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.
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
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.
An addendum to Will Jagy's answer:
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.
Answered by Jack D'Aurizio on November 14, 2021
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.
Answered by Will Jagy on November 14, 2021
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