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:
Allan Joseph Champneys Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.
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 . 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 ,Caldwell, Prime Pages found by R. Propper and S. Batalov.
Second series The second of these equations is:
Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259
which simplifies to . With a substitution it can also be written as .
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 Cubic function List of prime numbers Prime number
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