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.

First series
This is the first of these equations:


 * $$p = \frac{x^3 - y^3}{x - y},\ x = y + 1,\ y>0,$$

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

The formula for a general cuban prime of this kind can be simplified to $$3y^2 + 3y + 1$$. This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal.

the largest known has 3,153,105 digits with $$y = 3^{3304301} - 1$$, found by R.Propper and S.Batalov.

Second series
The second of these equations is:


 * $$p = \frac{x^3 - y^3}{x - y},\ x = y + 2,\ y>0.$$

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

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

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