A Jacobi Symbol Criterion Involving $k$-Fibonacci and $k$-Lucas numbers and Integer Points on Elliptic Curves

TL;DR

This paper extends a Jacobi Symbol Criterion to broader binary recurrence families and uses it to find all integer points on specific elliptic curves related to these sequences.

Contribution

It introduces a generalized Jacobi Symbol Criterion for binary recurrences and applies it to determine integer points on certain elliptic curves.

Findings

Established a new Jacobi Symbol Criterion for generalized binary recurrences.

Determined all integer points on elliptic curves of the form y^2=5x^2(x+3)^2+4(-1)^n.

Connected properties of Fibonacci-like sequences with solutions on elliptic curves.

Abstract

In 1989, Ming Luo \cite{L2} showed that the Fibonacci number $U_n$ is Triangular if and only if $n=\pm1,2,4,8,10$. For this, he established a Jacobi Symbol Criterion. Moreover, he observed that this problem is equivalent to finding all integer points on two elliptic curves. In this paper, we prove a Jacobi Symbol Criterion for more general families of binary recurrences. In addition, applying the criterion and elementary methods, we determine all integer points on the elliptic curves $y^2=5x^2(x+3)^2+4(-1)^n$.