3
$\begingroup$

Let $p,q$ be a pair of coprime positive integers. Let $S(N,p,q)$ be the number of integer solutions $(x,y)$ of $N=p x^2+q y^2$ such that $x$ and $y$ are coprime. If $(p,q)=(1,1)$, it follows from Fermat's two square theorem that $S(N,1,1)$ has no upper bound as $N\to\infty$. What will happens if $(p,q)\neq(1,1)$, i.e. is the sequence $\{S(N,p,q)\}_{N\geq1}$ still unbounded? Does there exist $(p,q)$ such that $\{S(N,p,q)\}_{N\geq1}$ is a bounded sequence?

$\endgroup$
2
$\begingroup$

Let $F$ be a positive definite binary quadratic form. For a positive number $Z$ put $N_F(Z)$ for the number of pairs of integers $(x,y)$ such that $F(x,y) \leq Z$, and let $R_F(Z)$ denote the number of integers $n \leq Z$ such that the equation $F(x,y) = n$ has solutions in integers $x$ and $y$. It is easy to see from a geometry of numbers argument that

$$\displaystyle N_F(Z) \sim A_F Z$$

for a positive number $A_F$, and it is known (first by Landau in the case $F(x,y) = x^2 + y^2$ and in general by the thesis of one of his students) that there exists a positive number $C_F$ such that

$$\displaystyle R_F(Z) \sim C_F Z (\log Z)^{-1/2}.$$

From these asymptotic formulas one can conclude, in your notation, that $S(n,p,q)$ behaves like $\sqrt{\log n}$ on average; in particular, it is unbounded.

$\endgroup$
  • $\begingroup$ Thanks! Is there any reference for the results of Landau and his student? $\endgroup$ – Huiping Pan Feb 2 '18 at 12:47

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.