A Brocard-Ramanujan-type equation with Lucas and associated Lucas sequences

Abstract

This paper deals with a Brocard-Ramanujan-type equation of the form un 1 un 2 . . . un k + 1 = u 2 m in unknown nonnegative integers k, n 1 , n 2 , . . . , n k and m with k ≥ 1, where u = (un) ∞ n=0 is either a Lucas sequence or its associated sequence. For certain infinite families of sequences we completely solve the above equation, extending some results of Marques [15], Szalay [21] and Pongsriiam [18]. The ingredients of the proofs are factorization properties of Lucas sequences, the celebrated result of Bilu, Hanrot and Voutier on primitive divisors of Lucas sequences and elementary estimations concerning the terms involved.

Diophantine equations with balancing-like sequences associated to Brocard-Ramanujan-type problem

“…In this section, we study some Brocard-Ramanujan-type equations that involve balancing-like and associated balancing-like numbers. These results are variants of the works done in [14,31,23,24,21] for other sequences. In the proof of our main results, we use factorizations of balancing-like and associated balancing-like numbers, some results from [7,33] on the existence of primitive prime divisors of Lucas and Lehmer numbers and Lemma 2.5.…”

On a variant of the Brocard--Ramanujan equation and an application

“…Szalay [25] and Pongsriiam [22] worked on another version of the Brocard-Ramanujan problem with Fibonacci, Lucas and balancing numbers, extending the result of Marques [18]. Taşci and Sevgi [26] studied Pell and Pell-Lucas numbers associated with the Brocard-Ramanujan equation, while Pink and Szikszai [21] investigated the Brocard-Ramanujan problem with Lucas and associated Lucas sequences.…”