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.

The name ”cuban prime” has to do with the rôle cubes (third powers) play in the equations, and has nothing to do with the prime minister of Cuba.