3 problems

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