Cuban prime
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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 (sequence A002407 in the OEIS)
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.
As of July 2023[update] 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 (sequence A002648 in the OEIS)
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
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
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- Fermat (22n + 1)
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- Double Mersenne (22p−1 − 1)
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- Euclid (pn# + 1)
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- Quartan (x4 + y4)
- Solinas (2m ± 2n ± 1)
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- Woodall (n·2n − 1)
- Cuban (x3 − y3)/(x − y)
- Leyland (xy + yx)
- Thabit (3·2n − 1)
- Williams ((b−1)·bn − 1)
- Mills (⌊A3n⌋)
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- k-tuple
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- Chen
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