586 problems

Dickson's conjecture. There should be infinitely many such pairs of primes $p,q$.

A Markov number is a positive integer appearing in a solution to the Markov equation … A triple satisfying this equation is a Markov triple. Frobenius' uniqueness conjecture. Every…

For a fixed $c\in\mathbb{Z}^+$, consider positive-integer solutions $(x,y,n,m)$ of … with $n,m\geq 2$. Pillai's conjecture. For every fixed $c\in\mathbb{Z}^+$, the equation … has o…

Let $(a,b,c)\in\mathbb{N}^3$ be a triple of coprime integers satisfying $a+b=c$, and let $\operatorname{rad}(n)=\prod_{p\mid n}p$ denote the product of the distinct prime factors o…

Schinzel's conjecture.

Catalan's conjecture. The only such solutions occur for $m=2$ and $n=3$, with

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

Let $A,B,C\in\mathbb{Z}$ satisfy $\gcd(A,B,C)=1$. For prime exponents $p,q,r$ with … a non-trivial primitive integer solution of $Ax^p+By^q+Cz^r=0$ is a solution with $xyz\ne0$ and…

Jeśmanowicz's conjecture. The Diophantine equation

Let $p$ be a prime with $p\equiv 4,7,8\pmod 9$. A prime is a rational cube sum if there exist $a,b\in\mathbb Q$ such that $p=a^3+b^3$. Sylvester's rational cube-sum conjecture. It…

Erdős–Moser conjecture. The equation

Terai's conjecture. For every $k>1$, this equation has only one solution,

Let $\mathcal{R}$ and $m:\mathcal{R}\to\mathbb{N}_{+}$ be as above, with $m$ defined as the minimal quotient for a proper solution and taking fourth power-free values. Limit-point…

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…

Markov Uniqueness conjecture. If $\max(a,b,c)=\max(a',b',c')$, then there exists a permutation $\sigma\in\mathfrak{S}_3$ such that

Schäffer's conjecture. If $(k,n)\notin E$, then the only nontrivial integer solution of

Tyszka's conjecture. There exists an integer $\delta\geq 9$ such that

Let $P_j\in\mathbb{Z}[u]$ for $j=1,2,\ldots,d-1$, and consider the parametric Thue equation … The trivial solutions are $(1,0)$, $(0,1)$, and $(P_j,1)$ for $j\in\{1,2,\ldots,d-1\}$…

Consider the generalized Fermat equation … A solution $(a,b,c)\in\mathbb{Z}^3$ is primitive if $\operatorname{gcd}(a,b,c)=1$ and trivial if $abc=0$. The Beal Prize conjecture. If…

Let $a,b,c$ be nonzero coprime integers satisfying $a+b+c=0$, let $N:=\operatorname{rad}(abc)$, and let $\omega(ab)$ denote the number of distinct prime factors of $ab$. Baker's ep…

Let $r,s,k$ be positive integers. Suppose that … where $a_1,\ldots,a_r,b_1,\ldots,b_s$ are positive integers satisfying $a_i\neq b_j$ for every $1\leq i\leq r$ and $1\leq j\leq s$.…

Caporaso–Harris–Mazur uniformity conjecture. There is a constant $C_g$ such that every curve of genus $g$ defined over $\mathbb{Q}$ has at most $C_g$ rational points:

Let $S_k(n)=\sum_{j=1}^{n-1}j^k$ for positive integers $k$ and $n$. Kellner's conjecture. Let $k\ge 1$ and $n\ge 3$ be integers. Then the ratio … is an integer if and only if $n=3$…

Consider the prefixes $\exists\forall\exists$ and $\exists\exists$, with all quantified variables ranging over $\mathbb{N}$. The prefix $\exists\exists$ is equivalent to the two-va…

Let $\mathcal{R}$ be the set of labels used for generalized $k$-Markov numbers, and let $m^{(k)}_{(p,q)}$ denote the generalized $k$-Markov number attached to $(p,q)\in\mathcal{R}$…