Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$

Question

I started like this :

$a^2+c^2=b^2(a^2-1)\\c^2 +1=(a^2-1)(b^2-1)$

but it's leads to nowhere. can you help please ?

Answer 1

Suppose that this Diophantine equation has an integral solution $(a,b,c)\neq(0,0,0)$. Then, $a\neq 0$ and $b\neq 0$, whence $a^2-1\geq 0$ and $b^2-1\geq 0$. If $a$ or $b$ is even, then $a^2-1\equiv 3\pmod{4}$ or $b^2-1\equiv3\pmod{4}$. Hence, $\left(a^2-1\right)\left(b^2-1\right)=c^2+1$ is divisible by a prime natural number $p\equiv 3\pmod{4}$. Thus, $c^2\equiv -1\pmod{p}$, which is a contradiction as $\left(\frac{-1}{p}\right)=(-1)^{\frac{p-1}{2}}=-1$. Therefore, $a$ and $b$ must be odd. Thus, $$c^2=a^2b^2-a^2-b^2\equiv1-1-1=-1\pmod{8}\,,$$ which is a contradiction.