The diophantine equation $ m = x^2 + 7y^2 $ I found this theorem.

A prime number $m \ne 7$ can be written as $x^2 + 7y^2$ for $x,y$ integers
iff $m$ is one of these residues modulo $28$
$1, 9, 11, 15, 23, 25$

It is stated in the first pages of this book.
https://www.amazon.co.uk/Primes-Form-ny2-Multiplication-Mathematics/dp/1118390180/
So far so good. But what does that imply for composite numbers $m$? And how does it imply it?
Is there some simple statement of this kind for composite numbers $m$?
I read some theory about all this but it all talks only about primes.
How do we make the leap to composites from there?
I think it's related to this
https://en.wikipedia.org/wiki/Brahmagupta%27s_identity
but I cannot quite make the leap to composites.
Is the leap to composites more complicated than just knowing this theorem and this identity?
E.g. is this following true: if we take $m$ and divide it by its largest divisor $M^2$, then what's left must be factored only into primes of the above mentioned residues?! I thought this is true but seems it's not. I am checking it computationally and it seems to me it is false.
 A: 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.
A: 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.
