Cuban prime

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 cuban prime 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. == See also ==