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.
This is the first of these equations:
i.e. the difference between two successive cubes. The first few cuban primes from this equation are
The formula for a general cuban prime of this kind can be simplified to . This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal.
As of January 2006[update] the largest known has 65537 digits with , found by Jens Kruse Andersen.
The second of these equations is:
which simplifies to . With a substitution it can also be written as .
The first few cuban primes of this form are:
The name "cuban prime" has to do with the role cubes (third powers) play in the equations.
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