25 problems

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…

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…

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…

Let $G$ be a finite group. Write $|G|$ for its order and let $\mathsf{E}(G)$ denote its Gao constant, namely the least integer such that every sequence over $G$ of that length cont…

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 $x,y$.

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…

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

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…

Short-progression exclusion conjecture. One has

Weak Loneliness Spectrum Conjecture. For every integer $n\geq4$, there exists a function $f_n:\mathbb{N}\to\mathbb{N}$ such that, whenever $v_n>f_n(v_{n-1})$, either

Symmetry conjecture. If there is an LP of length $n$ and exact order $m$, then there is one that is $(-1)^m$-symmetric. Alternatively, if there is an equipowerful bipartion of leng…

Let $r\ge 1$, let $N_1,\ldots,N_r$ be positive integers, and let $x_1,\ldots,x_r,y_1,\ldots,y_r$ be variables. For each nonnegative integer $n$, write $s_2(n)$ for the sum of the b…

Let $r\ge 1$, let $N_1,\ldots,N_r$ be positive integers, and let $x,y_1,\ldots,y_r$ be variables. For each nonnegative integer $n$, write $s_2(n)$ for the sum of the binary digits…

Let $t$ be an integer with $t\geq 2$, and let $n$ and $d$ be positive integers. For an element of ${\mathbb Z}/(t^n-1){\mathbb Z}$, write its standard $t$-ary expansion with digits…

Let $k$ and $l$ be positive integers, and let $p_1,\ldots,p_k,q_1,\ldots,q_l$ be positive integer gap lengths. A set of $k+l+1$ integers has gap sequence…

Let $W(r,k)$ denote the van der Waerden number. Quadratic growth-ratio conjecture. As $k$ grows large with $k>r$, … This is the simplified form obtained in the source under the add…

Let $W(r,k)$ denote the van der Waerden number, and write $W(r,k) = (r)^{n_r}$ and $W(r+1,k)=(r+1)^{n_{r+1}}$ for the corresponding exponents. Growth-ratio conjecture. As $r$ grows…

Sun's conjecture. There exists a permutation $\sigma$ of $\{1,\ldots,n-1\}$ such that $n\nmid s m_{\sigma(s)}$ for every $s=1,\ldots,n-1$. Theorem 1.1 in the paper confirms this co…

Let $\Lambda$ be a numerical semigroup, meaning a cofinite subsemigroup of the non-negative integers. Let $e(\Lambda)$ be the size of its minimal generating set, let…

Main conjecture. The first index for which there is a constant class is $m_{0}-1$: every class is part of the $j$-level for $1\leq j\leq m_{0}-2$. For every $m\geq m_{0}$, the $m$-…