

pi, modulo 12 pi modulo 100.000. , 18, 6, pi pipi 18 12, 245.000 Km, 45.000 Km, pi pipi 245.000 100.000.: pi pi 217

pi ; (modulo) pi ; pi pi ;

2.1 pi pipi A =13232741 8.

:

132 = 169 9 1 (mod 8).pi

1323 = 13211+1 = 13 (132)11 13 5 (mod 8).pi 27 3 (mod 8), pi pi 272 9 1 (mod 8). pi

2741 = 27220+1 = 27 (272)20 27 3 (mod 8). A 15 7 (mod 8) pi pi a Z , A = 8a + 7.pi pipi 7.

2

2.2 22225555 + 55552222 0 (mod 7).pi : 2222 3 (mod 7) 5555 4 (mod 7). 22225555 + 55552222 35555 + 42222 (mod 7).pi

35555 = (35)1111 = (2)1111 = 21111 (mod 7)

42222 = (42)1111 21111 (mod 7) pi , pi pi .

. 23

2.1

2

2.3 N, pi A = 45+ 3 -.

pi : a . pi a 0, 1, 2, 3, 4 (mod 5)

a2 0, 1, 4 (mod 5) a4 0, 1 (mod 5).pi 5 + 3 3 (mod 5), pi 5 + 3 a4

a Z. A . ( A4 = 5+ 3 3 (mod 5), pi).: ( 1997) pi a =

5n2 + 10

n Z.2

2.4 () pi 121 n2 +3n+ 5 n Z.pi : pi 121|n2+3n+5. 11|n2+3n+5. n2+3n+5 0 33

(mod 11). n2 + 3n 28 0 (mod 11) 11|(n 4)(n + 7) 11|n 4 11|n+ 7 (n = 11k+ 4 n = 11k7) n n2 + 3n+ 5, pi 121+ 0 < < 121 pi pi.

2

2.5 ( 1997) a, b, c ,

(a b)(b c)(c a) = a+ b+ c. pi a+ b+ c 27.

pi : a, b, c a, b, c 0,1 (mod 3).

pipi 3, a + b + c 1 +0 + 1 = 0 (mod 3) (a b)(b c)(c a) 6 0 (mod 3). pipi, a, b a b 0 (mod 3) (a b)(b c)(c a) 0 (mod 3) a + b + c 6 0 (mod 3). pi a, b, c pipi mod3. a b b c c a 0 (mod 3), (a b)(b c)(c a) 0 (mod 27). (a b)(b c)(c a) = a+ b+ c pi a+ b+ c 0 (mod 27) pi .

2

2.6 ( 1995) pi pi p, q pi pp+1 + qq+1 pi.

:

24

2

p, q pi, pp+1 + qq+1 > 2 pi. 2 pi. pi pi p = 2 q 0, 1,1(mod 3) pi. q 1 (mod 3), pp+1 + qq+1 8 + 1 0 (mod 3),pi. q 1 (mod 3), pp+1 + qq+1 8 + 1 0 (mod 3) ( q pi q + 1 ), pi. , q 0 (mod 3) pi q pi q = 3 pp+1 + qq+1 = 89 pi pi. p = 2, q = 3 ( p = 3, q = 2).

2

2.7 ( 1994) pi pi x3 +1995x2 1994x+ ;: pi 1, 2, 3. pi

pi V ieta

A = 1 + 2 + 3 = 1995 0 (mod 3) (1)

B = 12 + 23 + 31 = 1994 1 (mod 3).

(1) :

1, 2, 3 0 (mod 3) 1, 2, 3 1 (mod 3) 1, 2, 3 1 (mod 3) 1 0 (mod 3), 2 1 (mod 3), 3 1 (mod 3) ( pipi pipi ). pi B B 0 (mod 3) B 1 (mod 3), B 1 (mod 3). pi pi .

2

- : a (mod 10n) 0 < 10n, n a , n. pi, pi 992007 + 3 (1)2007 + 3 = 2 (mod 102), 992007 + 3 02.

2

2.8 ( 1988) 270.: pi 224 pi 223; pi

223 pi 2. pipi- 2 pi pi . ,

. 25

2.1

, pi .

02, [04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52] pi 20

,

04, 08, 26, 32 . . .

pi pi 20.

270 = (220)3 210,pi 270 210, 24.

: (pi ) pi pi pi pi pi 4, [2, 4, 8, 6], 2, 4, . . .

270 = (24)17 22.

