how to solve this equation with the unknown at the powers

Question

Please help me to solve this equation:

Find $n \in \mathbb{N}$ such that: $\sqrt{1+5^n+6^n+11^n} \in \mathbb{N}$.

$0$ is a particular solution, and does it have other one ?

Answer 1

We show there are no solutions other than $0$.

Suppose that $n$ is odd. Work modulo $3$. We have $$1+5^n+6^n+11^n\equiv 1+(-1)^n+0+(-1)^n\equiv-1\pmod{3}.$$ But no square is congruent to $-1$ modulo $3$.

Suppose that $n\ge 2$ is even. Then $1+5^n+6^n+11^n$ is congruent to $3$ modulo $4$, so cannot be a square.