43 problems

Let $k\geq 1$ be an integer, and let $v_1,\ldots,v_k$ be distinct integers. For a real number $x$, write $\|x\|$ for the distance from $x$ to the nearest integer. Lonely runner con…

Let $f_{p,q}(n)$ be the Birch–Erdős representation function, with $p$ and $q$ relatively prime integers satisfying $q>p>2$. Sign-change conjecture. The sign of … changes infinitely…

Craigen–Holzmann–Kharaghani conjecture. Complex Golay pairs do not exist in length $23$.

Let $C_n$ be a cyclic group of order $n$, and let $\operatorname{\mathsf{g}}(G)$ denote the smallest integer $k$ such that every squarefree sequence over a finite abelian group $G$…

Divisibility conjecture. The cardinality $|L|$ divides

Let $f_{p,q}(n)$ be the Birch–Erdős representation function. Equality-frequency conjecture. … for infinitely many $n$. This conjecture accompanies the proposed sign-change phenomen…

For integers $d\geq2$ and $g\geq d+1$, let $S_{1,g,g+d}$ be the greedy 3-subsumfree sequence beginning with $1,g,g+d$. The meta-conjecture. For every $g\geq d+1$, … In particular,…

For $g\geq 2$, let $S_{1,g,g+1}$ be the greedy 3-subsumfree sequence beginning with $1,g,g+1$, where each subsequent term is the smallest larger integer that is not the sum of thre…

Let $S_{1,2,3}$ be the greedy 3-subsumfree sequence beginning with $1,2,3$: after the initial values, each term is the smallest larger integer that is not the sum of three distinct…

Let $X_n$ be the sequence of anti-$k$-naccis, defined as sums of $k$ consecutive missing numbers. A sequence is $k$-automatic when its base-$k$ value sequence can be generated by a…

FKM's pairwise partition-regularity conjecture. If at least one of $ac$, $bc$, or $(a+b)c$ is a square, then $a x^2+b y^2=c z^2$ is partition regular with respect to $x,y$.

Kimberling's conjecture. The sequence $(S(n))_{n\geq 1}$ consists precisely of the positive integers not belonging to the Beatty sequence

Let $c(x)$ be a monic integer polynomial with coefficients $0$ or $1$, and write … Assume that $a(x)$ and $b(x)$ are monic polynomials with non-negative real coefficients. The unfa…

Kubertin–Gao–Han conjecture. The optimal bound in the equality

For an integer $n\geq 1$, the central binomial coefficient is $\binom{2n}{n}$. An integer is squarefree if it is not divisible by the square of any prime. Erdős's squarefree conjec…

Let $\\{H_n\\}$ be an incomplete positive linear recurrence sequence generated by $[c_1,\dots,c_L]$, and let $B_{H,n}$ be its $n$th Brown's gap. The $2L-1$ conjecture. If…

Let $[c_1,\dots,c_L]$ define a positive linear recurrence sequence (PLRS) $\\{H_n\\}$, and let $B_{H,n}$ denote its $n$th Brown's gap. The $2L-1$ conjecture. The sequence…

For each length $L\geq 2$, let $R_L^{\mathrm{comp}}$ denote the set of principal roots of complete positive linear recurrence sequences generated by $L$ coefficients. Density conje…

Restricted smallest-root conjecture. If the sequence generated by $[c_1,\ldots,c_L]$ is incomplete and satisfies

Smallest-root conjecture. If $[c_1,\ldots,c_L]$ generates an incomplete sequence, then its principal root is at least $\lambda_L$. Equivalently, the incomplete sequence of length…

Let $(H_n)$ be the positive linear recurrence sequence defined by the coefficient list $[c_1,\dots,c_L]$, and let $B_{H,n}$ denote the quantity in Brown's criterion, so that comple…

Monotonicity conjecture. If the sequence generated by $[\underbrace{1,\dots,1}_{g},\underbrace{0,\dots,0}_{k},N]$ is complete, then so is the sequence generated by…

Let $(H_n)$ be the positive linear recurrence sequence generated by the coefficient list $[1,1,\underbrace{0,\dots,0}_{k},N]$. Let $f_1=1$ and $f_2=2$, with $f_n$ denoting the Fibo…

Let $f$ be a coloring of $\mathbb N$ by the two colors $\{-1,+1\}$. A finite set $A=\{a_1<\cdots<a_n\}$ is divinely colored if at least half of $a_2,\dots,a_n$ have a color differe…

For a positive integer $k$, define the arithmetic proximity of positive integers $a,b$ by $\max\{a/d,b/d\}$, where $d=\gcd(a,b)$. The arithmetic graph $B_k$ has vertex set…