67 problems

Given a finite starting sequence of numbers, extend it by repeatedly appending the curling number of the sequence so far. The curling number is the largest $k$ for which the sequen…

Let $a$, $b$ and $d$ be positive integers such that $d$ is not a square. Set … and let $\varepsilon=(t+u\sqrt{d})/2$ be a unit in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ with $t,u$ po…

Let $l\geq 1$, let $F\in\mathbb{C}[x_0,\dots,x_{l-1}]$, and let $\{A_n\}_{n\geq 0}$ be a recurrence sequence satisfying … for all $n\geq 0$. The nonlinear Skolem–Mahler–Lech conjec…

Let $A(n,p,q)$ be the perturbed Gale–Robinson Laurent polynomial defined by … with initial variables $A(n,p,q)=x_{n,p,q}$ for $0\leq n<m$. Thus each $A(n,p,q)$ is a Laurent polynom…

Let $a(x)$ be the continuous-case function defined by the preceding maximization problem, and let $(a_i)_{i\geq 1}$ be the sequence defined by $a_1=1$, $a_2=2$, $a_3=4$, and, for…

Normal-set representation conjecture. If $S$ is a $p$-normal set, then there exists a field $K$ of characteristic $p$ and a $K$-recurrence sequence $a\in K^{\mathbb N}$ such that

Let $(a(n))$ be a recurrence sequence, and let its dominating characteristic roots be the characteristic roots of maximal modulus among those occurring with nonzero coefficient in…

Let $R$ be a second-order linear recurrence whose characteristic polynomial is reducible over $\mathbb Q$, so that, in the notation of the source, its general term has the form…

Let $\beta$ be a nonzero integer and define the sequence $P=(P_n)$ by … A term $P_n$ has a primitive divisor if it has a prime divisor that divides no earlier nonzero term of the s…

Let $\alpha=(\alpha_0,\alpha_1,\dots)$ and $\beta=(\beta_0,\beta_1,\dots)$ be sequences with $\alpha_0=\beta_0=\gamma_0$ satisfying linear recurrence relations of orders $a$ and…

Let $d\geqslant 1$ be the dimension of the base $\mathbb{R}^d$, let $k\geqslant 1$ be a differential order, and set … Let $W_d^k$ denote the Wronskian $N$-ary bracket on…

Infinitely many Cullen primes conjecture. There are infinitely many Cullen primes.

The unrestricted four-squares conjecture. There are at most four distinct integer squares among the $y_k'$. If $|N_{\alpha}|$ is a prime power or a perfect square, then there are a…

Let $A_n,B_n,C_n,D_n,E_n$ be the five complementary sequences obtained from the mex construction, with $E_n$ the anti-Teranacci sequence and $A_n,B_n,C_n,D_n$ the missing-number se…

Let $A_n$ be the first of the four complementary sequences arising from the mex construction of the anti-Tribonacci sequence. Kimberling's conjecture. For every relevant positive i…

Let $A_n,B_n,C_n,D_n$ be the four sequences obtained by the mex construction for the anti-Tribonacci sequence, with $D_n=A_n+B_n+C_n$. Kimberling's conjecture. For the correspondin…

Unrestricted-power four-squares conjecture. There are at most four distinct integer squares among the $y_k'$. If $|N_{\alpha}|$ is a prime power or a perfect square, then there are…

Let $\{g_n\}_{n\geq 0}$ be the sequence defined earlier in the paper through the rank-three Witten zeta function of type $G$. Nonvanishing conjecture for $g_n$. If…

Cerlienco's conjecture. If $K=\mathbb{Q}$, then the property

Let $(a,b)$-Fibonacci sequences be defined by $\mathcal{U}_0=0$, $\mathcal{U}_1=1$, and $\mathcal{U}_n=a\mathcal{U}_{n-1}+b\mathcal{U}_{n-2}$. For $m>1$, let $\omega_{(a,b)}(m)$ be…

Let $(a,b)$-Fibonacci sequences be defined by $\mathcal{U}_0=0$, $\mathcal{U}_1=1$, and $\mathcal{U}_n=a\mathcal{U}_{n-1}+b\mathcal{U}_{n-2}$. Let $\omega_{(a,-1)}(m)$ be the numbe…

Let $K$ be even, and let $F_{K,n}$ be the $K$-Fibonacci sequence. For a modulus $m$, let $\pi_K(m)$ denote its Pisano period, let $\omega_K(m)$ denote the number of zeros in that p…

Fixed-multiple conjecture. There exists an integer $C(N)$ such that for every $n$ there exists an integer index $j(n;N)$ satisfying

Hone's conjecture. The sequence $(a_k)_{k\in\mathbb N}$ contains infinitely many primes.

Simplicity conjecture. The roots of $\chi_S(x)$ are all simple whenever $d_t\equiv 1\pmod{2}$ and $d_i\equiv 0\pmod{2}$ for all $1\leq i\leq t-1$.