Cuban Prime

The cuban primes, named after differences between successive cubic numbers, have the form n^3-(n-1)^3. The first few are 7, 19, 37, 61, 127, 271, ... (OEIS A002407), which are also the prime hex numbers. They correspond to indices n=2, 3, 4, 5, 7, 10, 11, 12, 14, 15, 18, 24, 25, ... (OEIS A002504; Cunningham 1912).

The numbers of cuban primes less than 1, 10, 10^2, ... are 0, 1, 4, 11, 28, 64, 173, 438, 1200, ... (OEIS A113478), which is well-approximated by


Cuban primes are cyclotomic in nature, being the evaluation of the third homogeneous cyclotomic polynomial, x^3-y^3, at values (x+1) and x. The form therefore can only have primitive factors of the form 6n+1. Also, by construction, 2 and 3 are excluded as non-primitive factors. Therefore, this form has a slightly higher density than would arbitrary numbers of the same size (P. Carmody, pers. comm., Jan. 8, 2006).

See also

Cubic Number

Portions of this entry contributed by Phil Carmody

This entry contributed by Ed Pegg, Jr. (author's link)

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Cunningham, A. J. C. "On Quasi-Mersennian Numbers." Mess. Math. 41, 119-146, 1912.Sloane, N. J. A. Sequences A002407/M4363, A002504/M0522, and A113478) in "The On-Line Encyclopedia of Integer Sequences."

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Cuban Prime

Cite this as:

Carmody, Phil; Pegg, Ed Jr.; and Weisstein, Eric W. "Cuban Prime." From MathWorld--A Wolfram Web Resource.

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