When can $n$ be written as $a^2 +2b^2$?

Question

I've seen a few posts asking about the same form but for prime number $p$. I want to know how this can be extended to a general integer $n$. This property is certainly not exclusive to primes since $6= 2^2 + 2 \times 1^2$.

I’m not sure if prime decomposition would help because we would need to multiply with the remaining primes and that would mess up the form.

I can sense a slight relation to Fermat’s theorem on how $p=x^2 +y^2$ iff $p \equiv 1 \pmod 4$ but again this works for primes.

Any help would be much appreciated.

By the Brahmagupta-Fibonacci identity, the product of two numbers so expressible is also expressible. Squares are clearly so expressible. As with Fermat's Christmas Theorem, which you quote, determining the primes for which it works will then suffice. For sums of two squares, for instance, the answer is that it is precisely those positive integers whose prime factorization does not contain any non-expressible prime raised to an odd power. That's why people concentrate on primes. — Arturo Magidin, Oct 05 '22 at 05:50
See my answer here: https://math.stackexchange.com/a/4403873/348926 — Lukas Heger, Oct 05 '22 at 06:08
I would strongly recommend a great book of David Cox "Primes of the form $x^2+ny^2$" for more information about similar questions. https://dacox.people.amherst.edu/primes.html — richrow, Oct 05 '22 at 07:17
Does this answer your question? Is there any variation known to the sum of two squares theorem? — Parcly Taxel, Oct 05 '22 at 08:25

vvg · Accepted Answer · 2022-10-05T06:14:24.953

Kenneth Hardy, Joseph B. Muskat and Kenneth S. Williams. "A Deterministic Algorithm for Solving $n = fu^2 + gv^2$ in coprime integers $u,v$". Mathematics of Computation, Vol 55, No.191. July 1990, pp. 327-343. url (accessed Oct 5, 2022): https://people.math.carleton.ca/~williams/papers/pdf/165.pdf

This paper describes a method to write $n$ in the form $n = fu^2 + gv^2$. It requires factoring $n$.

One may also use Cornacchia's algorithm to solve $x^2 + dy^2 = m$ where $d, m$ are coprime. See: https://en.wikipedia.org/wiki/Cornacchia%27s_algorithm

As mentioned in comments below, this provides methods for representing a specific $n$ in the form.

The existence of a representation in the form derives from the Brahmagupta Identity:

$$ \begin{align} N & = (a^2+nb^2 )(c^2+nd^2 ) \\ & = (ac-nbd)^2+n(ad+bc)^2 \\ & = (ac+nbd)^2+n(ad-bc)^2 \\ \end{align} $$

i.e., $N$ is represented in the form if and only if the cofactors are reperesented in the same form.

This reference does not characterize all $n$ of a certain form, but rather presents a method to find a representation for a fixed $n$. — WimC, Oct 05 '22 at 05:51

When can $n$ be written as $a^2 +2b^2$?

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