Lovellss

2022-06-25

If $d$ is a positive integer, then the $\sqrt{9{d}^{4}+132{d}^{2}-16}$ is an integer or an irrational number? This question appears in my attempt to prove that the only triangle with integers for sides and area in arithmetic progression has sides $\left(3,4,5\right)$ and area $6$.

Schetterai

Expert

$9{d}^{4}+132{d}^{2}-16=\left(3{d}^{2}+22{\right)}^{2}-500$. As the next biggest square below ${n}^{2}$ that is of the same parity as ${n}^{2}$ is $\left(n-2{\right)}^{2}={n}^{2}-4\left(n-4\right)$, we conclude that $9{d}^{4}+132{d}^{2}-16$ can at most be a square when $4\left(3{d}^{2}+22-1\right)\le 500$, i.e., $d<6$. This leaves finitely many cases for manual checking.