16 problems

Cohn's cubic conjecture. The only solutions with $n=3$ are

Let $a,b,c$ be fixed relatively prime positive integers, all greater than $1$, and consider positive integer solutions $(x,y,z)$ of … Assume that none of $a,b,c$ is a perfect power…

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…

A purely exponential Diophantine equation is an equation whose terms involve fixed integer bases raised to unknown nonnegative integer exponents. Skolem's conjecture. If such an eq…

Erduvan–Keskin conjecture. This equation has no solutions when $y$ is a prime number $>7$. This conjecture extends the known results for the fixed bases $y=2,3,5$ and leaves the ge…

Bennett's conjecture. The equation has at most one solution $(n,m)\in\mathbb{Z}_{>0}^2$ for all but 11 specific exceptional triples $(a,b,c)$.

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

Goormaghtigh finiteness conjecture. There are only finitely many solutions $(x,y,m,n)$. This is explicitly presented as a weaker conjecture than the assertion that the two known so…

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.