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.
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 July 2023 [update] the largest known has 3,153,105 digits with , [2] found by R.Propper and S.Batalov.
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. [4]
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