On Diophantine equations involving intersection of Thabit and Williams numbers base b and some ternary recurrent sequences

2025/10/14 by Tripathy, Bibhu Prasad, Satapathy, Asutosh, Dutta, Utkal Keshari +1
#11B39 #11D61 #11J86 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.12145

Abstract

Let P_n be the n-th Padovan number, E_n be the n-th Perrin number and N_n be the n-th Narayana's cows number. Let b be a positive integer such that b ≥ 2. In this paper, we study the Diophantine equations P_n = (b ± 1)⋅ b^l ± 1, E_n = (b ± 1)⋅ b^l ± 1, and N_n = (b ± 1)⋅ b^l ± 1, in non-negative integers n, b and positive integer l. As a result, we determine the Padovan, Perrin and Narayana's cows numbers that are Thabit and Williams numbers base b. Moreover, we determine all solutions of the above equations within the range 2 ≤ b ≤ 10.

Citations

• Thabit and Williams Numbers Base b as a Sum or Difference of Two g-Repdigits
• Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers

Related

• Common terms of generalized Pell and Narayana's cows sequences
• Repdigits in Narayana's Cows Sequence and their Consequences
• Identities involving Narayana numbers
• Narayana numbers that are products of two Fibonacci numbers
• Members of Narayana's cow sequence that are concatenations of two repdigits