# Can we find integers x and y such that f,g,h are strictely positive integers Let a>2 and b>2 two strictely positive integers. Let us consider the following quantities: f= (xy+ay+a^(2))/(by) g=a(y+a)(xy+ay+a^(2))/(by^(2)x) h=(y+a)/(b)

Can we find integers x and y such that f,g,h are strictely positive integers
Let $a>2$ and $b>2$ two strictely positive integers. Let us consider the following quantities:
$f=\frac{xy+ay+{a}^{2}}{by}$
$g=a\left(y+a\right)\frac{xy+ay+{a}^{2}}{b{y}^{2}x}$
$h=\frac{y+a}{b}$
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.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

rcampas4i
If we let,
$a=\left(bk-1\right)y,\phantom{\rule{1em}{0ex}}x=by$
then the three polynomials lose their denominators,
$\left(b{k}^{2}-k+1\right)y,\phantom{\rule{1em}{0ex}}\left(bk-1\right)\left(b{k}^{2}-k+1\right)ky,\phantom{\rule{1em}{0ex}}ky$
and are positive integers if $b,k,y$ are positive integers.