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18.03.2013

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.


, apa (mod p), p - .



: a p; a p.

1) a p;

:

a ≡ 0 (mod p);

ap ≡ 0 (mod p);

apa (mod p).

.

2) a p;

a, 2a, 3a,...,(p - 1)a (*).

, p. , 0.

.

- ka, na p ( k > n). ka - na p. (k - n)a p. a p, k - n p , p. - , (*) p . :

ar1 (mod p);

2ar2 (mod p);

...

(p - 1)arp - 1 (mod p);

. p - 1, p , (p - 1)!

ap - 1(p - 1)! ≡ (p - 1)! (mod p);

(ap - 1 - 1)(p - 1)! ≡ 0 (mod p);

(p - 1)! p, p - , p. (ap - 1 - 1) p.

(ap - 1 - 1) ≡ 0 (mod p);

ap - 1 ≡ 1 (mod p);

apa (mod p);

.



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