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:

$p = \frac{x^3 - y^3}{x - y},\ x = y + 1,\ y>0$

and the first few cuban primes from this equation are (sequence A002407 in OEIS):

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 $\tfrac{(y+1)^3 - y^3}{y + 1 - y}$, 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.

As of January 2006^[update] the largest known has 65537 digits with y = 100000845⁴⁰⁹⁶[1], found by Jens Kruse Andersen.

The second of these equations is:

$p = \frac{x^3 - y^3}{x - y},\ x = y + 2.$

It simplifies to 3y² + 6y + 4. The first few cuban primes on this form are (sequence A002648 in 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

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 Cuba.

See also

• Cubic function
• List of prime numbers
• Prime number

References

• Phil Carmody, Eric W. Weisstein and Ed Pegg, Jr., "Cuban Prime" from MathWorld.