Bilu et al.'s conjecture on p-adic valuations of third-order recurrence sequences

Let $(x_n)$ be a third-order linear recurrence sequence, let $p$ be a prime number, and let $u_p$ denote the $p$-adic valuation. Bilu et al.'s conjecture. There \exists a positive integer $Q$ such that, for every $i\in\{0,1,\dots,Q-1\}$, one of the following holds:

• There exists $\kappa_i\in\mathbb{Z}_{\geq 0}$ such that, for all but finitely many $n\in\mathbb{Z}$ satisfying $n\equiv i\pmod{Q}$,

$$\nu_p(x_n)=\kappa_i.$$

• There exist $a_i\in\mathbb{Z}$, $\kappa_i\in\mathbb{Z}$, and $\mu_i\in\mathbb{Z}_{>0}$ satisfying

$$\nu_p(a_i-i)\geq\nu_p(Q),$$

सuch that, for all but finitely many $n\in\mathbb{Z}$ satisfying $n\equiv i\pmod{Q}$,

$$\nu_p(x_n)=\kappa_i+\mu_i\nu_p(n-a_i).$$

The paper presents this as equivalent to the Marques–Lengyel conjecture and notes that it fails for infinitely many primes in the Tribonacci setting; its validity for arbitrary third-order linear recurrence sequences is the subject of the article.