Cuban prime

From Infogalactic: the planetary knowledge core
Jump to: navigation, search

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[1]

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 3y^2 + 3y + 1. 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 y = 100000845^{4096},[2] found by Jens Kruse Andersen.

The second of these equations is:

p = \frac{x^3 - y^3}{x - y},\ x = y + 2,\ y>0.[3]

This simplifies to 3y^2 + 6y + 4. With a substitution y = n - 1 it can also be written as 3n^2 + 1, \ n>1.

The first few cuban primes of 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

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

Notes

  1. Cunningham, On quasi-Mersennian numbers
  2. Caldwell, Prime Pages
  3. Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259

References