# 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:

^{[1]}

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, ... (sequence A002407 in the OEIS)

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.

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

The second of these equations is:

^{[3]}

This simplifies to . With a substitution it can also be written as .

The first few cuban primes of this form are (sequence A002648 in the 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.

## Generalization

A **generalized cuban prime** is a prime of the form

In fact, these are all the primes of the form 3*k*+1.

## See also

## Notes

## References

- 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