14 problems

Ma's conjecture. The following assertions hold:

Let $a$ and $b$ be integers with $2\leq a<b$, and consider the equation … Here $(z,n)$ denotes a solution of the equation. Cohn's conjecture. Every solution $(z,n)$ satisfies … The…

Let $D$ be a fixed positive nonsquare integer, let $p$ be a fixed prime with $\gcd(D,p)=1$, and let $N(-D,2)$ denote the number of solutions $(x,n)$ in $\mathbb{N}^2$ of … The exce…

Cubic plus-equation conjecture. The only positive integer solutions are

No-solution conjecture for the plus equation. This equation has no solutions in positive integers for $n\geq4$. The conjecture is proposed as an analogue of Cohn's conjectures for…

Le's uniqueness conjecture. The equation has at most one solution in integers $x,y,z>1$. This conjecture concerns the uniqueness of solutions to a class of exponential Diophantine…

Let $a,b,c$ be positive integers. Le's conjecture. The equation … has at most one solution $(x,y,z)$ satisfying $\min(x,y,z)>1$. This conjecture is proposed after counterexamples t…

Let $a,b,c,p,q,r\in\mathbb{N}$ satisfy … with $a,b,c,p,q,r\geq 2$ and $\gcd(a,b)=1$. Terai–Jeśmanowicz' conjecture. The equation … has only the solution $(x,y,z)=(p,q,r)$ with…

Soydan–Németh–Szalay conjecture. Apart from these trivial solutions, the only solutions are

Nagell–Ljunggren conjecture. The only solutions in integers $x,y>1$, $n>2$, and $q\geq 2$ are

Let $Q\in\mathbb{Q}\cap\left(0,e^{-e}\right]$ with $Q\neq 1/16$, and let $h_o(Q)$ and $h_e(Q)$ denote the two solutions identified in the preceding proposition, with…

Let $T,R\in\mathbb{R}^+$ satisfy $T^R=R^T=N\in\mathbb{N}$ and $T\neq R$. Suppose $N\neq 16$, so that at least one of $T$ and $R$ is transcendental. Both-bases transcendence conject…

Nested exponential transcendence conjecture. If

Stronger transcendence conclusion. Under the hypotheses of Proposition $(i)RT=TR=A$, both $s$ and $t$ are transcendental.