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:

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

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.

Generalization

A generalized cuban prime is a prime of the form

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

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

See also

• Cubic function
• List of prime numbers

Notes

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

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