It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that $\chi(d)=1$ if $d\equiv 1\pmod{4}$, $\chi(d)=-1$ if $d\equiv 3\pmod{4}$, and $\chi(2k)=0$. My question: Can we extend this to the integers of other imaginary […]

I need to describe all numbers of the form $x^2 + 2y^2$. So far I’ve reduced the problem to primes, and showed p=2 satisfies it. I’ve also shown that any primes mod 5 or 7 can’t be a written in this form. How do I proceed to show that it holds for all primes mod […]

I’m writing a code in C that returns the number of times a non negative integer can be expressed as sums of perfect squares of two non negative integers. R(n) is the number of couples (x,y) such that x² + y² = n where x, y, n are all non negative integers. How can I […]

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