A cuban prime (from the role cubes (third powers) play in the equations) 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:
and the first few cuban primes from this equation are:
The general cuban prime of this kind can be rewritten as , which simplifies to . This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal.
The second of these equations is:
This simplifies to .
- 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
With a substitution , the equations above can also be written as follows:
A generalized cuban prime is a prime of the form
In fact, these are all the primes of the form 3k+1.
- Cunningham, On quasi-Mersennian numbers
- Caldwell, Prime Pages
- Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259
- Caldwell, Dr. Chris K. (ed.), "The Prime Database: 3*100000845^8192 + 3*100000845^4096 + 1", Prime Pages, University of Tennessee at Martin, retrieved June 2, 2012
- Phil Carmody, Eric W. Weisstein and Ed Pegg, Jr. "Cuban Prime". MathWorld.
- Cunningham, A. J. C. (1923), Binomial Factorisations, London: F. Hodgson, ASIN B000865B7S
- Cunningham, A. J. C. (1912), "On Quasi-Mersennian Numbers", Messenger of Mathematics, England: Macmillan and Co., 41, pp. 119–146