Saturday, February 24, 2018

The long compute

Around the middle of 2017, I started looking for Leyland primes between L(40210,287) and L(40945,328). L(x,y) = x^y + y^x and we assume x ≥ y > 1. L(40210,287) is Leyland #324766365 and L(40945,328) is Leyland #349812824. So there are 25046458 Leyland numbers between them which I want to check for probable primality. Any given check may take a significant amount of time since we are dealing with numbers that are roughly 100000 decimal digits long.

I haven't been totally committed to the task for the entire period but I may perhaps have spent six months on it. Since I've only covered about one fifth of the territory, I have two years to go! I was going to add some processors to the task but my intended purchase of a new machine has (sadly) been stymied. There was a second issue. My list of sorted consecutive Leyland numbers only went up to #331682621, having been computed with the sole objective of reaching 100000-digit numbers (which it did). I thought I was going to need the new computer to calculate more terms because my indexing computation was limited by available RAM and my current machines can't take any more than 64 GB. Fortunately, I recently discovered that that was sufficient to extend the indexing to L(40945,328).

The good news is that I have so far found eight previously unknown Leyland primes, ranging in size from 98889 to 99659 decimal digits. By this summer I should have scored my first Leyland prime with more than 100000 decimal digits. There are currently only nine known Leyland primes <decimal digits> larger than this:

L(40945,328)  <103013>     Norbert Schneider  Dec 2014
L(41507,322)  <104094>     Norbert Schneider  Dec 2014
L(222748,3)   <106278>     Anatoly Selevich   Dec 2010
L(45405,286)  <111532>     Norbert Schneider  Apr 2015
L(48694,317)  <121787>     Norbert Schneider  Aug 2015
L(234178,9)   <223463>     Anatoly Selevich   Jul 2011
L(255426,11)  <265999>     Serge Batalov      May 2014
L(314738,9)   <300337>     Anatoly Selevich   Feb 2011
L(328574,15)  <386434>     Serge Batalov      May 2014

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