28 problems

Let $n$ be square-free, let $d(n)$ be the divisor function, and let $\omega(n)$ be the number of distinct prime divisors of $n$. Square-free divisor-function conjecture. There is a…

For a positive integer $n$, let $\Omega(n)$ denote the total number of prime factors of $n$, counted with multiplicity, and let $\omega(n)$ denote the number of distinct prime fact…

Let $\mathcal{P}_3$ be the prime set used in the paper, let $\Phi_{4p}(2)$ be the cyclotomic value, and let $\omega(N)$ denote the number of distinct prime divisors of $N$. Divisor…

Let $A(x)=\#\bigl(\mathcal{S}_3^{(\leq 3)}\cap[1,x]\bigr)$, where $\mathcal{S}_3^{(\leq 3)}$ is the bounded-exponent semigroup associated with the prime set $\mathcal{P}_3$. Sublog…

Let $\mathcal{P}_3$ and $\mathcal{S}_3^{(\leq 3)}$ be as used in the paper, and define … … Here $\Phi_{4p}(2)$ is the value of the cyclotomic polynomial $\Phi_{4p}$ at $2$. Divisor…

Let $\mathcal{P}_3$ be the set of odd primes used in the paper whose shifted-prime closure has the stated 3-Higgs property, and let $\mathcal{S}_3^{(\leq 3)}$ be the associated bou…

Let $\mathcal{S}_3^{(\leq 3)}$ be the bounded-exponent semigroup used in the paper, and let $\operatorname{ord}_r(2)$ denote the multiplicative order of $2$ modulo a prime $r$. Hyb…

The classification conjecture. The five integers above are the only unitary perfect numbers. No further unitary perfect number is known; this is the classical Subbarao–Warren probl…

Let $\omega^*(n)$ count the shifted primes $p-1$ dividing $n$, and define its second moment by … Assume the moment asymptotic $M_2(x)\sim C_2\log x$, where $C_2>0$, and let $\zeta$…

Pomerance–Fan moment conjecture. For each $k\ge 2$, there exists a constant $C_k>0$ such that

Adleman–Pomerance–Rumely conjecture. The constant $\frac{1}{2}\log 2$ in this lower bound can be upgraded to $\log 2$, so that for infinitely many $n$,

For a positive integer $n$, let $c(n)$ denote the normalized number of residue classes modulo $n$ covered by congruences whose moduli are divisors of $n$, including the class corre…

Minimal-order conjecture. As $n\to\infty$,

Chor–Lemke–Mador conjecture. There exists a positive constant $C$ such that

Verstraëte's dichotomy conjecture. For some constant $\rho=\rho(k,f)$ depending only on $k$ and $f$, the maximal size of such a set $A$ is either

Let $M_a(n)$ be the partition functions defined by … Let $\vec{P}(x):=(p_1(x),p_2(x),\dots,p_a(x))\in\mathbb{Z}[x]^a$ be a vector of relatively prime integer polynomials. For integ…

Iwaniec–Sárközy conjecture. The upper bound should be replaceable by

Let $m(A)$ denote the largest number of distinct subsets of a finite set $A$ of natural numbers having the same sum. For a random set $\mathbf{A}$ of natural numbers, with each $a$…

Magic sign-smoothing conjecture. If $f(n)$ has constant sign for all $n\geq 1$, then $M_1[f^{-1}]$ and $M_2[f^{-1}]$ are bounded, with eventual signs respectively satisfying

Let $\Omega(n)$ be the number of prime factors of $n$, counted with multiplicity, and let $\mathbb{N}_k=\{n:\Omega(n)=k\}$. For a set of odd primes $\mathcal Q$, write…

Let $d(x)$ denote the number of positive divisors of $x$. Conjecture on three consecutive integers with four divisors. There are infinitely many positive integers $x$ such that … T…

Primitive non-deficient number component conjecture. There is an $i$, with $1\leq i\leq k$, such that

A multiplicative $2$-coloring is a multiplicative function $f:\mathbb N\to\{-1,+1\}$. The bounded multiplicative coloring conjecture. There exists a constant $C$ and a multiplicati…

A set $B\subseteq\mathbb N$ is balanceable if there are a constant $C$ and a $2$-coloring of $\mathbb N$ such that every cascade whose length belongs to $B$ is $C$-balanced. The po…

A golden seed is a finite sequence of signs satisfying the balance and multiplicativity conditions maintained by Rejmer's greedy construction. Rejmer's algorithm conjecture. Rejmer…