Cube sums of form $3p$ and $3p^2$ II

TL;DR

This paper proves that numbers of the form 3p and 3p^2 are cube sums for primes p ≡ 2,5 mod 9, and explores related elliptic curves' properties including explicit Gross-Zagier formulas and the 3-part BSD conjecture.

Contribution

It establishes the cube sum property for 3p and 3p^2 with primes p ≡ 2,5 mod 9, and derives explicit formulas and conjecture investigations for related elliptic curves.

Findings

Both 3p and 3p^2 are proven to be cube sums.

Explicit Gross-Zagier formulas are established.

Investigation of the 3-part BSD conjecture for related elliptic curves.

Abstract

Let $p\equiv 2,5\mod 9$ be a prime. We prove that both $3p$ and $3p^2$ are cube sums. We also establish some explicit Gross-Zagier formulae and investigate the 3 part full BSD conjecture of the related elliptic curves.