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# 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

$p=\frac{x^{3}-y^{3}}{x-y},$ |

with $x=y+1$ and $y>0$. The first few cuban primes from this equation are: 7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919.

The general cuban prime of this kind can be rewritten as

$\frac{(y+1)^{3}-y^{3}}{y+1-y},$ |

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 numbers.

This kind of cuban primes has been researched by A. J. C. Cunningham, in a paper entitled ”On quasi-Mersennian numbers”.

As of January 2006 the largest known cuban prime has 65537 digits with $y=100000845^{{4096}}$, discovered by Jens Kruse Andersen, according to the Prime Pages of the University of Tennessee at Martin.

The second of these equations is

$p=\frac{x^{3}-y^{3}}{x-y},$ |

with $x=y+2$. It simplifies to $3y^{2}+6y+4$. The first few cuban primes on this form are: 13, 109, 193, 433, 769.

This kind of cuban primes have also been researched by Cunningham, in his book Binomial Factorisations.

## Mathematics Subject Classification

11N05*no label found*

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