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 the 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 $y=100000845^{4096}$, found by Jens Kruse Andersen.

The second of these equations is:

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

It 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 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.

See also

• Cubic function
• List of prime numbers
• Prime number

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.{{cite web}}: CS1 maint: multiple names: authors list (link)
• Cunningham, A. J. C. (1923), Binomial Factorisations, London: F. Hodgson, ASIN B000865B7S
• Cunningham, A. J. C. (1912), "On Quasi-Mersennian Numbers", Messenger of Mathematics, vol. 41, England: Macmillan and Co., pp. 119–146