236 problems

Digital-anomaly conjecture for $k=2$. The only digital anomalies $(x,y,B,k)$ with $k=2$ are

Let $k\geq 0$ be an integer. An upper $k$-gap balancing number is a balancing number associated with a gap of $k$, and solutions are grouped into classes as in the paper. Divisor-c…

Hyperbola-based factoring conjecture. The set $\Gamma$ has exactly three elements and satisfies

Nonexistence conjecture. The equation has no integer solutions $(x,n,m)$ for $x>1$ and $n\geq 1$ when $x\equiv 0,1,$ or $3\pmod 4$. Equivalently, the corresponding Diophantine equa…

Jeśmanowicz' conjecture. The equation has only the positive integer solution $(x,y,z)=(2,2,2)$. Jeśmanowicz' conjecture is a classical uniqueness problem for exponential Diophantin…

Conjecture. The only values of $n\in\mathbb{N}_{\geq 3}$ such that $M(n)=2$ are

Let $m$ be a positive integer with $m\equiv 1\pmod{24}$. Suppose that all prime divisors of $m$ are congruent to $1$, $5$, $7$, or $11$ modulo $24$, and write … where…

For every integer $n\ge 10$ and every homogeneous cubic form $C\in\mathbb{Z}[x_1,\ldots,x_n]$, there exists a nonzero vector $\mathbf{x}\in\mathbb{Z}^n$ such that $C(\mathbf{x})=0$…

For every number field $K$ and every integer $g\ge 1$, there exist constants $C_{K,g},C'_{K,g}>0$ such that every $g$-dimensional abelian variety $A/K$ satisfies…

For every $\varepsilon>0$, there are only finitely many triples $(a,b,c)\in\mathbb{Z}_{>0}^3$ such that $a+b=c$, $\gcd(a,b,c)=1$, and $c>\operatorname{rad}(abc)^{1+\varepsilon}$, w…

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 $n,k\geq 2$. Euler's sum of powers conjecture. If … then $n\geq k$. The conjecture was refuted by a single counterexample, as reported in the source, so the claim is no longer…

Fermat's conjecture over an arithmetic function field. For every arithmetic function field $K$, there exists a positive integer $N_0$ depending on $K$ such that $F_N$ has Fermat's…

A Markov number is a positive integer occurring in a Markov triple, that is, a triple of positive integers satisfying the Markov equation. A Markov triple is ordered here as…

Denef–Lipshitz conjecture. For every number field $L$, the extension

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…

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

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 $(p,q)\in\mathcal{R}$…

Tijdeman–Zagier conjecture for products. The integer $x^n\pm y^m$ is not a product of degree $d$ satisfying

Infinitude conjecture. For each choice of the signs, the equation

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…

Classical abc conjecture. For every $\varepsilon>0$, there is a constant $K(\varepsilon)$ depending only on $\varepsilon$ such that whenever $a$, $b$, and $c$ are three coprime non…

Weak form of Szpiro's conjecture. There exists a constant $s>0$ such that, for mutually coprime integers $A$, $B$, and $C$ satisfying $A+B=C$,

Let $K$ be a number field, and let $A,B,C$ be non-zero elements of $\mathcal{O}_K$. Let $\Omega$ be the subgroup of roots of unity inside $\mathcal{O}_K^{\times}$. Suppose … for an…

A positive integer $n$ is powerful if $p^2\mid n$ for every prime $p$ dividing $n$. Three positive integers are consecutive powerful numbers if they have the form $n,n+1,n+2$ and e…