28 problems

Fix an arbitrary integer $k\geq 2$, and let $a_1,a_2,\ldots,a_k$ be any permutation of $0,1,\ldots,k-1$. Let $P^+(m)$ denote the largest prime factor of $m$. De Koninck–Doyon conje…

Let $P^+(n)$ denote the largest prime factor of $n$. For $a,b\in[0,1]$, define … Let $\rho(t)$ denote the Dickman–de Bruijn function. Erdős–Pomerance conjecture. For every…

Let $P^+(n)$ denote the largest prime factor of $n$, with $P^+(1)=1$. Erdős–Turán conjecture. As $x$ tends to infinity, … This conjecture concerns the expected symmetry between the…

For $n\in\mathbb{N}$, let $\gamma(n)=\prod_{p\mid n}p$ and let $\omega(n)$ denote the number of distinct prime factors of $n$. Define … The convention $0^0=1$ is used when $n=1$. T…

For coprime positive integers $a$ and $b$, put $c=a+b$ and define … where $\gamma(n)=\prod_{p\mid n}p$ is the radical of $n$. Conjecture A. There exists a real number $C_1$ such th…

Let $p_m$ denote the $m$-th prime, set $n_m=p_{m-1}+1$, and let $L_m$ be the largest prime factor of $n_m$ when $L_m>m$; define $R_m=n_m/L_m$. For a fixed positive integer $r$, wri…

Let $\omega(n)$ denote the number of distinct prime factors and define … where $\log_2 n=\log\log n$ and $\log_3 n=\log\log\log n$. Short-interval prime-factor conjecture. For some…

For $x>0$, define … where $\log_2 n=\log\log n$. Short-interval conjecture. For $\varepsilon>0$, there is a constant $C$ such that, for sufficiently large $x$, … and … This conject…

Let $\omega(n)$ denote the number of distinct prime factors and let $\Omega(n)$ count prime factors with multiplicity. Erdős's conjecture. For $\varepsilon>0$, there are infinitely…

Let $\omega(n)$ denote the number of distinct prime factors of $n$. Erdős's conjecture. There are infinitely many $n$ such that … for all integers $1\leq k<n$. This is Erdős Proble…

For a finite set $A\ne\textstyle\varnothing$ of positive integers, let $\Sigma^*(A)$ denote the set of nonempty subset sums, and let $P(A)$ be the greatest prime factor of…

For a finite set $A etextstylevarnothing$ of positive integers, let $\Sigma^(A)$ denote the set of nonempty subset sums, and let $P(A)$ be the greatest prime factor of…

The abc conjecture. For each positive real number $\varepsilon$ there is a positive number $c(\varepsilon)$, which depends on $\varepsilon$ only, such that for all pairwise coprime…

Chen–Chen conjecture. For any integer $k\ge 1$ and any $c\in [\frac{1}{k+1},\frac{1}{k})$, we have

Let $X$, $y$ and $H$ be parameters satisfying the conditions denoted by … , and let $\alpha_y(n)$ be the indicator of integers without prime factors at most $y$, with mean density…

The $D(n)$ bump conjecture. There are arbitrarily large $n$ such that

The $C(n)$ bump conjecture. There are arbitrarily large $n$ such that

The $B(n)$ bump conjecture. There are arbitrarily large $n$ such that

The $A(n)$ bump conjecture. There are arbitrarily large $n$ such that

The generalized growth conjecture. For any $\alpha>1$ and $C>0$, for sufficiently large $n$ one has

Let $\g\in\operatorname{GL}_2(\mathbb{Q})$ be a hyperbolic matrix, that is, one having two distinct real eigenvalues; equivalently … Let $\G=\langle\g\rangle^+:=\{\g^n:n\ge0\}$ be…

Let $p_{1},\ldots,p_{k}$ denote the first $k$ odd primes in succession, and let $n$ be an even integer such that $n>p_{k}$. Williamson's prime-factor conjecture. If, for all…

For a positive integer $n$, let $s_p(n)$ denote the sum of the base-$p$ digits of $n$, and define … Let $P(m)$ be the largest prime factor of $m$. Kellner's conjecture. For $n>192$…

For an integer $a$ and a positive integer $r$, define … where $\operatorname{ord}_p(a)$ is the multiplicative order of $a$ modulo the prime $p$. Erdős's conjecture. For every…

Sárközy–Stewart conjecture. For every $\epsilon$ satisfying $0<\epsilon<1$, there exist $N(\epsilon)$ and $C(\epsilon)>0$ such that, for every integer $N\geq N(\epsilon)$ and every…