Is every integer a solution to the generalized pell-like equation

Question

Is every integer a solution to a generalized Pell-like equation, like, can we find integer solutions to $ax^2-by^2=n$, $a,b,x,y\in\mathbb{N}$ for any integer $n$?

Specifically, it is difficult to determine the existence to a particular Pell equation. But, I think any integer can satisfy an equation of the form $ax^2-by^2$, that is, any integer can be in the vector space of integer squares. Is this true? I think this has a lot to do with quadratic forms. Any hints? Thanks beforehand.

Answer 1

Setting $a=n+1$ and $x=y=b=1$ gives you $$ax^2-by^2=(n+1)\cdot 1 - 1\cdot 1 = n$$

Answer 2

1. If $n$ is not a power of $2$ then $n=2^k(2m+1) \space k \ge 0, m > 0$ so $n=2^k(m+1)^2 - 2^km^2$.

2. If $n$ is a power of $4$ then $n = 2^{2k} = (2^{k+1})^2 - 3(2^k)^2$.

3. If $n$ is a power of $2$ but not a power of $4$ then $n=2^{2k+1} = (2^{k+1})^2 - 2(2^k)^2$.