Cuban prime

Last updated August 20, 2023

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

^[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 $3y^{2}+3y+1$ . 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^[update] the largest known has 3,153,105 digits with $y=3^{3304301}-1$ ,^[2] 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.

^[3]

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 (sequence A002648 in the OEIS )

The name "cuban prime" has to do with the role cubes (third powers) play in the equations.^[4]

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". PrimePages. University of Tennessee at Martin. Retrieved 2022-10-06.

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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 , England: Macmillan and Co., vol. 41, pp. 119–146

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[1] Allan Joseph Champneys Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.

[2] Caldwell, Prime Pages

[3] Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259

[4] Caldwell, Chris K. "cuban prime". PrimePages. University of Tennessee at Martin. Retrieved 2022-10-06.

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v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
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 Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) 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/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 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 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers