the number $\sqrt{1+5^n+6^n+11^n}$ is an integer

Question

How can i prove that $\sqrt{1+5^n+6^n+11^n}$ is an integer??

PS : i shouldn't use induction

Answer 1

$n > 0 \Rightarrow 1+5^n+6^n+11^n \equiv 3 \mod 5,$ hence this is never an integer except when $n=0.$