Cuban prime

File:Visual proof centered hexagonal numbers sum.svg

Proof without words that the difference between two consecutive cubes is a centered hexagonal number, shewn by arranging n³ balls in a cube and viewing them along a space diagonal Template:Ndashcolors denote horizontal layers and the dashed lines the hexadecimal 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><ref>Allan Joseph Champneys Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.</ref>

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 <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.

Template:As of the largest known has 3,153,105 digits with <math>y = 3^{3304301} - 1</math>,<ref>Caldwell, Prime Pages</ref> 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><ref>Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259</ref>

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 (sequence A002648 in the OEIS)

The name "cuban prime" has to do with the role cubes (third powers) play in the equations.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

See also

• Cubic function
• List of prime numbers
• Prime number

References

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