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''.

First series

This is the first of these equations:

:$p = \frac,\ x = y + 1,\ y>0,$

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

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.

the largest known has 65537 digits with $y = 100000845^$, found by Jens Kruse Andersen.

Second series

The second of these equations is:

:$p = \frac,\ x = y + 2,\ y>0.$

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

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

Notes

References

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{{Prime number classes, state=collapsed

Classes of prime numbers