primes of the form |sphenic^3 - sphenic^2|

[Geoffrey Caveney]

In the discussion of so-called "boring" integers that do not currently appear in any OEIS sequence (not including b-files), Hugo Pfoertner asked about the smallest such prime number 48973, and I provided a possible sequence in which it would appear prominently. Out of curiosity I checked for the next smallest such "boring" prime, and I find it is 50129.

50129 is potentially interesting for the following property: it can be represented as |70^3 - 627^2|, and both 70 and 627 are products of three distinct primes, which are called sphenic numbers (https://oeis.org/A007304).

This is a rather rare property, because all six prime factors of the two sphenic numbers must be distinct. If they shared a prime factor, then the resulting difference would be divisible by the square of that factor, and thus the difference could not be prime itself. This restriction requires the cube and square to be relatively large, making it less likely for their difference to be small.

For comparison, the most famous difference of a cube and square is 2 = 3^3 - 5^2, where the bases of the cube and square are both prime themselves. When the bases are both required to be semiprimes, the smallest prime difference is 11 = 15^3 - 58^2. When the bases are both required to be the products of three not necessarily distinct primes (https://oeis.org/A014612), the smallest prime difference I find is 151 = |105^3 - 1076^2|, where 1076 = 2^2 * 269. 

But when the bases are both required to be sphenic numbers, each the product of three distinct primes, the smallest prime difference that I have found so far is 1439 = 590^3 - 14331^2, followed by 3709 = 385^3 - 7554^2, 10061 = |455^3 - 9706^2|, 10487 = 246^3 - 3857^2, and 21269 = 105^3 - 1066^2. 

Checking all cubes of sphenics up to 2001^3, I find a total of only 18 such primes less than 50129. As the value of the cube term grows larger, examples of such primes become quite scarce, with 38231 = 1986^3 - 88505^2 the only example with a cube term between 1000^3 and 2000^3. I cannot rule out that other such needles in the haystack may occur, but I expect them to be few and far between for larger values of the cube term.

Geoffrey

[Hugo Pfoertner]

I'm sorry to be so blunt, but we undermine the credibility of OEIS as a trustworthy mathematical resource if we now start searching for new sequences whose primary motivation is the (non-)occurrence of numbers in the DATA of existing sequences. It might be a fun topic for the SeqFan list or for the YouTube world, but please don't use it as justification for new content in OEIS itself.