cuban prime

This is the first of these equations:

:$p\; =\; \backslash frac\{x^3\; -\; y^3\}\{x\; -\; y\},\backslash \; x\; =\; y\; +\; 1,\backslash \; y>0,$Allan Joseph Champneys Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.

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 {{OEIS|id=A002407}}

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|2023|alt=As of July 2023}} the largest known has 3,153,105 digits with $y\; =\; 3^\{3304301\}\; -\; 1$,Caldwell, Prime Pages found by R. Propper and S. Batalov.

The second of these equations is:

:$p\; =\; \backslash frac\{x^3\; -\; y^3\}\{x\; -\; y\},\backslash \; x\; =\; y\; +\; 2,\backslash \; y>0.$Cunningham, Binomial Factorisations, Vol. 1, pp. 245-259

which simplifies to $3y^2\; +\; 6y\; +\; 4$. With a substitution $y\; =\; n\; -\; 1$ it can also be written as $3n^2\; +\; 1,\; \backslash \; n>1$.

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 {{OEIS|id=A002648}}

The name "cuban prime" has to do with the role cubes (third powers) play in the equations.{{Cite web|url=https://primes.utm.edu/glossary/page.php?sort=CubanPrime|title=cuban prime|last= Caldwell|first=Chris K.|website=PrimePages|publisher=University of Tennessee at Martin|access-date=2022-10-06 }}

• Cubic function
• List of prime numbers
• Prime number

{{reflist}}

• {{Citation

| url = https://t5k.org/primes/page.php?id=136214

| title = The Prime Database: 3^4043119 + 3^2021560 + 1

| editor-last = Caldwell

| editor-first = Dr. Chris K.

| editor-link = Chris Caldwell (mathematician)

| work = Prime Pages

| publisher = University of Tennessee at Martin

| accessdate = July 31, 2023

}}

• {{MathWorld|title=Cuban Prime|urlname=CubanPrime|author=Phil Carmody, Eric W. Weisstein and Ed Pegg, Jr.}}
• {{Citation

| title = Binomial Factorisations

| last = Cunningham

| first = A. J. C.

| author-link = A. J. C. Cunningham

| publisher = F. Hodgson

| publication-place = London

| year = 1923

| asin = B000865B7S

}}

• {{Citation

| title = On Quasi-Mersennian Numbers

| last = Cunningham

| first = A. J. C.

| author-link = A. J. C. Cunningham

| work = Messenger of Mathematics

| volume = 41

| pages = 119–146

| publisher = Macmillan and Co.

| publication-place = England

| year = 1912

}}

{{Prime number classes|state=collapsed}}

Category:Classes of prime numbers