Cuban prime


A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y. The first of these equations is:
and the first few cuban primes from this equation are:
7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241,...
The general cuban prime of this kind can be rewritten as, which simplifies to. This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal.
the largest known has 65537 digits with, found by Jens Kruse Andersen.
The second of these equations is:
This simplifies to.
The first few cuban primes of this form are :
With a substitution, the equations above can also be written as follows:

Generalization

A generalized cuban prime is a prime of the form
In fact, these are all the primes of the form 3k+1.