270 22 4.

2

: ( 1989) 61989.

2.9 (pi )

7777...7

1001 7: pi 74 1 (mod 10). pi,

72k 1 (mod 4) 72k+1 3 (mod 4)

a := 7777...7

1000 7 3 (mod 4) pi a = 3k + 4, k Z.

,

7777...7

1001 7= 7a = 74k+3 = 73 (74)k 73 3 (mod 10).

2

26

2

2.10 ( 1994) pi pi pi 4 1994 1993.

pi :

anan1an2 . . . a4 A

1994 = 10000A+ 1994 = A(5 1993 + 35) + 1993 + 1

35 A+ 1 (mod 1993)

35 A+1 0 (mod 1993) pipi 35A+1 = 1993k, k Z A = 57k 2k + 1

35(1) pipi 2k + 1 pipi 35.

k = 17 A = 968. pi 9681994 pipi 1993.pi pi (1) pi .

2

2.11 pi n N 17|34n+2 + 2 43n+1.

pi : 34 = 81 13 (mod 17). pi

34n+2 = 9 81n 9 13n (mod 17).pi 43 = 64 13 (mod 17), pi 43n+1 = 4 (43)n 4 13n (mod 17).

34n+2 + 2 43n+1 9 13n + 8 13n = 17 13n 0 (mod 17).

: pipi pi pi pi pi.

2

2.12 ( 1990) an a1 = 1, a2 = 3 an+2 = (n+ 3)an+1 (n+ 2)an, n pi 11|an.:

an+2 an+1 = (n+ 2)(an+1 an)an+1 an = (n+ 1)(an an1)

...

a3 a2 = 3(a2 a1)a2 a1 = 2

. 27

2.2 pipi

pipi pipi pipi

an+2 an+1 = (n+ 2)!

a1 = 1, a2 = 3, a3 = 9 11 . pi a4 = 33 11|a4.a5 = a4 + 5! 11 6 | 5! 11 6 | a5. 11 6 | a6 11 6 | a7.

a8 = a4 + 5! + 6! + 7! + 8! = a4 + 5!(1 + 6 + 6 7 + 6 7 8)= a4 + 5!(7 7 + 6 7 8) = a4 + 5!7 55,

11|a8. pi 11 6 | a9 11|a10

a10 = a8 + 9! + 10! = a8 + 9!(1 + 10).

a11 = a10 + 11!, 11|a11. n N n 11 11|n! pi 11|an

an = a10 + 11! + 12! + + n!

, 11|an n = 4, n = 8 n 10.

2

pi 1.10 pi

2.3 ( )

(i) p 6= 3 pi p2 1 (mod 3).(ii) p 6= 2 pi p2 1 (mod 8).

(iii) p > 3 pi p2 1 (mod 12).(iv) pi p > 3 p 1 (mod 6) (pi

1.10).

2.2 pipi

2.2 n a1, . . . , an pi pi-pi modn, pi a1, . . . , an pipi {0, 1, . . . , n 1} 5.

5 a1, . . . , an pi pipi n {0, 1, . . . , n 1}.

28

2

pipi modn pi pi :

(i) pipi modn: {0, 1, . . . , n 1}.(ii) pipi modn: {1, . . . , n}

(iii) pi pipi modn:

n pi {n 1

2, . . . ,1, 0, 1, . . . , n 1

2

} n , {

(n

2 1), . . . ,1, 0, 1, . . . , n

2

} 2.13 S = {14, 24, 9,11, 34, 68,21, 87} pipipi mod8. ,

14 6 (mod 8), 24 0 (mod 8), 9 1 (mod 8), 11 5 (mod 8),34 2 (mod 8), 68 4 (mod 8), 21 3 (mod 8), 87 7 (mod 8). {0, 1, . . . , 7} pi pipimod8

S pi pi pipi mod8.

2

2.14 T = {1, 22, 32, . . . , n2} pi pipi modn.

pi : (n 1)2 1 = n2 2n, (n 1)2 1 (mod n) pi

T ( (n 1)2 1) pipi {0, 1, . . . , n} ( 1).

