Cuban Primes

Investigation of today's prime day, 24571, revealed that it is a cuban prime, a prime that is a difference of two consecutive cubes (OEIS A002407). In the case of 24571, the two cubes are \(91^3\) and \(90^3\). 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 is \(\displaystyle \frac{x^3-y^3}{x-y} \) where \(x=y+1\) and \(y>0\)
• the second is \( \displaystyle \frac{x^3-y^3}{x-y}\) where \(x=y+2\) and \(y>0\)

24571 obviously belongs to the first type with \(y\) = 90 and \(x\) = 91. To quote from Wikipedia: "the name "cuban prime" has to do with the role cubes (third powers) play in the equations, and has nothing to do with Cuba". 

Returning to the first equation, it can be seen that it simplifies to \( (y+1)^3-y^3 \) and then to \( 3y^2+3y+1 \) which is the equation of centred hexagonal numbers. Thus every cuban prime of the first type is a centred hexagonal number. These latter numbers begin 1, 7, 19, 37, 61, 91, 127, 169, ... etc. Many of the centred hexagonal numbers are prime because they all end in 1, 7 or 9.

Because 24571 is more distant than usual from its neighbouring primes, it also gains entry into the OEIS via A137875: prime numbers, isolated from neighbouring primes by more than 16 and A163111: prime numbers with gaps larger than 18 towards both neighbouring primes. The nearest primes are 24551 and 24593.

There are also quartan, quintan and sextan primes. For example quintan primes can be of the form:$$\frac{x^5-y^5}{x-y} \text{ or } \frac{x^5+y^5}{x+y}$$An example of a quintan prime is \( 4651 = \displaystyle \frac{6^5-5^5}{6-5}\). 

Another example is \( 26321 = \displaystyle \frac{11^5-5^5}{11-5} \).

It should be noted that \( x-y \) will always divide \(x^n-y^n\) for \(n \geq 1 \) and in the case of \(x^5-y^5\) this division yields \(x^4 + yx^3 + y^2x^2 + y^3x + y^4\). Thus the situation of \(x=y\) is possible and in the case of \(x=y=1\) yields the quintan prime 5. Primes of this form make up OEIS A002649. 

A002649

Quintan primes: \(p = \dfrac{x^5 - y^5 }{x - y} \)

If we return to the definition of a cuban prime as being the difference of two consecutive cubes, then an equivalent definition of a quintan prime as being the different of two consecutive fifth powers does not necessarily hold. It does in the case of 4651 where 5 and 6 differ by 1 but in the case of 26321 the difference between 5 and 11 is 6. This makes for some degree of confusion which I might try to clarify further at some future date.

on April 27th 2021 and

on October 4th 2023