6 problems

Conjecture on trigonometric identities. If

Van de Lune's conjecture. The records of $S_n(\sqrt 2)$ satisfy a Pell-like recurrence. The source states that this specific conjecture was confirmed in prior work, so it is includ…

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 $(Q_n)_{n\geq 0}$ 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…

Siar–Keskin's Pell-family conjecture. For every odd integer $k>3$, the equation has only the solution

Let $THK(m,n)$ denote the Turk's Head link obtained by alternating a braid representation of the $(m,n)$ torus link. Here $m,n>1$, and $THK(m,n)$ is a knot when $m$ and $n$ are rel…