2

2.15 ( 1990) pi a7 a. a , pi pi 42.

pi :a7 a = a(a6 1) = a(a3 1)(a3 + 1) = a(a 1)(a+ 1)(a2 + a+ 1)(a2 a+ 1).

a a, a + 1 pipi 2. pi a 1, a, a + 1 pipi 3 pi (2, 3) = 1 6|a(a 1)(a + 1). pi (6, 7) = 1 pi pi 7. pipi

. 29

2.2 pipi

a. a = 7k, pipi 7. a = 7k + 1, 7k 1 7|a 1, 7|a+ 1 , pi. a = 7k+2 a = 7k3 a2 +a+1 pipi 7 a = 7k + 3 a = 7k 2 a2 a+ 1 pipi 7. ( pi {3,2,1, 0, 1, 2, 3} pi pipi mod7).

2

2.4 {x0, x1, . . . , xn1} pi pipi modn a, b Z (a, n) = 1. {ax0 + b, . . . , axn1 + b} pi pi pipi modn.

30

2

2.3 Wilson - Fermat Euler

pi p pi.

2.2 ( Wilson) p > 1 pi, ,

(p 1)! 1 (mod p).

2.3 pi p

(p 2)! 1 (mod p).

2.16 0 < s < p, pi p pi , pi

(s 1)!(p s)! + (1)s1 0 (mod p).

pi : s = 1 Wilson. pi

(s 2)!(p (s 1))! + (1)s2 0 (mod p)

pi

(s 2)!(p s+ 1)! + (1)s2 0 (mod p) (s 2)!(p s)!(p s+ 1) + (1)s2 0 (mod p) (s 2)!(p s)!p (s 2)!(p s)!(s 1) + (1)s2 0 (mod p) (s 2)!(s 1)(p s)! + (1)s2 0 (mod p) (s 1)!(p s)! (1)s2 0 (mod p) (s 1)!(p s)! + (1)s1 0 (mod p)

s 0 < s < p.

: s = p. ,

(p 1)!0! + (1)p1 = (p 1)! + 1 0 (mod p).

2

2.5 p pi pi. [(p 1

2

)!

]2=

{ 1 (mod p), p 1 (mod 4)1 (mod p), p 3 (mod 4)

. 31

2

2

() Fermat Euler.

2.4 ( Euler) n > 1 a , (a, n) = 1.

a(n) 1 (mod n).

2.19 pi pipi 106k+4, pi k N, 7.: (10, 7) = 1, Fermat 106 1 (mod 7), pi pi 106k 1

(mod 7). pi104 34 = 92 22 = 4 (mod 7).

106k+4 4 (mod 7)

pi pipi 4.

2

2.20 7 19681968 3 6878

10 .

pi :

10|7 19681968 3 6878. Fermat 34 1 (mod 5). pi

19681968 31968 = (34)492 1 (mod 5).pi,

6878 378 = 9 (34)19 9 4 (mod 5),pi

7 19681968 3 6878 7 3 4 = 5 0 (mod 5).

5|7 19681968 3 6878. 7 19681968 3 6878 (2, 5) = 1 pi

10|7 19681968 3 6878.

. 33

2

p = 5 pi . pi p 7. 5p 4p 1 0 (1)p 1 = 0 (mod 5)5p 4p 1 1p 0 1 = 0 (mod 4) , 2.4, pi 5p 5 (mod p) 4p 4 (mod p) 5p 4p 1 5 4 1 = 0 (mod p) pi (4, 5, p) = 1 pi 20p|5p 4p 1.: (2 ... 1989) p pi pi

42p|3p 2p 1. (pi : 7|3p 2p 1, pi 1.10).

2

2.24 p 7 pi. pi

11 . . . 1 p1

pi p.

pi :

11 . . . 1 p1

=10p1 1

9

pi pipi pi Fermat 6.

2

2.25 p pi p > 5. pi p8 1(mod 240).

pi : pi 240 240 = 24 3 5. pi

Fermat, p2 1 (mod 3) p4 1 (mod 5). pi pi pi 24 pi (24) = 23 Euler, p8 1 (mod 16). pi p8 1 (mod m) m = 3, 5, 16 pi ... 240 p8 1 (mod 240).: n4 1 (mod 16) n 1,3,5,7

(mod 16). pi pi pi p4 1(mod 240) pi p > 5.

2

6 (10, p) = 1

. 35


2.26 pi n

n2 1|2n! 1.

pi : m = n + 1. m(m 2)|2(m1)! 1. pi

(m)|(m 1)!, 2(m) 1|2(m1)! 1 pi Euler m|2(m)1. , pipi m|2(m1)!1. ,m2|2(m1)!1 pi m pi, pi (m,m 2) = 1 pi.

