46 problems

Graham's conjecture. The set $S(t,\alpha)$ is complete for every

Given a binary sequence $\mathbf{b}=(b_j)_{j=0}^{n-1}$ with $b_j\in\{\pm1\}$, let $F$ denote its merit factor. Barker-sequence merit-factor conjecture. The values $169/12$ and…

Let $BS(m,n)$ denote four sequences $A,B,C,D$ of $71$ of lengths $m,m,n,n$ with zero combined non-periodic autocorrelation, and let $BS(n+1,n)$ be the corresponding case. Base sequ…

Logarithmic-phase asymptotic conjecture. For all $n>0$,

Let $BS(n+1,n)$ be the set of base sequences of the indicated lengths. A base sequence $(A;B;C;D)\in BS(n+1,n)$ is near-normal when $a_i=(-1)^{i-1}b_i$ for every $i$ with…

Let $p$ be a permutation of $[n]$. A $k$-modal subsequence is a subsequence with at most $k$ changes in direction, where each change is between increasing and decreasing. Chung's c…

A comma sequence in base $b$ is the lexicographically earliest sequence of positive integers satisfying the comma rule for a fixed positive initial value: the difference of consecu…

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

Burr–Erdős–Graham–Li conjecture. For any $s\ge1$, $\operatorname{Pow}(A;s)$ is complete if and only if

Monotonicity conjecture for $S_3^{(3)}$. The sequence

Yang's conjecture. There is an $NNS(n)$ for each even integer $n$. The conjecture was verified for $n\le40$, but exhaustive search found no $NNS(n)$ for $n=42$ and $n=44$, giving c…

Chung's conjecture. For fixed $k$, this upper bound is asymptotically tight as $n$ tends to infinity; equivalently, $\rho(n;k)\sim\sqrt{(2k+1)n}$. This extends the Erdős–Szekeres m…

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 $\lambda_n=\log n$…

Wythoff's losing-state characterization. The states $langle p(n),q(n)\rangle$ for $n\geq 1$ characterize all losing states of Wythoff's game.

Infinite-prime conjecture. There are infinitely many primes of this form.

Let $p>3$ be a non-Mersenne prime, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind. Consider the sequence of values $E_n(1,\frac14)$ modulo $p$. Non-Me…

Let $S$ be a subset of $\mathbb{Z}_n\setminus\{0\}$ with sum $\Sigma S=0$. A rotational sequencing is an ordering of the elements of $S$ whose partial sums are distinct except that…

Let $G$ be an abelian group and let $S$ be a subset of $G\setminus\{0\}$. An ordering of the elements of $S$ is a sequencing if its partial sums, including the initial sum $0$, are…

Let $(h_n)_{n\in\mathbb{N}}$ be the sequence studied in the paper. For $k\in\mathbb{N}$ and $i\in\{0,\ldots,2^k-1\}$, consider the ratios of absolute values of consecutive terms al…

Let $n_1,n_2,n_3,\ldots$ be the sequence defined by $\sigma_{n_\ell}=\ell+1$ for all $\ell\geq 1$, where the values of $\sigma_n$ arise from the paper's preceding construction. Quo…

The conjectures. (A) For every integer $k\ge0$, the set $\{n\ge1:a_n=k\}$ has a positive density $\delta_k$, and

Erdős discrepancy conjecture. For every such sequence $X$, the quantity $D_X(n,k)$ can be made arbitrarily large by choosing appropriate positive integers $n$ and $k$. This is a ce…

Limiting-ratio conjecture. The ratio

Sign-change conjecture. The sequence $(c_n)_{n\geq0}$ changes sign infinitely often.

Growth conjecture for $AP_k$-covering sequences. (i) For every integer $k\geq 3$, there exists an $AP_k$-covering sequence $A$ such that