22 problems

Let $L=F_{q^{n}}$ be a finite field, let $P$ be the $A$-characteristic of $L$, let $m=[L,A/P]$, and let $d=\operatorname{deg}P$. Let $C(d,m,q)$ and $C_{0}(d,m,q)$ denote the two qu…

Density conjecture for sums of distinct powers. If every pair $a_1,a_2\in A$ satisfies $\gcd\{a_1,a_2\}=1$ and

Exponential word-length growth conjecture. There exists a constant $c_r>1$ such that

Let $F_r$ be a free group of rank $r\geq 2$, and interpret “almost all” using asymptotic density on the set of elements of $F_r$ ordered by word length. An element is Whitehead min…

Erdős–Granville–Pomerance–Spiro conjecture. If $\mathcal{A}$ has asymptotic density zero, then $s^{-1}(\mathcal{A})$ also has asymptotic density zero.

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

Asymptotic-density-zero conjecture. The set $A$ has asymptotic density $0$ in the natural numbers. The corresponding density statement is known for the dimensions…

Let $\mathcal{W}_2$ be the set of all positive integers $a$ for which there is no integer $m>1$ such that $(a-2^k,m)>1$ for every positive integer $k$. The $\mathcal{W}_2$ density…

EGPS conjecture. The preimage $s^{-1}(\mathcal{A})$ also has asymptotic density zero.

Let $\mathcal{S}$ be the set of odd positive integers $n$ such that $2n/\sigma(n)-1=1/x$ for some positive integer $x$. Its asymptotic density is the limit, when it exists, of … wh…

Density-one conjecture. The asymptotic density of $T$ in $S$ is $1$. The paper proves that an explicitly described infinite family has $c(n)=\infty$, and asks for a simple way to d…

Let $\mathcal{W}{2,n}$ denote the set of binary words of length $n$, and let $Vn=\mathcal{U}({0,1})capmathcal{W}{2,n}$. The density of the binary Ulam set is measured by…

Density conjecture. The lower asymptotic density satisfies

Sign-exception density conjecture. The sets $\mathcal{A}_{m,j}$ are infinite and have asymptotic density zero:

Generalized Uniformness Conjecture. The asymptotic density of $S$ equals $2^{-k-1}$.

Let $F_{1,n,n}(r)$ denote the sum of the reciprocals of the absolute determinants of all full-rank $n\times n$ integer matrices whose entries have Euclidean norm at most $r$, and l…

Density-minimum conjecture.

Density conjecture. If $\mathfrak{N}_Q$ and $\mathfrak{N}_{Q'}$ are non-empty and $\mathfrak{n}_Q<\mathfrak{n}_{Q'}$, then

Let $C(n)$ and $W(n)$ denote the numbers of Carmichael numbers and weak Carmichael numbers in the interval $[1,n]$, respectively. Asymptotic equality conjecture. … The conjecture p…

Limiting lower-bound conjecture. The sequence of lower bounds satisfies

Limiting $V_n$-density conjecture. The limiting density is