On Certain Diophantine Equations Involving Lucas Numbers

2024/09/16 by Priyabrata Mandal, Mandal, Priyabrata
Physics and Astronomy · Mathematics · #Advanced Mathematical Theories and Applications #Advanced Mathematical Identities #Advanced Mathematical Theories

paper · pdf · doi:10.48550/arxiv.2409.10152

Abstract

This paper explores the intricate relationships between Lucas numbers and Diophantine equations, offering significant contributions to the field of number theory. We first establish that the equation regarding Lucas number L_n = 3x² has a unique solution in positive integers, specifically (n, x) = (2, 1), by analyzing the congruence properties of Lucas numbers modulo 4 and Jacobi symbols. We also prove that a Fibonacci number F_n can be of the form F_n=5x² only when (n,x)=(5,1). Expanding our investigation, we prove that the equation L_n²+L_n+1²=x² admits a unique solution (n,x)=(2,5). In conclusion, we determine all non-negative integer solutions (n, α, x) to the equation L_n^α+ L_n+1^α= x², where L_n represents the n-th term in the Lucas sequence.

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