4 problems

Finite-supremum conjecture. The supremum of the number of solutions of this equation, as $D$ ranges over integer-coefficient polynomials with $\deg(D)\leq k$, exists; denote it by…

Bounded-solution conjecture. Equation (5) has at most $5$ positive integer solutions, where $D$ is an integer-coefficient polynomial of $n$.

Let $x,y,n\in\mathbb{N}$ satisfy $\gcd(x,y)=1$ and $n>2$. The conjecture for $x^2-2=y^n$. The equation … has no solutions $(x,y,n)$. The source presents this as an unresolved conje…

Let $D$ be an odd positive integer, let $p$ be an odd prime, and let $N(-D,4,p)$ denote the number of solutions $(x,n)\in\mathbb{N}^2$ of $x^2-D=4p^n$. Let cases $(i)$ and $(ii)$ b…