On the 𝑝-adic zeros of the Tribonacci sequence

Let ( T n ) n ∈ Z (T_n)_{n\in {\mathbb Z}} be the Tribonacci sequence and for a prime p p and an integer m m let ν p ( m ) \nu _p(m) be the exponent of p p in the factorization of m m . For p = 2 p=2 Marques and Lengyel found some formulas relating ν p ( T n ) \nu _p(T_n) with ν p ( f ( n ) ) \nu _p(f(n)) where f ( n ) f(n) is some linear function of n n (which might be constant) according to the residue class of n n modulo 32 32 and asked if similar formulas exist for other primes p p . In this paper, we give an algorithm which tests whether for a given prime p p such formulas exist or not. When they exist, our algorithm computes these formulas. Some numerical results are presented.