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.

First series

This is the first of these equations:

${\displaystyle p={\frac {x^{3}-y^{3)){x-y)),\ x=y+1,\ y>0,}$^[1]

i.e. the difference between two successive cubes. 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, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227 (sequence A002407 in the OEIS)

The formula for a general cuban prime of this kind can be simplified to ${\displaystyle 3y^{2}+3y+1}$. 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^[update] the largest known has 65537 digits with ${\displaystyle y=100000845^{4096))$,^[2] found by Jens Kruse Andersen.

Second series

The second of these equations is:

${\displaystyle p={\frac {x^{3}-y^{3)){x-y)),\ x=y+2,\ y>0.}$^[3]

which simplifies to ${\displaystyle 3y^{2}+6y+4}$. With a substitution ${\displaystyle y=n-1}$ it can also be written as ${\displaystyle 3n^{2}+1,\ n>1}$.

The first few cuban primes of this form are:

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, 69313 (sequence A002648 in the OEIS)

The name "cuban prime" has to do with the role cubes (third powers) play in the equations.^[4]