234 problems

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…

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…

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…

Let $N\in\mathbb{N}$ and $y\in\Gamma_N$. Let $\eta\in X_{N,y}(\mathbb{Z})$ be an integral point. For $\widehat{w}\in\Sigma W^0$ and $g\in G$, define the action $g\cdot\widehat{w}$…

Infinitude conjecture. There are infinitely many values of $c'\not\equiv 3\pmod 7$ such that every prime other than $2$ and $7$ occurring in the factorization of $3^{3+6c'}+1$ has…

Ma's conjecture. The following assertions hold:

Let $a$ and $b_{1}$ be relatively prime positive integers and put $b=-b_{1}^{2}$. Define the binary recurrence sequence by $u_{0}(a,b)=0$, $u_{1}(a,b)=1$, and…

Let $a$ and $b$ be integers with $2\leq a<b$, and consider the equation … Here $(z,n)$ denotes a solution of the equation. Cohn's conjecture. Every solution $(z,n)$ satisfies … The…

Let $D$ be a fixed positive nonsquare integer, let $p$ be a fixed prime with $\gcd(D,p)=1$, and let $N(-D,2)$ denote the number of solutions $(x,n)$ in $\mathbb{N}^2$ of … The exce…

Let $n$ be a nonnegative integer, and let $k$ be an integer. Consider the Diophantine equation … A positive integer solution means $x,y>0$, and a positive odd integer solution mean…

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

A number $t$ is represented as a ratio of two differences of fourth powers if there exist integers or rational numbers $a,b,c,d$ such that … with $c^4-d^4\neq 0$. Ratio of differen…

Congruence ABC conjecture for $N$. For each $\epsilon>0$, there exists a constant $C_\epsilon$ such that

A Diophantine prefix is generically decidable when an algorithm decides the corresponding positive-integer Diophantine sentences on the restricted collection of inputs specified by…

Fix an integer $d\geq2$. For a monic polynomial $f(x)\in\mathbb Z[x]$, consider integers $b$ such that whenever $p^2$ divides $b$, the prime $p$ does not divide the discriminant of…

Relative undecidability conjecture. The finiteness problem for integral points is undecidable relative to the existence problem.

Let $(a,b,c)$ be an admissible triplet, meaning a triplet of positive integers satisfying the admissibility conditions for the three-variable Frobenius problem, and let $f(a,b,c)$…

Generalized Pythagorean partition-regularity conjecture. The equation

Let $f_2(k)$ be the smallest positive integer $n$ such that every 2-coloring of $\{1,2,\ldots,n\}$ has a monochromatic solution, with the variables not necessarily distinct, to … C…

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…