Cuban prime - Citizendia

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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The first of these equations is:

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

and the first few cuban primes from this equation are (sequence A002407 in OEIS):

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 $\tfrac{(y+1)^3 - y^3}{y + 1 - y}$ , which simplifies to $3 y 2 + 3 y + 1$ . The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences In mathematics Seven is the fourth Prime number. It is not only a Mersenne prime (since 23 &minus 1 = 7 but also a 19 ( nineteen) is the Natural number following 18 and preceding 20. 37 ( thirty-seven) is the Natural number following 36 and preceding 38. 61 ( sixty-one) is the Natural number following 60 and preceding 62. This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal. A centered hexagonal number, or hex number, is a centered Figurate number that represents a Hexagon with a dot in the center and all other dots

This kind of cuban primes has been researched by A. J. C. Cunningham, in a paper entitled On quasi-Mersennian numbers. The Mathematician Allan Joseph Champneys Cunningham ( Delhi 1842 – London 1928 started a military career with the East India Company 's Bengal

As of January 2006 the largest known has 65537 digits with $y = 100000845 4096$ [1], found by Jens Kruse Andersen. Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar.

The second of these equations is:

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

It simplifies to $3 y 2 + 6 y + 4$ . The first few cuban primes on this form are (sequence A002648 in 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

This kind of cuban primes have also been researched by Cunningham, in his book Binomial Factorisations. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

The name "cuban prime" has to do with the role cubes (third powers) play in the equations, and has nothing to do with Cuba. In Arithmetic and Algebra, the cube of a number n is its third power &mdash the result of multiplying it by itself three times The Republic of Cuba (ˈkjuːbə or) consists of the island of Cuba (the largest and second-most populous island of the Greater Antilles) Isla de la

References

Phil Carmody, Eric W. Weisstein and Ed Pegg, Jr., Cuban Prime at MathWorld. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld Ed Pegg Jr is an expert on mathematical Puzzles and is a self-described recreational mathematician MathWorld is an online Mathematics reference work created and largely written by Eric W

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