To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# 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.

## Notes

1. ^ Cunningham, On quasi-Mersennian numbers
2. ^ Caldwell, Prime Pages
3. ^ Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259

## 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.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, England: Macmillan and Co., 41, pp. 119–146