Let $X = a^2 +b^2$ where all the terms are positive integers and $X$ is a composite number and $\gcd(a,b)=1$ . Do there exist positive integers $c$ and $d$ other than $a$ and $b$ such that $X = c^2+d^2$ ?
By Fermat's Two Square Theorem, since $\gcd(a,b) =1$ , all prime factors of $X$ (other than $2$) must be of the form $4k+1$. I can prove that primes of the form $4k+1$ can be uniquely written as a sum of two squares, but can composites also follow the same property?
Any help will be appreciated.