Thread: A Couple Problems with Primes

1) Show that (mod p) when p is an odd prime

2) Show that if p is a prime and p > 3, then (mod p)

For 1), I really don't understand which theorem or identity I'm supposed to put into use to prove this. Obviously it works every time I just don't understand how this works in terms of technicality.

2) is the same story. I don't see how it works and therefore I can't prove it formally.

Originally Posted by Janu42

1) Show that (mod p) when p is an odd prime

2) Show that if p is a prime and p > 3, then (mod p)

For 1), I really don't understand which theorem or identity I'm supposed to put into use to prove this. Obviously it works every time I just don't understand how this works in terms of technicality.

2) is the same story. I don't see how it works and therefore I can't prove it formally.

For #1, use Fermat's Little Theorem.

For #2, I'm not sure if the following is valid, but if not then I think it's close. Using Fermat's Little Theorem,

The form of the terms in both sums suggests using Fermat's Theorem.
Let . Then .
Now Fermat's Theorem implies that - show this!. Use the fact that - again, why?
Therefore, you can cancell to get .