Cuban prime

A cuban prime is a prime number that is also a solution to one of two different specific equations involving differences between third powers of two integers x and y.

This is the first of these equations: i.e. the difference between two successive cubes.

The first few cuban primes from this equation are The formula for a general cuban prime of this kind can be simplified to

This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal.

As of July 2023[update] the largest known has 3,153,105 digits with

The second of these equations is: which simplifies to

The first few cuban primes of this form are: The name "cuban prime" has to do with the role cubes (third powers) play in the equations.

Proof without words that the difference of two consecutive cubes is a centered hexagonal number by arranging n 3 semitransparent balls in a cube and viewing along a space diagonal – colour denotes cube layer and line style denotes hex number