Cuban prime

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.

This is the first of these equations:

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

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 (sequence A002407 in the OEIS)

The formula for a general cuban prime of this kind can be simplified 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 July 2023^[update] the largest known has 3,153,105 digits with ${\displaystyle y=3^{3304301}-1}$,^[2] found by R.Propper and S.Batalov.

The second of these equations is:

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

which simplifies to ${\displaystyle 3y^{2}+6y+4}$. With a substitution ${\displaystyle y=n-1}$ it can also be written as ${\displaystyle 3n^{2}+1,\ n>1}$.

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 (sequence A002648 in the OEIS)

The name "cuban prime" has to do with the role cubes (third powers) play in the equations.^[4]

• Cubic function
• List of prime numbers
• Prime number

• Caldwell, Dr. Chris K. (ed.), "The Prime Database: 3^4043119 + 3^2021560 + 1", Prime Pages, University of Tennessee at Martin, retrieved July 31, 2023
• Phil Carmody, Eric W. Weisstein and Ed Pegg, Jr. "Cuban Prime". MathWorld.{{cite web}}: 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, vol. 41, England: Macmillan and Co., pp. 119–146