Is $n=26$ the only integer satisfying $n-1$ being a square and $n+1$ being a cube? I guess it is possible to solve it by ring theory, but I would prefer an elementary method instead. I can only figure out that $n$ is an even integer, otherwise there will be contradiction w.r.t. modulo $4$. I have also tried to modulo several other integers(3, 5, 8, 9, 10) but haven't got any contradiction yet. Any discussion is welcomed!
I wont give you the whole solution but try fuctoring n-1 and n+1, and compare them for some specific modulus
Alse you might want to consider quadratic reciprocoty
I wont give you the whole solution but try fuctoring n-1 and n+1, and compare them for some specific modulus
Alse you might want to consider quadratic reciprocoty