Solving the Diophantine equation $t^n + 2 \equiv 0 \bmod s^n - 1$

Question

My problem is this. find the maximal integer n, so the equation:

$t^n+2\equiv0 \mod (s^n-1). $

has a solution (s,t>1 have to be integers). I would like to read your solution and even just an opinion. I'm not even sure this problem can be solved.

Answer 1

It seems to be a difficult problem. I will write $(n,t,s)\in S$ to indicate that the triplet is a solution. Some easy observations follow.

• $(1,t,d+1)\in S$ for all $t$ and $d$ dividing $t$.
• $(2,t,2)\in S$ if $t$ is not divisible by $3$ (there are other solutions with $n=2$, like $(2,14,10)$.)
• If there is a solution for a number $n$, then there is a solution for any divisor of $n$.

Based on the last observation, I have started a search for solutions with $n$ prime. The only one I found is $(5,8860,19)$.