A cuban prime (from the role cubes (third powers) play in the equations) 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, ... (sequence A002407 in the OEIS)
The general cuban prime of this kind can be rewritten as
As of January 2006 the largest known has 65537 digits with
The second of these equations is:
This simplifies to
The first few cuban primes of this form are (sequence A002648 in the OEIS):
13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313With a substitution
Generalization
A generalized cuban prime is a prime of the form
In fact, these are all the primes of the form 3k+1.