Sum and Product of Root

Question

Suppose the roots of the polynomial $x^2+mx+n$ are positive prime integers (not necessarily distinct). Given that $m<20$ how many possible values of $n$ are there?

Answer 1

Assuming $m < 20$ means $|m| < 20$,

Say the roots are $p_1$ and $p_2$ (needn't be distinct)

then, the quadratic becomes $x^2 - (p_1 + p_2)x + p_1p_2$

$m = - (p_1 + p_2)$ and $n = p_1p_2$

So, $p_1$ and $p_2$ can take all combinations (repeat) from$ ({2, 3, 5, 7, 11, 13, 17}\ |\ p_1 + p_2 < 20)$

number of $n = 18$