One number-theoretic result mentioned in the course was Wilson’s Theorem : If p is a prime then (p − 1)! is congruent to −1 mod p . — The purpose of this exercise is to show the reverse implication....

One number-theoretic result mentioned in the course wasWilson’s


Theorem:If p is a prime then (p − 1)! is congruent to −1 mod p. — The purpose of this exercise is to show the reverse implication.

(b) The preceding part of the problem proves the reverse implication unless n = p²
where p is a prime. Prove that if p > 2 is prime then (p²
− 1)! is congruent to 0 mod p², and find k ∈ {0, 1, 2, 3} such that (2²
− 1)! is congruent to k mod 4.