Proof of Fermat's Little Theorem
Jason Holt
BYU Internet Security Research Lab
25 Sept. 2002
This document is in the public domain.

Fermat's Little Theorem states that

   p-1
  a    mod p  = 1

For all 0<a<p and prime p.

Lemma: xa congruent to ya, (0 < a,x,y < p) implies that x = y (mod p).
That is, the set P=(1a, 2a, 3a, ... (p-1)a) has unique elements, which
are therefore the same elements as the set N=(1,2,3, ... (p-1)).

Proof of lemma by contradiction:
Assume x != y.  Let x > y.  xa congruent to ya means that:

xa = kp + z       and     ya = lp + z      (k != l)
Subtracting,

xa - ya = kp - lp

(x-y)a  = (k-l)p

Note that x-y and a are each < p.  Now consider the factorization of
each side, which must be the same if both sides are equal.  Either
(x-y) or a must be a multiple of p, since p has no factors.  But since
x-y and a are < p, we have a contradiction.

Proof of theorem:

Multiply the elements of P and N together, recalling that they have
the same elements and therefore must have the same product:

1a * 2a * 3a * ... (p-1)a   =   1 * 2 * 3 * ... (p-1)     (mod p)

                    p-1
(1*2*3*...(p-1)) * a        =   (1*2*3*...(p-1))          (mod p)

Dividing out (p-1)!, we get that:

 p-1 
a    =  1  (mod p).