Problems & Puzzles: Puzzles

Puzzle 974. More twins like these... Carlos Rivera asks this: Find large twin of the following model: p#/2+/-2 The first four of these are for p=5, 7, 13 and 31. Example: for p=31, the twins are: 100280245065+/-2 Q. Send your largest twin of these.

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Contributions came from Oscar Volpatti, Jan van Delden.

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Oscar wrote on October 17, 2019:

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.

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Jan wrote on October 17, 2019

A number of the form p#/2+/-2 is not divisible by the primes 2,3,…,p.
This increases the probability that these numbers are primes themselves.

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 2,3,…,p.

 

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 known.

The given solutions are registered at: Sloane's integer sequences: A178648

 

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