A question about points on an elliptic curve with prime denominator

TL;DR

This paper investigates the frequency with which the denominator of rational points on elliptic curves, expressed in a specific coprime form, is a prime number, addressing a fundamental question in number theory.

Contribution

The paper formulates and explores the question of how often the denominator of rational points on elliptic curves is prime, providing new insights into the distribution of such points.

Findings

Initial results suggest primes occur infrequently as denominators.

Heuristic arguments indicate possible density patterns for prime denominators.

Open problems are proposed for further research.

Abstract

Let E be an elliptic curve defined by a Weierstrass equation with integer coefficients. Any rational point on E other than the identity is of the form $ ( x(P) / z(P)^2 , y(P) / z(P)^3 ) $ where $ x(P), y(P) \in \mathbb Z $ and $ z(P) \in \mathbb N $ and in addition both $ \gcd( x(P) , z(P) ) = 1 $ and $ \gcd( y(P) , z(P) ) = 1 $ hold. The question is: how often is $ z(P) $ a prime number?