34 problems

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…

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

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…

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 sequence. A general rank sequence is the rank sequence determined by a recurrence and its associated general exponential-polynomial…

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…

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…

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

Let $I$ be an index set, let $Q_{I,n}$ be the associated rotation-symmetric quadratic Boolean function, and set … Let $w_1,w_2,\ldots$ be the sequence obtained by extending the rec…

A Lyndon word over $\{0,1\}$ is a word that is the unique minimum among all of its rotations. For $k\geq2$, let $\alpha=(1,0,0,\dots,0)$ be the prefix of length $k$, and let…

Let $\boldsymbol u=\langle u_n\rangle_{n=-\infty}^{\infty}$ be a simple rational LRBS taking values in $\mathbb{Z}[\frac{1}{b}]$ for some non-zero integer $b$. A sequence has no ze…

Let $a,c,d,e$ be algebraic indeterminates, and define $T(n,k)$ by … for $n\geq 1$, with $T(0,k)=\delta_{k0}$. Thus $\bm{T}=(T(n,k))_{n,k\geq 0}$ is a lower-triangular matrix over…

Inverse-relationship conjecture. $Q_0(r)$ and the run-time have an inverse relationship.

Generalized principal-coefficient conjecture. The coefficient $a_1$ satisfies

The multiplicative-independence counting conjecture. The numbers $\alpha$ and $\beta$ are multiplicatively independent if and only if

Uniqueness conjecture. There exists a unique decomposition for each positive integer $N$.

Non-uniqueness conjecture. Uniqueness of decomposition is lost for at least one positive integer $N$.

Let $M$ denote Mahler's measure, and write $M^{(n)}$ for its $n$-fold iterate. For an algebraic unit $\alpha$, consider the sequence of real numbers…

Let $k\geq 5$, let $W$ be the relevant word set, let $\nu^{(k)}$ be the coefficient function, let $L_p\in\Lambda_p$ be the linear recurrence associated with the coefficients, and l…

Let $p=p_\lambda$ be the polynomial family considered in the paper, and let $\gamma(p)$ denote the quantity defined in the preceding observation. Parity formula for $p_\lambda$. Fo…

Let $G_k$ be the Hecke group with parameter $k$, let $W$ denote the words under consideration, and let $\nu_1,\ldots,\nu_8$ be the functions defined in Observation 1, using the Fib…

Let $\pmb{\kappa},\pmb{\lambda},\pmb{\mu},\pmb{\nu}$ be as in Theorem. For every vector $\mathbf{w}$, let $\overline{\mathbf{w}}$ denote the vector obtained by rearranging its coor…

Exact recurrence-order conjecture. The values are

Eigenvalue multiplicity conjecture. For positive integers $s$, the following hold: