Prove that every prime number other than 2, can be expressed as a difference of two squares.
- Princeton Math Club Website
This is the first problem I think I've figured out on the site!
To begin, x, and a are positive integers, so
x^2-a^2>0
x^2>a^2
x>a
p=our prime number, also pos. int.
p= x^2-a^2=(x+a)(x-a)
Primes are only divisible by themselves and 1, so
p=x+a (1) and
1=x-a-->x=1+a (2)
because x+a>x-a.
Substituting our value from (2) into (1) we obtain
p=a+1+a=2a+1
2a is always even, so 2a+1 is always odd, and thus our prime p must be an odd number if it can be expressed as the difference of two squares. This obviously excludes 2, the only even prime, from the rule.
solution
