On Certain Diophantine Equations Involving Lucas Numbers

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 $L_n = 3x^2$ 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 $F_n=5x^2$ only when $(n,x)=(5,1)$. Expanding our investigation, we prove that the equation $L_n^2+L_{n+1}^2=x^2$ admits a unique solution $(n,x)=(2,5)$. In conclusion, we determine all non-negative integer solutions $(n, \alpha, x)$ to the equation $L_n^\alpha + L_{n+1}^\alpha = x^2$, where $L_n$ represents the $n$-th term in the Lucas sequence.