Large gaps between Wagstaff prime exponents

[Bobby Jacobs]

The last few Wagstaff prime exponents have large gaps between them. The next Wagstaff prime exponent after 374321 is 986191, which is more than twice the previous number. Then, it jumps to 4031399. That is more than quadruple the last exponent. Is the next number 13347311? Then, there would be 3 times in a row with a number more than twice the last number. That is very weird.

[GP2]

Originally Posted by Bobby Jacobs

The last few Wagstaff prime exponents have large gaps between them. The next Wagstaff prime exponent after 374321 is 986191, which is more than twice the previous number. Then, it jumps to 4031399. That is more than quadruple the last exponent. Is the next number 13347311? Then, there would be 3 times in a row with a number more than twice the last number. That is very weird.

In 2013 there was a project by several people to check the range below 10M. They found no new Wagstaff primes. That project calculated so-called Vrba-Reix residues, which aren't really mathematically proven. Nonetheless, it is likely that the absence of Wagstaff primes in this range is real.

Also in 2013, Ryan Propper discovered the Wagstaff primes 13347311 and 13372531. He searched at least parts of the 10, 11, 12 and 13 million ranges, but it wasn't clear from his posts whether he covered those ranges exhaustively or only in part. His post mentioned that he calculated ordinary PRP residues.

I am currently double-checking this range, finding and publishing PRP residues (see this mini-website). I am using 128 Skylake cores for this and the latest mprime version 29.5. So far I've reached 8.7 million and will probably get to 10 million sometime next month.

I'm hoping to find something, but at this point I suspect that there are no new Wagstaff primes to be found below 14 million.

[Bobby Jacobs]

374321, 986191, 4031399, 13347311, 13372531

There are 3 big gaps and a very small gap. It is amazing.