TITLE: show that $\frac{a^2+b^2+c^2}{15}$ is non-square integer
QUESTION [5 upvotes]: Let $a,b,c$ be non-zero integers. Show that
$\dfrac{a^2+b^2+c^2}{15}$ is a not a perfect square for all $a,b,c$
My approach is the following:
Assume there exists an integer $d$ such that
$$a^2+b^2+c^2=15d^2$$
It follows that
$$a^2+b^2+c^2=15d^2\equiv -d^2\pmod 8\Longrightarrow a^2+b^2+c^2+d^2\equiv 0\pmod 8$$
Well and now I'm stuck and don't know how to proceed

REPLY [3 votes]: You are almost there.  Now note that all squares are $0,1$ or $4 \pmod 8$, so $a,b,c,d$ must be even because if one is odd you can't sum to zero.  Given a solution with $a,b,c,d$ even, you can divide them all by $2$ and have a new solution.  Infinite descent.  Of course, you need to exclude $a=b=c=d=0$