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:
- [math]\displaystyle{ p = \frac{x^3 - y^3}{x - y},\ x = y + 1,\ y\gt 0, }[/math][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 [math]\displaystyle{ 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.
(As of 2023) the largest known has 3,153,105 digits with [math]\displaystyle{ y = 3^{3304301} - 1 }[/math],[2] found by R.Propper and S.Batalov.
Second series
The second of these equations is:
- [math]\displaystyle{ p = \frac{x^3 - y^3}{x - y},\ x = y + 2,\ y\gt 0. }[/math][3]
which simplifies to [math]\displaystyle{ 3y^2 + 6y + 4 }[/math]. With a substitution [math]\displaystyle{ y = n - 1 }[/math] it can also be written as [math]\displaystyle{ 3n^2 + 1, \ n\gt 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.[4]
See also
Notes
- ↑ Allan Joseph Champneys Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.
- ↑ Caldwell, Prime Pages
- ↑ Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259
- ↑ Caldwell, Chris K.. "cuban prime". University of Tennessee at Martin. https://primes.utm.edu/glossary/page.php?sort=CubanPrime.
References
- Caldwell, Dr. Chris K., ed., "The Prime Database: 3^4043119 + 3^2021560 + 1", Prime Pages (University of Tennessee at Martin), https://t5k.org/primes/page.php?id=136214, retrieved July 31, 2023
- Phil Carmody, Eric W. Weisstein and Ed Pegg, Jr.. "Cuban Prime". http://mathworld.wolfram.com/CubanPrime.html.
- Cunningham, A. J. C. (1923), Binomial Factorisations, London: F. Hodgson
- Cunningham, A. J. C. (1912), "On Quasi-Mersennian Numbers", Messenger of Mathematics (England: Macmillan and Co.) 41: pp. 119–146
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