Prove that every prime except one has the form $a^2 − b^2$ for some positive integers $a$ and $b$

Question

This is a homework question, so I'm looking for some help to get started with the problem since I cannot think of a good way to do so. I know that we can say the integers a and b can be written as $a = p_1p_2...p_k$ and $b = q_1q_2...q_k$ where $p_i$ and $q_i$ are prime numbers. After this I'm stuck and don't know how to continue. Any help is appreciated, thank you.

Answer 1

Hint: Try showing that every odd number can be written as the difference of two squares. Handle the prime number 2 separately.

Answer 2

Note that $a^{2}-b^{2} = (a + b)(a - b)$. Note that every odd number, and thus every odd prime, can be written this way by taking $a-b = 1$. Thus, every prime except $2$ can be written in the form $a^{2}-b^{2}$. $\blacksquare$

Answer 3

Every odd number $2n+1$ can be written as the sum of two consecutive numbers: $n,n+1$. The difference between two consecutive numbers is simply $1$. We see that $((n+1)+n)\cdot ((n+1)-n)=(n+1)^2-n^2=2n+1$, so every odd number, to include every odd prime number, can be readily represented as the difference of two squares. Can you finish by showing why the only even prime, $2$, cannot be so represented?