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 (sequence A002407 in OEIS):

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, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227

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.

As of January 2006 the largest known has 65537 digits with , found by Jens Kruse Andersen.

The second of these equations is:

This simplifies to . With a substitution it can also be written as .

The first few cuban primes of this form are (sequence A002648 in OEIS):

The name "cuban prime" has to do with the role cubes (third powers) play in the equations, and has nothing to do with Cuba.