39 problems

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 $F=F(A)$ be a free group of rank $k\geq 2$ with free basis $A=\{a_1,\dots,a_k\}$, and let $\alpha:F\to\mathbb Z^k$ be the abelianization homomorphism. For $w\in F$, define…

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…

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…

Let $\mathscr{M}$ be the collection of maximal $\mathsf{d}$-independent families, and for a family $\mathcal{A}$ write $\mathsf{d}[\mathcal{A}]$ for its set of asymptotic densities…

Let $A$ and $B$ be infinite sequences of positive integers with positive lower asymptotic densities, and let $\alpha$ be irrational. Sárközy's conjecture. Each of the equations … a…

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 $A$ be an infinite set of natural numbers. A set $B$ of natural numbers is a sparse additive complement of $A$ if $\bigcup_{a\text{ sufficiently large?}}$ — more precisely, if…

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…

Let $\mathcal{P}$ be the set of all positive primes, let $\mathbb{N}$ be the set of all positive integers, and let $\mathcal{U}$ be the set of positive odd integers that cannot be…

Let $M=(\mu_n)_{n\in\mathbb{N}^*}$ be the sequence defined by … The preceding result gives $b(\Lambda)=+\infty$ for $\Lambda=(\lambda_n)_{n\in\mathbb{N}^*}$ with $\lambda_n=\log n$…

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

Ordowski's density conjecture. The set $\mathcal S$ has an asymptotic density. Counts up to $10^8$ suggest that this density may be about $5/8$. The source proves that the asymptot…

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…

Let $\mathcal{S}$ be the set of odd positive integers $n$ such that $2n/\sigma(n)-1=1/x$ for some positive integer $x$, and let $\pi_{\mathcal{S}}(n)$ denote the number of elements…

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…

Conjecture 4. The lower asymptotic density $\underline{\delta}(\mathcal{N}_d)$ is $0$ for all but finitely many integers $d$ that are neither a square nor the negative of a square.

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 $2^{-k-1}$.