The Uniform Primality Conjecture for the Twisted Fermat Cubic

TL;DR

This paper proves that the number of prime terms in the elliptic divisibility sequence on the twisted Fermat cubic is uniformly bounded, with all but the first terms non-prime under certain conditions.

Contribution

It establishes a uniform bound on prime terms in the sequence and characterizes the primality of terms when related by a 3-isogeny.

Findings

Number of prime terms is uniformly bounded.

All terms beyond the first are non-prime under specific isogeny conditions.

Provides evidence for the uniform primality conjecture in this context.

Abstract

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point under a certain 3-isogeny, all terms beyond the first fail to be primes.