A Family of Elliptic Curves With Rank $\geq5$

TL;DR

This paper constructs a family of elliptic curves with rank at least 5 using a novel approach involving Heron’s formula and a specific quartic Diophantine equation, expanding understanding of elliptic curve ranks.

Contribution

It introduces a new method to generate elliptic curves with high rank by linking Heron’s formula to solutions of a quartic Diophantine equation.

Findings

Constructed explicit family of elliptic curves with rank ≥ 5

Established connection between Heron formula and elliptic curve parameters

Solved a specific quartic Diophantine equation related to curve construction

Abstract

In this paper, we construct a family of elliptic curves with rank $\geq 5$. 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 $A^4+D^4=2(B^4+D^4)$.