Find all pairs $(a, b)$ of positive integers such that

$$
1+5^{a}=6^{b} .
$$
(Based on work by Fengning Ding) The only solution is $(1,1)$.

It is clear that if $b=1$ then $a=1$, and that $(1,1)$ is a solution. Consequently assume $b>1$. Then $6^{b}$ is divisible by 4 . On the other hand, since $5^{a} \equiv 1^{a}=1 \bmod 4$ for all $a$, the left side is $2 \bmod 4$. Thus there are no solutions for $b>1$.