4 problems

Terai's conjecture. The equation has only the solution

Three-solution conjecture. For any such $k$ and $B$, the equation has at most $3$ solutions in non-negative integers $(x,n)$. The conjecture is motivated by numerical experiments f…

The second four-solution conjecture. This equation has exactly four solutions in integers $(x,n)$, with

The four-solution conjecture. This equation has exactly four solutions in integers $(x,n)$, with