7 problems

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–…

Let $[c_1,\dots,c_L]$ define a positive linear recurrence sequence (PLRS) $\\{H_n\\}$, and let $B_{H,n}$ denote its $n$th Brown's gap. The $2L-1$ conjecture. The sequence…

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

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…