Can we find integers $x$ and $y$ such that $f,g,h$ are strictely positive integers

Question

Let $a>2$ and $b>2$ two strictely positive integers. Let us consider the following quantities: $$ f= \dfrac{xy+ay+a^{2}}{by}$$ $$g=a(y+a)\dfrac{xy+ay+a^{2}}{by^{2}x}$$ $$h=\dfrac{y+a}{b}$$

My question is:

Can we find integers $x$ and $y$ (not necessarly positive) such that $f,g,h$ are strictely positive integers. Or at lest how one can proves that they are exist.

Answer 1

If we let,

$$a = (bk-1)y,\quad x = by$$

then the three polynomials lose their denominators,

$$(bk^2-k+1)y,\quad (bk-1)(bk^2-k+1)ky,\quad ky$$

and are positive integers if $b,k,y$ are positive integers.