13 problems

Hong–Yang conjecture. The integer

A distinct covering system is a covering system whose moduli are distinct integers greater than $1$. Let $m\geq 2$, let $a$, $b$, and $c$ be positive integers, and let $L$ denote t…

Let $g(k,x)$ be the largest integer $s$ for which there exist integers … with no $k$ of the $a_i$ forming an LCM-$k$-tuple, where an LCM-$k$-tuple consists of distinct integers who…

Erdős's stronger conjecture. There exist integers $k$ and $0<x<y$ with $y>x+k$ such that

Let $f\in\mathbb{Z}[X]$, let … and let … A prime is said to divide $Q_f(N)$ exactly once when its exponent in the prime factorization of $Q_f(N)$ is one. Sah's conjecture. The size…

Sah's conjecture. As $N\to\infty$,

Let $q=p^k$ be a prime power, and let $f\in\mathbb{F}_q[T][X]$ be a fixed irreducible polynomial with $X$-degree $d\geq 2$. Define … Let $V_f:=\{g\in\mathbb{F}_q[T]:f(X+g)=f(X)\}$…

The conjecture for $\psi_F$. One should have

Let $k\ge 3$ and define … The notation $[n_1,\ldots,n_k]$ denotes the least common multiple of $n_1,\ldots,n_k$. Let $P_{2^k-1}(t)$ be a polynomial in $t$ of degree $2^k-1$, and le…

Shevelev–Moses conjecture. The sequence $a_n$ is given by

Farhi's conjecture. The exponent $n-1$ in the known lower bound should be replaceable by $n$, yielding the optimal exponent obtainable in such a lower bound. This conjecture was co…

Let $f\in\mathbb Z[x]$ be irreducible over $\mathbb Q$ of degree $d\ge 2$, and define … where $\operatorname{rad}(n)$ is the product of the distinct primes dividing $n$. Radical LC…

Let $f(x)=adx^d+dots+a0 i mathbb{Z}[x]$ be an irreducible polynomial of degree $d a 3$, with $ad0$, and let $psif(n)$ denote the least common multiple of the values of $f$ up to…