30 problems

Classification conjecture. A $\Phi_{\kappa}$-sequence $(a_n)_n$ is complete if and only if

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 $\beta$ 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 $d\geqslant 1$ 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…

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…

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 $a$, $b$ and $d$ be positive integers such that $d$ is not a square, let $\alpha=a+b^{2}\sqrt{d}$ have norm … and let $\varepsilon=(t+u\sqrt{d})/2$ be a unit in…

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…

Let $(a,b)$ be an $(a,b)$-Fibonacci sequence, and suppose it falls under case (vi) of the finite-orders conjecture, namely the case where the range of $\omega_{(a,b)}(m)$ is conjec…

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…

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 $1\leq i\leq t-1$.

Let $a$, $b$, and $d$ be positive integers with $d$ nonsquare, let $\alpha=a+b^{2}\sqrt{d}$ have norm $N_{\alpha}=a^{2}-b^{4}d$, and let $\varepsilon=(t+u\sqrt{d})/2$ be a unit in…

Let $a$, $b$, and $d$ be positive integers with $d$ nonsquare, let $\alpha=a+b^{2}\sqrt{d}$ have norm … and let $\varepsilon=(t+u\sqrt{d})/2$ be a unit in…

Let $s\in\mathbb Z$. The sequence $(P_n(s,0))_{n\geq 1}$ is called polynomially recurrent if it satisfies a polynomial recurrence in $n$. Polynomial-recurrence conjecture. The sequ…

Let $l\geq 1$, let $g,a_i\in\mathbb{C}(x)$ for $i=0,\dots,l-1$, and let $\alpha\in\mathbb{C}$ satisfy $a_i(g^n(\alpha))\neq\infty$ for every $i=0,\dots,l-1$ and $n\geq 0$. Let…

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 $(Q_n)_{n\geq 0}$ be the associated Pell sequence. For a prime $p$, let the rank of apparition (or entry point) $e_Q(p)$ be the smallest positive integer $r$ such that $p$ divi…

Let $L/K$ be a finite extension of number fields and let $m\in\mathbb{N}$. Let … be a polynomial map from $L^m$ to $L^m$. For an element $a=(a_1,\ldots,a_m)\in L^m$, define…

Let $(N_n)$ be the Narayana sequence, and let an $m$-block repdigit be a positive integer formed by repeating the same decimal block of length $m$ at least twice. Block-repdigit ra…

Let $G$ and $H$ be expansions whose corresponding recurrence relations have multiplicatively independent dominant roots. Let $H_G(n)$ and $H_H(n)$ denote the relevant expansion com…

Let $L=(1,-1)$, let $K$ be the parameter appearing in the definition of the generalized sequence $b_T$, and let $h\in\mathbb{N}$. Infinitely-many-instances conjecture for $b_T$. If…

Let $L=(1,-1)$ and $R=(1)$, and let $K$ be the parameter appearing in the definition of the generalized sequence $b_T$. Write $\operatorname{\underline{\deg}} K$ for its indicated d…

A 4-free Fibonacci sequence is a Fibonacci-type sequence defined by the paper's 4-free divisibility rule. Non-cycling conjecture. With probability $1$, a 4-free Fibonacci sequence…

A 3-free Fibonacci sequence is a Fibonacci-type sequence in which each newly appended term is obtained by dividing the sum of the preceding two terms by a power of $3$ whenever the…