2

2.27 p pi 3k+2 pi a2+ab+b2

pi a, b. pi a, b pi p.

pi : pi p a. pi p|a2 + ab+ b2, p

a3 b3 = (a b)(a2 + ab+ b2) pi a3 b3 (mod p). a3k b3k (mod p) (1)

pi p b. pi Fermat ap1 bp1 1 (mod p),

a3k+1 b3k+1 (mod p) (2)

pi p pi pi a, (1), (2) pi a b (mod p). a2 + ab + b2 0 (mod p) 3a2 0 (mod p)., p 6= 3 p|a, pi.

2

pi pi pipi pi, 3 - 13 pi 1996.: p pi 4k + 3 (k N). x, y Z

p|x2 + y2, pi p|x p|y.2

2.28 ( pi 2005) - a1, a2, . . . pi pi

an = 2n + 3n + 6n 1

n. pi pi pi .

36

2

: pi 1. pi p an

pi n. p = 2 p = 3 a2 = 48.

pi p 5. pi Fermat 2p1 3p1 6p1 1 (mod p).

3 2p1 + 2 3p1 + 6p1 3 + 2 + 1 0 (mod 6)

6(2p2 + 3p2 + 6p2 1) 0 (mod p), p|6ap2. pi (p, 6) = 1, ap2 pi p pi .

2

. 37

2.4

2.4

2.3 n > 1 a, b .

ax b (mod n),pi x pi , . x0 pi pi pipi

ax0 b (mod n).

: pipi, y x0 y (mod n), pipi . , ax b(mod n) pipi y x0 y (mod n) pi x0 pipi pi pipi ., pi pi . .. 6x 1(mod 8) (8|6x 1, pi 6x 1 pi).

2.29 pi 2x 10(mod 6).

: 10 4 (mod 6), pi 2x 4

(mod 6). 0, 1, 2, 3, 4, 5 pi pi pipimod6 pi :

2 0 = 0 6 4 (mod 6), 2 1 = 2 6 4 (mod 6), 2 2 4 (mod 6)

2 3 = 6 6 4 (mod 6), 2 4 = 8 6 4 (mod 6), 2 5 = 10 4 (mod 6).pi x 2, 5 (mod 6).: pi pi pi-

pi pi pi n pi pipi. pi .

2.4 a modn ( pi) b pi ab 1 (mod n). a1 1a.

pi 3 mod7 5 3 5 1 (mod 7).

2.7 a Z a 6= 0. pi a, , (a, n) = 1.

2.8 (a, n) = 1. ax b (mod n) .

38

2

2.30 pi 137x 4(mod 102)

: 137 35 (mod 102), pi 35x

4 (mod 102). (102, 35) = 1 ( pipi ) 12 102+35 35 = 1.pi 35 35 1 (mod 102). pi 35 mod102, , pipi 35x 4(mod 102) 35, pi x 4 35 38 (mod 102).

2

: (a, n) = 1 pi amodn pi : r ar 1 (mod n) (pi.. r = (n)). a modn, ar1 modn. pi (a, n) = 1, ax b(mod n) x bar1 (mod n).

2.31 7x 8 (mod 30).: (7, 30) = 1, . 7(30) 1

(mod 30). (30) = 8 7 mod30 7, (30) = 8. pi 72 = 49 19 11 (mod 30) 74 (11)2 = 121 1(mod 30). ord30(7) = 4 , pipi 73 pi

x 738 (77)8 (17)8 13 8 = 104 14 (mod 30).

2

2.5 ax b (mod n) d|bpi d = (a, n). , x0 pi , pi d

x x0, x0 + nd, x0 + 2

n

d, . . . , x0 + (d 1)n

d(mod n)

2.32 pi

21x 6 (mod 33).

:

7 (a, n) = 1 a modn r pi ar 1 (mod n). ordn(a) pi (pi ) ordn(a)|(n).

. 39

2.5

(21, 33) = 3 3|6. pipi , 3 . x pi pipi 7x 2 (mod 11). 0,1,. . .,10 pi pi pipimod11. pi pipi pi 5 pi. 7x 2(mod 11) x 5 (mod 11). 5 pi 21x 6 (mod 33).pi, pipi 21x 6(mod 33) x 5, 16, 27 (mod 33).

2

2.33 pi

2086x 1624 (mod 1729).

