55 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 $T_c(x)$ denote the number of primes $p\le x$ such that the largest prime factor $P^+(p-1)$ satisfies $P^+(p-1)\ge p^c$, where $0<c<1$: … and let $\pi(x)$ be the prime-counting…

Let $a,b,c$ be positive integers. Győry–Sárközy–Stewart's conjecture. The largest prime factor of … tends to infinity as $\max\{a,b,c\}\to\infty$. This conjecture concerns the prim…

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…

Pomerance's conjecture. For every $\theta\in(0,1)$,

Let $\theta$ be the supremum of real numbers $c$ such that there are $\gg x/\log x$ primes $p\leq x$ for which $p-1$ has a prime factor greater than $p^c$. Shifted-prime exponent c…

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 $P^+(n)$ 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 positive integers $a$ and $b$, let $B$ denote the dyadic scale of the larger variable, so that $\max(a,b)\in[B,2B)$, and define … where $\gamma(n)=\prod_{p\mid n}p$ and…

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 $n_k$ denote the least integer $n>2k$ such that … is not divisible by any prime in the interval $(k,2k)$. Erdős's conjecture. The quantity $n_k$ grows superpolynomially fast: f…

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…

For each positive integer $d$, let $\Phi_d(X)$ be the $d$th cyclotomic polynomial, and let $\omega(m)$ denote the number of distinct prime factors of $m$. Cyclotomic prime-factor c…

For each positive integer $n$, let $P(2^n-1)$ denote the largest prime factor of the Mersenne number $2^n-1$. Erdős's conjecture. As $n$ tends to infinity, … The conjecture concern…

Let $n>163$ be a positive integer. For positive integers $a$ and $b$, write $n=a+b$, and let $\Omega(ab)$ denote the total number of prime factors of $ab$, counted with multiplicit…

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…

Kontorovich–Lagarias conjecture.

Let $P(m)$ denote the largest prime factor of an integer $m$, with $P(0)=P(\pm 1)=1$. Erdős's conjecture. The quantity $P(2^n-1)$ grows quicker than $n$ as $n$ tends to infinity. T…