عدد أولي مكعبي

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.

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 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 (المتتالية A002407 في 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 (المتتالية A002648 في 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, University of Tennessee at Martin, https://t5k.org/primes/page.php?id=136214, retrieved on July 31, 2023 
• Phil Carmody, Eric W. Weisstein and Ed Pegg, Jr., Cuban Prime at MathWorld.
• Cunningham, A. J. C. (1923), Binomial Factorisations, London: F. Hodgson 
• Cunningham, A. J. C. (1912), On Quasi-Mersennian Numbers, 41, England: Macmillan and Co., pp. 119–146