Contributions came from Oscar Volpatti, Jan van Delden.
A 30-hours computation checked the
pairs from (37#)/2±2 to (73517#)/2±2, finding no twin PRPs.
The members of last checked pair are 31771 digits long. If a
larger pair of PRPs were found, primality proving would possibly
take some CPU years with Primo.
A number of the form p#/2+/-2 is not divisible by the primes
This increases the probability that these numbers are primes
Unfortunately there are a lot of other potential divisors of
p#/2+/-2, if p increases in size.; the probability of finding
another solution decreases with increasing p.
An estimate of this probability can be found by finding an
approriate heuristic, similar to the twin prime constant heuristic,
using an estimate of the size of p# and a correction for the primes
Lucky for us a test routine is rather easy to write, given a good
library for large integers.
I tested until prime number 9500, i.e. p=98947, where p# has 42833
digits and found no other solutions than the ones that are already
The given solutions are registered at: Sloane's
integer sequences: A178648