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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
<math>p = (x^3 - y^3) / (x - y), x = y + 1, y>0<math>
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, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227
The general cuban prime of this kind can be rewritten as ((y+1)3 - y3) / (y + 1 - y), which simplifies to 3y2 + 3y + 1. This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal.
This kind of cuban primes has been researched by A. J. C. Cunningham, in a paper entitled On quasi-Mersennian numbers.
The second of these equations is
<math>p = (x^3 - y^3 ) / (x - y), x = y + 2<math>
and the first few cuban primes from this equation 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
This kind of cuban primes have also been researched by Cunningham, in his book Binomial Factorisations.
The name "cuban prime" has to do with the role cubes (third powers) play in the equations, and has nothing to do with prime Cuban cigars or with Cuban prime minister Fidel Castro.
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