Cuban prime

Proof without words that the difference of two consecutive cubes is a centered hexagonal number by arranging n³ semitransparent balls in a cube and viewing along a space diagonal – colour denotes cube layer and line style denotes hex number

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:

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

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 $3y^{2}+3y+1$. 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 the largest known has 3,153,105 digits with $y=3^{3304301}-1$, found by R.Propper and S.Batalov.

Second series

The second of these equations is:

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

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

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.

See also

• Cubic function
• List of prime numbers
• Prime number

References

• 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