# Cuban prime

A cuban prime (from the role cubes (third powers) play in the equations) 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:

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

The general cuban prime of this kind can be rewritten as ${\displaystyle {\tfrac {(y+1)^{3}-y^{3}}{y+1-y}}}$, which simplifies 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.

The second of these equations is:

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

This simplifies to ${\displaystyle 3y^{2}+6y+4}$.

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

With a substitution ${\displaystyle y=n-1}$, the equations above can also be written as follows:

${\displaystyle 3n^{2}-3n+1,\ n>1}$.
${\displaystyle 3n^{2}+1,\ n>1}$.

## Generalization

A generalized cuban prime is a prime of the form

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

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