: 2086 357 (mod 1729) 1624 105 (mod 1729), pi

pipi 357x 105 (mod 1729). pi (357, 1729) = 7 pi 7 =19 1729 92 357. pi 92 357 7 (mod 1729). pi 15 pi (9215)357 105 (mod 1729). pipi 7

x 349, 596, 843, 1090, 1337, 1584, 102 (mod 1729).

2

2.5

2.5

a1x b1 (mod n1)...

akx bk (mod nk) pi pi pi . pi, 3x 9 (mod 10), 2x 1 (mod 5) 3. , pipi, pi pi pi pi . a modc pi pipi a modc. .

40

2

2.6 ( pipi pipi -) b1, . . . , bk n1, . . . , nk > 1, pi .

x b1 (mod n1)x b2 (mod n2)...

x bk (mod nk)

modn1 nk, pi : Nj = n1 nj1nj+1 nk Mj

Njx 1 (mod nj) 8 ( pi ). x0 (mod n1 nk), pi

x0 = b1N1M1 + + bkNkMk.

2.34

x 2 (mod 5), x 3 (mod 7), x 4 (mod 11).

:pi 5, 7, 11 pi , -

pipi, pipi mod385.

N1 = 7 11 = 77, N2 = 5 11 = 55, N3 = 5 7 = 35.

, pi

77x 1 (mod 5), 55x 1 (mod 7), 35x 1 (mod 11),

2x 1 (mod 5), 6x 1 (mod 7), 2x 1 (mod 11),

pi

x 3 (mod 5), x 6 (mod 7), x 6 (mod 11),

. pi,

x0 77 3 2 + 55 6 3 + 35 6 4 = 2292 367 (mod 385).

2

8 n1, . . . , nk pi (Nj , nj) = 1 j =1, . . . , k, modnj .

. 41

2.5

pipi pi

2.7

x b1 (mod n1)x b2 (mod n2)...

x bk (mod nk) , , (ni, nj)|bi bj i, j i 6= j. x0 , x0 (mod [n1 nk]).

pi pi pi pi pi pipi .

2.35

x 1 (mod 15), x 7 (mod 18).

: (15, 18) = 3 3|1 7. pi [15, 18] = 90. pi,

pipi mod90. pi, pi .

x = 1 + 15y pi

1 + 15y 7 (mod 18).

pi 15y 6 (mod 18) pi pi 5y 2 (mod 6). pi 4 pi pipi . pi

x = 1 + 15 4 61 (mod 90).2

2.36

x 3 (mod 8), x 11 (mod 20), x 1 (mod 15).

: (8, 20) = 4, (8, 15) = 1, (15, 20) = 5 4|3 11, 1|3 1, 5|1 11.

pi [8, 20, 15] = 120. pipi pipi (mod 120).

pi

x 3 (mod 8), x 11 (mod 20), [8, 20] = 40.

42

2

x = 3 + 8y

3 + 8y 11 (mod 20),

pi piy 1 (mod 5),

x 3 + 8 = 11 (mod 40).

x 11 (mod 40), x 1 (mod 15).

x = 11 + 40y pi

11 + 40y 1 (mod 15).

pi y 2 (mod 3).

x 11 + 2 40 = 91 (mod 120).

2

2.7

aix bi, i = 1, . . . , k.

pi

a1x b1 (mod n1)...

akx bk (mod nk). , pipi pi , pi di|bi pi di = (ai, ni), i = 1, . . . , k. pi di|bi, i = 1, . . . , k. pi Ai, Bi, Ni Z (Ai, Ni) = 1 ai = diAi, bi = diBi, ni = diNi, i = 1, . . . , k. pi pipi

A1x B1 (mod N1)...

Akx Bk (mod Nk).

. 43

2.5

(Ai, Ni) = 1, Aix Bi (mod Ni) x Ci (mod Ni), i = 1, . . . , k. ,

x C1 (mod N1)...

x Ck (mod Nk). pi pi , pi pi pi-.

2.37

8x 4 (mod 20), 15x 10 (mod 35), 9x 12 (mod 39).

: (8, 20) = 4, (15, 35) = 5, (9, 39) = 3. 4|4, 5|10, 3|12,

pi .

2x 1 (mod 5), 3x 2 (mod 7), 3x 4 (mod 13).

x 3 (mod 5), x 3 (mod 7), x 10 (mod 13).

5, 7, 13 pi . pi mod455. pi

x 3 (mod 5), x 3 (mod 7).

