19 problems

Index divisor conjecture. There exists a prime $p>k$ which divides $\Delta$. Moreover, the index at which $p$ occurs in the product $n(n-1)\cdots(n-k)$ does not divide the power to…

Erdős–Selfridge conjecture. If $n+k\geq p^{(k)}$, then there is a prime greater than $k$ which divides $(n+1)\cdots(n+k)$ to the first power.

Index divisor free prime conjecture. Except when $(d,k)=(27,3)$, there exists an index divisor free prime for $(d,k)$.

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…

Erdős–Turán conjecture for two-variable polynomials. There exists a constant $c$ depending only on $f$ such that

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…

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…

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 $p$:

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…

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…

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…

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,

Let $T_k$ denote the set of primes $p$ dividing an integer of the form $2^{a_1}+\dots+2^{a_k}+1$, where $a_1,\ldots,a_k\in\mathbb{N}$, and let $T_k(X)$ be the number of such primes…

Let $c_k$ be the limiting proportion of vertices of rank $k$, and let $denom(c_k)$ denote its denominator when $c_k$ is written in lowest terms. Denominator prime-divisor bound con…