Prove $x^4 + 131 = 3y^4$ has no integer solutions

Question

I tried proving that $\dfrac{\Delta}{4}$ isn't a perfect square, but reducing it modulo 4 doesn't lead anywhere. Any help is greatly appreciated.

Answer 1

$x^4\equiv0,1\pmod5$ for $x\in\Bbb Z$ and $131\equiv1\pmod5$. We now reduce the equation modulo 5 and check four cases, but all turn out to be impossible: $$0+1\not\equiv3\cdot0\pmod5$$ $$0+1\not\equiv3\cdot1\pmod5$$ $$1+1\not\equiv3\cdot0\pmod5$$ $$1+1\not\equiv3\cdot1\pmod5$$ $$x^4+131\not\equiv3y^4\pmod5$$ Hence the equation has no integer solution.