Cuban prime explained

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:

^[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

The formula for a general cuban prime of this kind can be simplified to

. This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal.

the largest known has 3,153,105 digits with

,^[2] found by R.Propper and S.Batalov.

Second series

The second of these equations is:

^[3]

which simplifies to

. With a substitution it can also be written as .

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.^[4]

See also

• Cubic function
• List of prime numbers
• Prime number

Notes and References

1. Allan Joseph Champneys Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.
2. Caldwell, Prime Pages
3. Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259
4. Web site: cuban prime. Caldwell. Chris K.. PrimePages. University of Tennessee at Martin. 2022-10-06 .