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:

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

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 $\tfrac{(y+1)^3 - y^3}{y + 1 - y}$, which simplifies 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.

As of January 2006 the largest known has 65537 digits with $y = 100000845^{4096}$,[2] found by Jens Kruse Andersen.

The second of these equations is:

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

This 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 (sequence A002648 in 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, 69313

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

Notes

1. ^ Cunningham, On quasi-Mersennian numbers
2. ^ Caldwell, Prime Pages
3. ^ Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259