Cuban prime

A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y. The first of these equations is:

p={\frac {x^{3}-y^{3}}{x-y}},\ x=y+1,\ y>0

^[1]

and 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, ... (sequence A002407 in the OEIS)

The general cuban prime of this kind can be rewritten as ${\tfrac {(y+1)^{3}-y^{3}}{y+1-y}}$ , which simplifies 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 January 2006 the largest known has 65537 digits with $y=100000845^{4096}$ ,^[2] found by Jens Kruse Andersen.

The second of these equations is:

p={\frac {x^{3}-y^{3}}{x-y}},\ x=y+2,\ y>0.

^[3]

This 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 (sequence A002648 in the OEIS):

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, and has nothing to do with Cuba.

Generalization

A generalized cuban prime is a prime of the form

p={\frac {x^{3}-y^{3}}{x-y}},x>y>0.

In fact, these are all the primes of the form 3k+1.

Notes

↑ Cunningham, On quasi-Mersennian numbers
↑ Caldwell, Prime Pages
↑ Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259

References

Caldwell, Dr. Chris K. (ed.), "The Prime Database: 3*100000845^8192 + 3*100000845^4096 + 1", Prime Pages, University of Tennessee at Martin, retrieved June 2, 2012
Phil Carmody, Eric W. Weisstein and Ed Pegg, Jr. "Cuban Prime". MathWorld.

Cunningham, A. J. C. (1923), Binomial Factorisations, London: F. Hodgson, ASIN B000865B7S
Cunningham, A. J. C. (1912), "On Quasi-Mersennian Numbers", Messenger of Mathematics, England: Macmillan and Co., 41, pp. 119–146

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 50 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229

List of prime numbers

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Cuban prime

Generalization

See also

Notes

References