29 problems

Conjecture for $274837012705$. Every solution $(x,y,n)$ has one of the following forms:

Let $H_{\mathrm{even}}$ be the set of even integers $m$ for which every prime divisor of $2^m+1$ is 3-Higgs. A prime is 3-Higgs when every prime divisor $q$ of $r-1$ satisfies…

Consider the purely exponential Diophantine equation … where the variables $x_{ij}$ are positive integers. Skolem's conjecture. If this equation has no solution, then there exists…

Let $a$, $b$, and $c$ be fixed coprime positive integers with $>1$ and let $N(a,b,c)$ denote the number of solutions in positive integers $(x,y,z)$ to … Assume that $a$, $b$, and…

Let $n$ be an even perfect number represented as $n=x^m+y^m$, where $x$ and $y$ are positive integers and $m$ is an integer with $m\geq 2$. Makowski's conjecture. Then $m=3$ and…

Let $k$ be a positive integer, and let $a,b$ be coprime positive integers with $\min\{a,b\}>1$. Consider the equation … The solution $(x,y,z)=(1,1,1)$ is always present, and any so…

Let $\mathbb{P}$ be the set of positive rational primes, and let $S(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of … where $a,b,c\in\mathbb{P}$, $a<b$, and th…

Let $k$ be a positive integer such that $4\Vert k$ and $2k-1$ is a prime power. Write … where $a$ is square-free. The form $(a^2b^2,0,2k-1)$ is the binary quadratic form…

Terai's conjecture. The equation has only the solution

One-solution conjecture. There is at most one solution $(x,y,z)$, except when $(a,b,c)$ or $(b,a,c)$ is one of $(5,2,3)$, $(7,2,3)$, $(3,2,11)$, $(3,2,35)$, $(3,2,259)$,…

Scott–Styer's prime-base conjecture. We have $N(a,b,c)\leq1$, apart from the six explicitly listed exceptional cases: $(2,3,5)$, $(2,3,11)$, $(2,5,3)$, $(2,7,3)$, $(3,5,2)$, and…

Scott–Styer's conjecture. If $a<b$, then $N(a,b,c)\leq 1$, apart from the thirteen explicitly listed exceptional cases and solutions in the source statement: the family…

An exponential Diophantine equation over $\mathbb Q$ is an equation built using variables, rational constants, arithmetic operations, and exponentiation with nonnegative base and e…

Let $a,b,c$ be relatively prime positive integers satisfying … A solution $(x,y,z)$ in positive integers is nontrivial if it is not $(2,2,2)$. Jesmanowicz's conjecture. The equatio…

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 $p$ be an odd prime, $a\in\mathbb{Z}$ with $a\geq0$, and $x,m,n\in\mathbb{N}$. Ma's second conjecture. The equation … has only the solution $(p,x,a,m,n)=(5,49,3,2,1)$. The sour…

Let $b,c,r,t$ be fixed positive integers satisfying … Let $x,y,z\in\mathbb{N}$ satisfy $x^2+b^y=c^z$. Terai's prescribed-exponent conjecture. The only solution is … The conjecture…

Corrected Cao–Dong conjecture. The equation

Let $(a,b,c)$ be a primitive Pythagorean triple with … where $a,b,c\in\mathbb{N}$, $\gcd(a,b)=1$, and $a\equiv0\pmod{2}$. Terai's conjecture. The equation … has only the solution…

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…

Let $D$ be a fixed positive nonsquare integer, let $p$ be an odd prime, and let $N(-D,p)$ denote the number of solutions $(x,n)\in\mathbb{N}^2$ of $x^2-D=p^n$. Beukers' odd-prime c…

Miyazaki's shuffle conjecture. If $q=r=2$ and $b+1=c$, the only solution is $(X,Y,Z)=(1,1,p)$; otherwise, there are no solutions. The source presents this as a final open conjectur…

Miyazaki's shuffle conjecture. If $f=g+1$, the only solution is $(x,y,z)=(1,1,2)$; otherwise, there are no solutions. The source describes this as unsolved, with verification in se…

Siar–Keskin's bounded-exponent conjecture. If this equation has a solution $n,x$, then

Siar–Keskin's Pell-family conjecture. For every odd integer $k>3$, the equation has only the solution