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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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what are all possible pairs (k,m) such that n=2k^2+ m^2

I am working on a problem in number theory and would like to count all possible ways to partition an integer $n\geq 1$ into pairs $(k,m)$ of positive integers such that $n=2k^2+m^2$ and $n=4k^2+m^2$. …