16 problems

Let $(F_n^{(k)})$ be the $k$-generalized Fibonacci sequence, and let $G$ be the Galois group associated with its characteristic polynomial, viewed as a subgroup of the symmetric gr…

Zero-set boundedness conjecture. The cardinality of $\mathcal S(u_m)$ is bounded in terms of $n$ only.

Let $f$ be an irreducible polynomial, and call it LRS-degenerate when it has the property defined in the paper. Write $k$ for an order for which $f$ is $k$-LRS-degenerate, and let…

Let $f$ be a polynomial, and call it LRS-degenerate when it has the property defined in the paper; let $k$ denote the minimal order for which $f$ is $k$-LRS-degenerate, and let…

Let $H^{(k)}$ be the $k$-generalized Fibonacci sequence with non-positive indices, and let $\mathcal{Z}(H^{(k)})=\{n\in\mathbb{Z}:H_n^{(k)}=0\}$ denote its zero set. The zero-multi…

Let $P,Qin\thinspace\mathbb Z$ and let $(u_n)$ be defined by … with initial conditions $u_0=0$ and $u_1=1$. A sequence satisfies condition (D) when its Dold congruences hold. Moss–…

For each length $L\geq 2$, let $R_L^{\mathrm{comp}}$ denote the set of principal roots of complete positive linear recurrence sequences generated by $L$ coefficients. Density conje…

Restricted smallest-root conjecture. If the sequence generated by $[c_1,\ldots,c_L]$ is incomplete and satisfies

Smallest-root conjecture. If $[c_1,\ldots,c_L]$ generates an incomplete sequence, then its principal root is at least $\lambda_L$. Equivalently, the incomplete sequence of length…

Let $(H_n)$ be the positive linear recurrence sequence defined by the coefficient list $[c_1,\dots,c_L]$, and let $B_{H,n}$ denote the quantity in Brown's criterion, so that comple…

Monotonicity conjecture. If the sequence generated by $[\underbrace{1,\dots,1}_{g},\underbrace{0,\dots,0}_{k},N]$ is complete, then so is the sequence generated by…

Let $(H_n)$ be the positive linear recurrence sequence generated by the coefficient list $[1,1,\underbrace{0,\dots,0}_{k},N]$. Let $f_1=1$ and $f_2=2$, with $f_n$ denoting the Fibo…

Let $alpha \in \bar{\Z}$, let $K=\Q(\alpha)$, and let $d_k(\alpha)$ denote the largest positive integer such that … Assume that either (a) $[\Q(\alpha^r):\Q]\geq 3$ for every…

A Baillie–PSW pseudoprime is an odd composite integer that passes both the strong probable-prime test and the Lucas primality test. Baillie–PSW pseudoprime infinitude conjecture. T…

Let $\{s_n\}_{n\in\mathbb N}$ and $\{t_n\}_{n\in\mathbb N}$ be linear recurrence sequences of complex numbers. Suppose that $s_n/t_n$ is an integer for every $n\in\mathbb N$. Pisot…

Juricevic's conjecture. There are infinitely many prime numbers $p>5$ such that $T(p)$ is a prime number. This conjecture is one of two conjectures assumed in Juricevic's classific…