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

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