A Family of Elliptic Curves With Rank $\ge 5$

Abstract

In this paper, we construct a family of elliptic curves with rank $\ge 5$. To do this, we use the Heron formula for a triple $(A^2,B^2,C^2)$ which are not necessarily the three sides of a triangle. It turns out that as parameters of a family of elliptic curves, these three positive integers $A$, $B$, and $C$, along with the extra parameter $D$ satisfy the quartic Diophantine equation $A^4+D^4=2(B^4+D^4)$.

arXiv comment9 pages, keywords: Diophantine equation, elliptic curve and Heron formula