The diophantine equation $ m = x^2 + 7y^2 $

An addendum to Will Jagy's answer:

• $x^2+7y^2$ is the only reduced binary quadratic form of discriminant $-28$, hence any odd prime $p$ such that $-7$ is a quadratic residue $pmod{p}$ can be represented by such a form; by quadratic reciprocity odd primes of the form $7k+1,7k+2,7k+4$ are good primes and primes of the form $7k+3,7k+5,7k+6$ are bad primes
• $x^2+7y^2$ does not represent $2$ but it represents $4,8,16,32,ldots$
• the norm on $mathbb{Q}[sqrt{-7}]$ gives us Lagrange identity $(x^2+7y^2)(X^2+7Y^2)=(xX+7yY)^2+7(xY-yX)^2$
• by Fermat's descent, if an odd squarefree number $m$ can be represented as $x^2+7y^2$ then all its divisors can be represented by such a form.

Summarizing, since $7$ is a Heegner number we have a minor variation on the problem of understanding which numbers can be represented as a sum of two squares.

A number $m$ that you are able to factor: there is an integer expression $m = x^2 + 7 y^2$ if and only if

(I) the exponent of the prime $2$ is not one: either that exponent is $0$ or it is at least 2, AND

(II) the exponent of any prime $q equiv 3,5,6 pmod 7$ is EVEN

the exponent of $7$ and the exponents of primes $o equiv 1,2,4 pmod 7$ are not restricted.