Well, I have noticed that for every integer a in a prime mod p, a^(p - 1) is congruent to 1 (mod p). I don't know why this is though, and if I had a proof for it, I could use the multiplicative property of congruency to show a^p is congruent to a (mod p). What should I do?
Let prime and an integer such as .
Consider the integers,
Now two are congruent to each other for that will yield,
Since, division yields,
Which is impossible for and so they have different remainders.
Thus,
form "a complete set of residues".
What does that mean?
Well it means,
...
Where are the remainders they leave.
But by Dirichelt's Pigeonhole Principle they are the integers (not necessarily in that order).
Multiply these congruences,
But, (as explained before).
Thus,
Also,
Division yields,
Q.E.D.
(Applause)