# Cuban prime

A cuban prime is a prime number that is also a solution to one of two different specific equations involving differences between third powers of two integers x and y.

## First series

The first of these equations is:

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

i.e. the difference between two successive cubes. 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, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227 (sequence A002407 in the OEIS)

The formula for a general cuban prime of this kind can be simplified to ${\displaystyle 3y^{2}+3y+1}$. 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 ${\displaystyle y=100000845^{4096}}$,[2] found by Jens Kruse Andersen.

## Second series

The second of these equations is:

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

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

The first few cuban primes of this form are:

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 (sequence A002648 in the OEIS)

The name "cuban prime" has to do with the role cubes (third powers) play in the equations, and has nothing to do with Cuba.[citation needed]

• Phil Carmody, Eric W. Weisstein and Ed Pegg, Jr. "Cuban Prime". MathWorld.{{cite web}}: CS1 maint: multiple names: authors list (link)