3 . pi (5, 7) = 1, (-) x 3 (mod 35). pi

x 3 (mod 35), x 10 (mod 13).

x = 3 + 35y 3 + 35y 10 (mod 13), pi pi 9y 7 (mod 13). pi y 8 (mod 13) pi

x 3 + 8 35 = 283 (mod 455).

2

44

2

1: ( Branmagupta, 7 ..) pi pi 2,3,4,5,6 , : 1,2,3,4,5 . pi 7 . pi pi pipi pi . 2: ( pi ) pi 17 pi

pi pi ( ). pi - pipi . 3 . pi . 4 . pi pi , pi . 5 .pi pi pi pi. pi pi;

. 45

3 Fermat

3.1 p pi a :

(i) ap a (mod p)(ii) ( Fermat) (a, p) = 1

ap1 1 (mod p).

pi:: pi pi pi Fermat .

pi pi - pi - .

(i) pi. a = 1 . pi p|ap a. pi p|(a+ 1)p (a+ 1).pi pi Newton (9),

(a+ 1)p = ap +

(p

1

)ap1 +

(p

2

)ap2 + +

(p

p 1)a+ 1.

pi

(a+ 1)p ap 1 =(p

1

)ap1 +

(p

2

)ap2 + +

(p

p 1)a.

p (10) . pi pi,

p | [(a+ 1)p ap 1] + (ap a) = (a+ 1)p (a+ 1).

(ii) pi (i) p | ap a p | a(ap1 1) pi (a, p) = 1

p | ap1 1.

9(a+ b)n =

nk=0

(n

k

)akbnk.

10 pi . pi pi :(a) n pi n! (b) p pi,

(p1

),(p2

), . . . ,

(p

p1) pi p.

46

4

2

3.1 (i) (2, 11) = 1 11 pi, 2111 1(mod 11). 210 = 1024 11, pipi1.

(ii) : pi.. 235112 = 4840 101 pi- 101 pi, 4840100 1 (mod 101).

2

3.1 p pi a (a, p) = 1, d pi

ad 1 (mod p) d | p 1.

pi .

2

4 Euler

n 1, (n) pi n pi pi pi n. pi

: N N

(n) = ]{k N\k n (a, n) = 1}(11).

4.1 (9) = 6 6 1, 2, 4, 5, 7, 8 pipi 9.

2

Euler

(i) (1) = 1

11 ]{. . .} pi {. . .}.

. 47

(ii) n = p pi, (p) = p 1 p, 1, 2, . . . , p 1, pi pi p.

(iii) pipi (m,n) = 1,

(m n) = (m) (n)( pi (21) = (3 7) = (3) (7) = (3 1) (7 1) = 12).

(iv) p pi,

(pk) = pk pk1 = pk1(p 1)[pi pi pi pk pi pi pk ( , pipi p pi pi pk1)].

(v) , n = pk11 pk22 pkll n pi ( ) pi, pi (iii) :

(n) = pk111 (p1 1) pk212 (p2 1) pkl1l (pl 1)= n

(1 1

p1

)(1 1

p2

) (

1 1pl

)= n

li=1

(1 1

pi

)2

4.2

(1200) = (22 34 52) = 1200(

1 12

)(1 1

3

)(1 1

5

)= 1200 1

2 2

3 4

5= 320

pi , pi pi pi, pi pi pi 1200 pi pi 320.

2

4.3 (i) pi n N\{4} pi (n) 2 (mod 4) n = pk n = 2pk, pi k N p pi p 3 (mod 4).

(ii) pi pi n (n) = 14.

:

48

4

(i) n pi , -pi pi pi pi pi 3. , pi ,

n = pk11 pk22 pkll , pi 3 i = 1, . . . , l l 2.

(n) = pk111 (p1 1)pk212 (p2 1) pkl1l (pl 1)

, l 2, pi 2 pi (p1 1), (p2 1), . . . , (pl 1). (n) 0 (mod 4), pi.

n = 2rpk.

r 3 (r 6= 2 n 6= 4) ,

(n) = 2r1pk1(p 1) 0 (mod 4), pi.

, r = 0, 1 (r 6= 2 n 6= 4) pi

n = pk n = 2pk.

p 3 (mod 4). p 1 (mod 4)(12), ( pipi n)

(n) = pk(p 1) 0 (mod 4), pi.

pi .

(ii) pi .

2

pi Euler . pi pi pi pi pi. pi Euler , pi ( pi pi ), pipi Fermat .

4.1 ( Euler ) a pi pi n

a(n) 1 (mod n).

