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:
:
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, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227
The general cuban prime of this kind can be rewritten as $frac\{(y+1)^3\; -\; y^3\}\{y\; +\; 1\; -\; y\}$, 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.
This kind of cuban primes has been researched by A. J. C. Cunningham, in a paper entitled ''On quasi-Mersennian numbers''.
As of January 2006 the largest known has 65537 digits with $y\; =\; 100000845^\{4096\}$[1], found by Jens Kruse Andersen.
The second of these equations is:
:
It simplifies to $3y^2\; +\; 6y\; +\; 4$. The first few cuban primes on 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
This kind of cuban primes have also been researched by Cunningham, in his book ''Binomial Factorisations''.
The name "cuban prime" has to do with the role cubes (third powers) play in the equations, and has nothing to do with Cuba.

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References

See also

★ List of case sensitive English words.

References

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