2,761 problems

For every positive integer $n\in\mathbb{Z}_{>0}$, define the Collatz map $T:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ by $T(n)=n/2$ if $n$ is even and $T(n)=3n+1$ if $n$ is odd. The conje…

Let $\operatorname{syr}$ be the Syracuse map on odd positive integers, and let $\operatorname{sft}(n)$ denote the Syracuse falling time of an odd integer $n\geq 3$, namely the leas…

For a finite set $A\subseteq\mathbb{Z}_{>0}$, define its subset-sum set by $H(A)=\left\{\sum_{a\in B}a:B\subseteq A\right\}$. Let $g_3(n)$ be the least integer $N$ such that there…

Let $\operatorname{sft}(n)$ denote the Syracuse falling time of an odd positive integer $n$. Large-number Syracuse falling-time conjecture. … for all odd $n\geq 2^{5000}$. This con…

Let $n$, $a$, and $d$ be positive integers, and set … Let $\varphi(n)$ denote Euler's totient function and let $\zeta_k$ be a primitive $k$-th root of unity. Arithmetic conjecture.…

Supercongruence conjecture modulo $p^3$. Under these conditions,

Let $f\in\mathbb{Z}[x,y]$, let $C_p$ be the curve modulo $p$ defined by $f(x,y)\equiv0\pmod p$, and write $e_{p^2}(u)=\exp(2\pi i u/p^2)$. Exponential-sum conjecture. There exists…

Let $\mathbb{K}$ be a finite field of characteristic different from $2$, and let $\mathbb{K}^{*}$ denote its multiplicative group. An element $x$ is a generator if it generates…

Bounded-gcd conjecture. For every Kurepa factorial sequence, $\mathcal{F}_n(a)$ is bounded above by $2$, with

Let $n\geq 0$, $i\geq 3$, and $k\geq 0$. The overcubic partition tuple function $\overline{b}_m(r)$ is considered at the indicated arithmetic progressions. Congruence conjecture. F…

Let $\overline{p}_{9}(n)$ denote the $9$-color overpartition function, and let $n\ge0$ and $\alpha\ge1$. Arithmetic congruence conjecture. For all $n\ge0$ and $\alpha\ge1$, … These…

Let $\overline{p}_{6}(n)$ and $\overline{p}_{8}(n)$ denote the $2$-color overpartition functions with indicated color parameters, and let $n\ge0$. Higher-modulus congruence conject…

Let $N=16n+6$ for some nonnegative integer $n$. Define $\a0\sigma_0(m)$ to be the number of positive divisors of $m$. The $N=16n+6$ congruence conjecture. … The authors report comp…

Let $S_m(n)=\sum_{j=1}^{n-1}j^m$, and let $k$ be a positive integer. Even-modulus power-sum conjecture. For every $k\ge 1$ and every even positive integer $n$, the congruence … hol…

Let $n,m\in\mathbb{N}$ satisfy $n>2$, $\lfloor\log_2(n)\rfloor+1\geq m>1$, and suppose there is no $k\in\mathbb{N}$ such that $k^m=n$. Then Shunia's conjecture. … The formula gives…

Let $a,n\in\mathbb{Z}^+$ satisfy $a>2$, $\left\lfloor\log_2(a)\right\rfloor+1\geq n>1$, and suppose there is no $k\in\mathbb{Z}^+$ such that $k^n=a$. Integer-part root conjecture.…

Let $\overline{p}_{\ell}(n)$ denote the $2$-color overpartition function, where $\ell$ is a positive integer, and let $n\ge0$. The mod 8 congruence conjecture. For all…

Let $d_k(n)$ denote the $k$-elongated plane partition function, and let $c,n\geq 0$. For a list of integers in an argument, interpret the corresponding congruence as holding for ea…

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

Generalized Xi identity conjecture. For all $s_0\in\mathbb C$ and all $a\in\mathbb C$,

General constraint conjecture.

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…

Let $G(x)$ denote the number of primes in the set $\left\{\left\lfloor x/n\right\rfloor:1\le n\le x\right\}$. Suppose that $x=p_1p_2p_3$, where $p_1,p_2,p_3$ are primes satisfying…

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

For a positive integer $n$, define … Let $p$ be the smallest prime factor of $n$. Lower-bound conjecture for $v(n)$. If … then … This would strengthen the proven bound…