: n = p, pi Fermat .12 p 6= 2, p pi pi p 0, 2 (mod 4)

. 49

4.4 pi (9) = 6, (9, 4) = 1 46 1 (mod 9).

4.1 a pi pi n, k l (mod (n)),

ak al (mod n).

pi: pi k l. k l

(mod (n)), pi pi pi k = pi(n) + l, Euler ,

ak = al(a(n)

)l al 1l al (mod n)2

5 Euler -

pi pi pi pipi .

6 pi 1989 :

5.1 pi n N 1n + 2n + 3n 7 ;

: pi (1 ) - . pi pi pi (2) pipi .

1 (. ) n = 1 pi 7, n = 2 7

n = 3 7.

n = 2k

12k + 22k + 32k = 1 + 4k + 9k = 1 + 4k + pio.7 + 2k = pio.7 + 1 + 2k + 4k (1)

{ k = 3l (1)

pio.7 + 1 + 23l + 43l = pio.7 + 1 + 8l + 64l

= pio.7 + 1 + pio.7 + 1l + pio.7 + 1l

= pio.7 + 3

50

5

{ k = 3l + 1 (1)

pio.7 + 1 + 23l+1 + 43l+1 = pio.7 + 1 + 2 8l + 1 64l= pio.7 + 1 + 2(pio.7 + 1l)

+ 4(pio.7 + 1l)

= pio.7 + 1 + 2 + 4 = pio.7

{ k = 3l + 2 (1)

pio.7 + 1 + 23l+2 + 43l+2 = pio.7 + 1 + 4 8l + 16 64l= pio.7 + 1 + pio.7 + 4 1l+ pio.7 + 16 1l= pio.7 + 1 + 4 + 16 = pio.7

n = 2k, pipi k = 3l + 1 k = 3l + 2, n = 6l + 2 n = 6l + 4.

n = 2k + 1

12k+1 + 22k+1 + 32k+1 = 1 + 2 4k + 3 9k = 1 + 2 4k + pio.7 + 3 2k= pio.7 + 2(1 + 2k + 4k) + 2k 1

k = 3l + 1 k = 3l + 2, 1 + 2k + 4k = pio.7, . 2k 1 k = 3l + 1 k = 3l + 2.

{ k = 3l + 1

2k 1 = 23l+1 1 = 2 8l 1 = pio.7 + 2 1l 1 = pio.7 + 1

pio.7

{ k = 3l + 2

2k 1 = 23l+2 1 = 4 8l 1 = pio.7 + 4 1l 1 = pio.7 + 3

pio.7

2k 1 pio.7 k = 3l

23l 1 = 8l 1 = pio.7 + 1l 1 = pio.7

1k+2k+4k pio.7 . pipi n pio.2 pio.3, pipi n = 6k + 2 n = 6k + 4 n = 6k 2.

. 51

2 7 pi (7, 2) = 1 = (7, 3), pi Fermat

271 1 (mod 7) pi 26 1 (mod 7)371 1 (mod 7) pi 36 1 (mod 7)

, pipi 2n 7, pi pi 6 pi ( 3.1), 6 (13)

( 1, 2, 3, 6). pipi 3n 7. pi pi pipi pipi, pipi 1n + 2n + 3n , pin = 6k + , = 0, 1, 2, 3, 4, 5, pi 6 pipi 2n 3n 7.

= n (mod 6) 0 1 2 3 4 5 pi 1n (mod 7) 1 1 1 1 1 1 12n (mod 7) 1 2 4 1 2 4 33n (mod 7) 1 3 2 6 4 5 6

1n + 2n + 3n (mod 7) 3 6 0 1 0 3 pi pipi pi 1n + 2n + 3n

pipi 7, n n = 6k + 2 6k + 4. ( , 1n + 2n + 3n, 7 pi pipi 2, 4, 5.)

2

pi pipi pi pi, pi pi, pi , pi.

5.2 pipi A = 2 3n + 3 7n+1 + 53n+1 7 11.

: pi pi pi pi

n+1 3n+1 pi pi.

(11, 3) = (11, 7) = (11, 5) = 1, pi 3n, 7n, 5n (11) = 10 ( pi 1, 2, 5 10) pi (14)

13 pi , pipi pi pipi pi n, pi pipi pi 6, pi 1.

14 3n (mod 11), 5n (mod 11), 7n (mod 11) pipi pi .

