thumb|[[Proof without words that the difference between two consecutive cubes is a centered hexagonal number, shown by arranging n<sup>3</sup> balls in a cube and viewing them along a space diagonal colors denote horizontal layers and the dashed lines the hexagonal number, respectively.]]
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
This is the first of these equations:
:<math>p = \frac{x^3 - y^3}{x - y},\ x = y + 1,\ y>0,</math>
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
The formula for a general cuban prime of this kind can be simplified to <math>3y^2 + 3y + 1</math>. This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal.
the largest known cuban prime has 3,153,105 digits with <math>y = 3^{3304301} - 1</math>, found by R. Propper and S. Batalov.
Second series
The second of these equations is:
:<math>p = \frac{x^3 - y^3}{x - y},\ x = y + 2,\ y>0.</math>
which simplifies to <math>3y^2 + 6y + 4</math>. With a substitution <math>y = n - 1</math> it can also be written as <math>3n^2 + 1, \ n>1</math>.
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
The name "cuban prime" has to do with the role cubes (third powers) play in the equations.
See also
- Cubic function
- List of prime numbers
- Prime number