Which integers can be represented as the most pair of difference of two squares? Let $f(x)$ be the number of $a,b,x\in \mathbb N$ where $a^2-b^2=x$.
For example, 1971 is not only $986^2-985^2$ but also $50^2-23^2$, $114^2-105^2$, $330^2-327^2$. So, $f(1971)=4$. Is there some sort of limit to $f(x)/x$ or $f(x)/ln(x)$ or similar? For which kind of $x$?

Partial answer. If $x$ is odd, then $a=(x+1)/2$, $b=(x-1)/2$, satisfy $x=a^2-b^2$.
If $x$ is even, and $a$, $b$ exist, so that $x=a^2-b^2$, then $x=(a-b)(a+b)$, and clearly, both $a+b$, $a-b$ are even, and hence $x$ is divisible by $4$. Then, $a=(x+4)/4$ and $y=(x-4)/4$, satisfy $x=a^2-b^2$.
Note that in the first case $x\ge 3$, and in the second $x\ge 8$.
So $x$ is either odd $\ge 3$ or multiple of 4 and $x\ge 8$.