52

5

n (mod 10) 0 1 2 3 4 5 6 7 8 9 pi n+ 1 (mod 10) 1 2 3 4 5 6 7 8 9 0 3n+ 1 (mod 10) 1 4 7 0 3 6 9 2 5 8

3n (mod 11) 1 3 9 5 4 1 3 9 5 4 52 3n (mod 11) 2 6 7 10 8 2 6 7 10 8 5

7n (mod 11) 1 7 5 2 3 10 4 6 9 8 103 7n+1 (mod 11) 10 4 6 9 8 1 7 5 2 3 10

5n (mod 11) 1 5 3 4 9 1 5 3 4 9 553n+1 (mod 11) 5 9 3 1 4 5 9 3 1 4 5

A 10 1 9 2 2 1 4 8 6 8

pi n n = 10k+2, A 11, pipi 1. , (pi) pipi pi pi:: pi n 2,

A pipi 1 11.

2

:

1. pi pipi pi pi pi . pi pi pi .. , , , , 1998. n 1 pi 24n+1 22n 1 pi 9. pi, pi- (2, 9) = 1 (9) = 6, pipi 2n 9 pi pi 6. pi pi (pi pi pi ), pi pipi, pi !

2. pipi pi pi pi pi pi pi pipi pi pipi :

1: n 6= 0 (mod 6)

1n + 2n + 3n + 4n + 5n + 6n 0 (mod 7).

pi pi - .

. 53

2

pi pi Forum (15): 2: n 6= 0 (mod (p 1))

p1i=1

in 0 (mod p)

: pi ( pi) , pi pi pi pi.

pi

pn+2 = p+

(n+ 2

1

)[1 + 2 + + (p 1)]

+

(n+ 2

2

)[12 + 22 + + (p 1)2]

+ +(n+ 2

n+ 1

)[1n+1 + 2n+1 + + (p 1)n+1]

p | [1k + 2k + + (p 1)k] k = 1, 2, , n, pi n {1, 2, , (p 3)}, p | [(n+ 2)(1n+1 + 2n+1 + + (p 1)n+1)]

pi n {1, 2, , (p3)} p (n+2). pip | [1n+1 + 2n+1 + + (p 1)n+1].

pi pi pi pi p | [1+2+ +(p1)] = p(p 1)2

p | [1k + 2k + + (p 1)k] k = 1, 2, , (p 2) a(p1)m+u (ap1)mau au (mod p) a (a, p) = 1 (16)pipi p | [1n+2n+ +(p1)n] n N n 6 0 (mod p1),

pi p pi 2.: , , Chevalley-

Warning pi pi pi pi , pi pipi . . pi .

2

pi pi pi pi . pi :

15www.mathlinks.ro/Forum/viewtopic.php?t = 11223416 pi Fermat ap1 1 (mod p) (a, p) = 1

54

5

5.1 pi p > 3, 6k+1 6k+5 ( pipi n. n = 6k + , = 0, 1, . . . , 5 () 6= 2, 3, 4 n pi) 5.3 (2 1989)

pi p pi, 42p | 3p 2p 1.pi: 42p = 237p A = 3p2p1 pipi

pi 2, 3, 7, p (17)

(i) 2: A A 0 (mod 2).(ii) 3: A = 3p (2p + 1) = 3p (2 + 1)(2p1 + 2p2 + + 2 + 1) 0

(mod 3)

(iii) p: pi 3.1(i)

3p 3 (mod p) 2p 2 (mod 2)

A = 3p 2p 1 3 2 1 = 0 (mod p).(iv) 7:

pi 5.1, pi p = 6k + 1 p = 6k + 5.

p = 6k + 1 Fermat , (2, 7) = 1,

26 1 (mod 7) 26k 1 (mod 7) 26k+1 2 (mod 7), (3, 7) = 1

36k+1 3 (mod 7)

A = 3p 2p 1 3 2 1 = 0 (mod 7) p = 6k + 5 pi pipi

26k+5 25 = 32 4 (mod 7)

36k+5 35 = 243 5 (mod 7)

A = 3p 2p 1 5 4 1 = 0 (mod 7) pipi pi A 0 (mod 7)

17 p, q pi n , p | n q | n p q | n.

. 55

: (2, 7) = 1 = (3, 7) (7) = 6 pi-pi 2k 3k 7, pi pipi,pi pi 6. pi pi, pi pipi pi, pipi p = 6k + 1, p = 6k + 5 A 0 (mod 7). pi pipi pi mod 6 pi ( pi ) 5.1.

2

56

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