# On the Rank of the Elliptic Curve y^2=x(x-p)(x-2)

## TL;DR

This paper calculates the rank of specific elliptic curves defined by y^2=x(x-p)(x-2) where p and p-2 are twin primes, using 2-descent, contributing to the classification of elliptic curves by rank.

## Contribution

It applies 2-descent to determine the rank of elliptic curves with twin prime parameters, a novel approach for this family.

## Key findings

- Determined the rank for the family of curves with twin prime parameters
- Identified conditions under which the rank is positive or zero
- Provided explicit rank calculations for specific prime pairs

## Abstract

An elliptic curve E defined over \Q is an algebraic variety which forms a finitely generated abelian group, and the structure theorem then implies that E = \Z^r + \Z_{tors} for some r \geq 0; this value r is called the rank of E. It is a classical problem in the study of elliptic curves to classify curves by their rank. In this paper, the author uses the method of 2-descent to calculate the rank of two families of elliptic curves, where E is given by E: y^2 = x(x-p)(x-2) with p, p-2 being twin primes.