29 problems

Let $f(x)=x^2-1$, and for an integer $n\geqslant 2$ let $P(n)$ denote the set of prime numbers arising in the associated sequence defined in the paper. Density-zero conjecture. For…

Murty–Murty moment conjecture. For every integer $k\ge2$, there exists a constant $C_k>0$ such that

For a prime $p$, let $s_p(n)$ be the sum of the base-$p$ digits of $n$, let $\omega(n)$ count the distinct prime divisors of $n$, and define … Kellner's conjecture. The following s…

Szalay–Ziegler conjecture. If $|S|=2$, then no $S$-Diophantine quadruple exists.

Let $b$ be an integer that is not a negative square, and define the sequence $P_n=n^2+b$. The claim concerns whether this sequence has infinitely many prime terms. Prime-term conje…

Let $b\in\frac{1}{2}\mathbb{Z}$ and let $\phi\in\mathbb{Z}[x]$ be separable and quadratic. Assume that the forward orbit $\{\phi^n(b):n=1,2,\ldots\}$ is infinite. Strong Dynamical…

Let $\phi\in\mathbb{Z}[x]$ be quadratic, with infinite critical orbit and all iterates irreducible. For an integer sequence $a_n=\phi(a_{n-1})$, let $P(a_n)$ denote the largest pri…

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 $n$, define the shifted prime divisor function by … Here $\asymp$ means that the two quantities are bounded above and below by positive constant multiples of…

Let $k$ be a Sierpiński number, meaning that $k\cdot 2^n+1$ is composite for every positive integer $n$. For each positive integer $n$, let the smallest prime divisor of…

Prime-divisor density conjecture. The result that the set of prime divisors of the orbit has density zero remains valid for rational initial conditions and the resulting almost int…

Pomerance–author conjecture. One has

Second-moment constant conjecture. One has

Distinct-prime-divisor conjecture. If $\ell\geq 3$, then $2^n-1$ is divisible by at least $2^\ell$ distinct primes.

The rogue divisor conjecture. The least prime divisor of $k$ is smaller than $p$:

Let $\mathcal Q_H(x)$, $\mathcal Q_{H^-}(x)$, and $\mathcal Q_{H^+}(x)$ count the prime divisors up to $x$ of the $H$-, $H^-$-, and $H^+$-sequences, respectively. Let…

Let $(T,Q)$ be a parameter pair for a recursive sequence, and let the sets of prime divisors of its even-indexed and odd-indexed terms be denoted by $P_{\mathrm{even}}$ and…

For the sequence $5\cdot2^n+1$, let $p_n$ denote its smallest prime divisor. The conjecture. The values $p_n$ are unbounded as $n$ varies. This is presented as a concrete instance…

Let $k$ be a power-free integer, and consider the sequence $k\cdot2^n+1$ as $n$ ranges over the natural numbers. Erdős's conjecture. Erdős's intuition is correct for all power-free…

Let $\mathbb{N}$ and $\mathbb{P}$ denote the non-negative integers and prime numbers, respectively. For $f,g,h\in\mathbb{Z}[X]$, let…

The non-geometric prime-divisor conjecture. If $\mathbf{a}\in\mathcal{R}$ is not a geometric progression, then the set $\mathcal{P}_{\mathbf a}$ is infinite.

Let $p$ and $q$ be a $\sigma_{2,2}$ pair, meaning that they are distinct odd primes satisfying the paper's definition of a $\sigma_{2,2}$ pair. Squarefreeness conjecture. The integ…

A number $n$ with at least two distinct prime divisors is strongly prime-additive if $n=\sum_{p\mid n}p^{a_p}$ for some integers $a_p>0$ and … for every prime $p\mid n$. Erdős–Hegy…

Large prime divisor conjecture. There exists a prime divisor $p$ of this integer such that $p>z$. The paper presents this as an observation suggested by computations and proposes i…

Density conjecture. The set $\mathcal{A}_r$ is infinite. Moreover,