mathschallenge.net

Problem

Prove that all primes greater than 2 can be written as the difference of two squares.

Solution

By considering the difference of two consecutive square numbers,
(n + 1)² n² = (n² + 2n + 1) n² = 2n + 1

Hence all odd numbers greater than 1, which must include all primes greater than 2, can be written as the difference of two square numbers.

For example, 83 = 241 + 1 = 42² 41²

Investigate primes that can be written as the difference of square numbers in general, including non-consecutive squares.