65 problems

Recurrence conjecture. The sequence $(\tau_{n,r})$ satisfies the linear recurrence of order $2r+2$

Let $s=(s(n))_{n\geq 0}$ be a constant-recursive sequence. A general rank sequence is the rank sequence determined by a recurrence and its associated general exponential-polynomial…

Let $p \geq 5$ be a prime with $\varrho_p=3$, let $\alpha$, $\beta$, and $\gamma$ be the roots of $X^3-X-1$ in $\mathbb F_p$, and define … A $\Phi_3$-sequence $(a_n)_n$ is complete…

Let … be a linear recurrence which, for some initial values $n_1,n_2,\dots,n_r$, defines an infinite increasing sequence of natural numbers $S=\\{n_1,n_2,\dots\\}$. Let $\Phi_S(n)$…

Let $k$ be a finite field with cardinality $q$, and let $K$ be a closed set of distributions $[?]$ on $k$. Assume that no $[?]$ is supported on a single point, nor on $ck_0$ for an…

Hollander's conjecture. If $L$ is regular, there exists $p$ such that the limit

Let $(a_n)_{n\in\mathbb{N}}$ be the sequence counting permutations $\pi\in S_n$ such that $\pi(i+2)-\pi(i)\neq 2$ for $i=1,\ldots,n-2$. A linear recurrence equation of order $8$ an…

Period Count for $x^n-1$.

Period Count for $\Phi_{p^j}$.

Let $v_0\in\mathbb{Z}^d\setminus\{0\}$, and suppose $T_{x,v_0}(n)$ is coboundary equivalent to a companion matrix whose recurrence $L$ satisfies the Poincaré–Perron conditions, wit…

Let $\boldsymbol{u}=(u_n)_{n\in\mathbb{Z}}$ be a simple rational LRBS, meaning a simple rational linear recurrence binary sequence, taking values in $\mathbb{Z}[1/b]$ for some inte…

Loxton–Van der Poorten conjecture. For every $\epsilon>0$, there is an effectively computable constant $C(\epsilon)$ such that

Quadratic Easy Coefficients Conjecture. For $n=1,2,\ldots$, the Hamming weight of the associated function $f_n$ satisfies

Exponential local-global principle. The sequence $u$ has no zero term $u_{n}$ if and only if there is an integer $m\geqslant 2$ such that $\gcd(b,m)=1$ and every term $u_{n}$ does…

Let a recurrence have characteristic polynomial with $k$ distinct roots, without multiplicity, and let $N\subseteq\mathbb{Z}^k$ be the subgroup of relations between these roots. Wr…

Let $s=(s(n))_{n\geq 0}$ be a constant-recursive integer sequence, and let its rank be the order of its minimal recurrence. A rank sequence is a sequence of integers of the form…

Let $P_n$) be the Pell numbers defined by $P_n=2P_{n-1}+P_{n-2}$ for $n>1$, with $P_0=0$ and $P_1=1$. Let $S^P_n$ denote the sum of the first $n$ Pell numbers, and let $m^P_n$ be t…

Define integers $G_n,F_n$ by … so that $G_{n+2}=2G_{n+1}+G_n$. Perfect-power conjecture. The equation $G_n=p^r$ has no solution $(n,p,r)$ with $p>2$ and $r\geq 2$. A theorem cited…

Let $p_2(t),\ldots,p_d(t)\in\mathbb{Z}[t]$ be monic polynomials, and consider the split family of Thue equations … Its trivial solutions are … where $\epsilon=1$ for odd $d$ and…

Let the Jacobsthal sequence be defined by $J_0=0$, $J_1=1$, and $J_n=J_{n-1}+2J_{n-2}$, and let $m>1$ be an integer. A modulus $m$ is a fixed point when its Pisano period equals…

Let $(x_n)$ be a third-order linear recurrence sequence, let $p$ be a prime number, and let $up$ denote the $p$-adic valuation. Bilu et al.'s conjecture. There exists a positi…

For a prime $p$, let $(T_n)$ be the Tribonacci sequence defined by … with $T_0=0$ and $T_1=T_2=1$, and let $up$ denote the $p$-adic valuation. Marques–Lengyel conjecture. The…

Marques and Lengyel's conjecture. Similar piecewise formulas should hold for $\nu_p(T(n))$ for primes $p$ other than $2$.

Kontorovich–Lagarias conjecture.

Let $(x_n)_{n\geq0}$ and $(y_n)_{n\geq0}$ be two distinct linear recurrence sequences in the integers. Assume that both grow exponentially in the sense that there is a constant…