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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.

Mathematics Asked by peter.petrov on November 14, 2021

2 Answers

2 Answers

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.

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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