42 problems

Jeśmanowicz's conjecture. The Diophantine equation

Let $x,y,m,n$ be positive integers with $x>y$ and $m,n\geq 2$. Goormaghtigh's conjecture. If … then … This conjecture describes the exceptional equalities between two geometric sum…

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…

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…

Powers-of-2 and powers-of-3 conjecture. The only solutions are

Let $m\in\mathbb Z^+$, and let $x$ range over positive odd integers. Say that an integer has at least two distinct prime factors when its prime factorization contains at least two…

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…

Coprime-case finiteness conjecture. Apart from the known cases listed in the source, there are no further triples $(a,b,c)$ for which the equation has more than one solution…

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 $a$, $b$, and $c$ be distinct primes with $a<b$, and consider the equation … A solution means a triple $(x,y,z)$ of positive integers. Le's conjecture. The equation has at most…

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…

Let $m$ and $p$ be exponents and let $k$ be a natural number. Generalised Catalan conjecture. For all natural numbers $k$, the equation … has only finitely many solutions. This gen…

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…

Sun Zhi-Wei's conjecture. For every integer $n>2$, neither $2^n+n$ nor $2^n-n$ is a triangular number.

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

For a natural number $n$, write $2^n$ in base $3$ and consider its ternary digits. Erdős's conjecture. For every natural number $n>8$, the base-$3$ representation of $2^n$ contains…

Bertók–Hajdu conjecture. If an exponential Diophantine equation has finitely many solutions and satisfies these other natural restrictions, then there exists an integer $M$ such th…

Let $(F_n)$ and $(L_n)$ be the Fibonacci and Lucas sequences, defined by … and … for $n\geq 2$. Let $n,m,r$ be nonnegative integers, with $(n,m)\neq(1,0),(2,0)$, and let $x\geq 4$.…

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…

Corrected Cao–Dong conjecture. The equation

Cao–Dong's conjecture. The equation