Saturday, January 23

Indexing the Leyland primes, reprise

When I wrote Indexing the Leyland primes last May, I suggested that (of the then-known Leyland primes) "the smallest 954 (or slighly more) are indexable". Letting a Leyland number L(x,y) = x^y+y^x, x>=y, and [excepting L(2,1)] y>1. Two and a half weeks ago, I found that L(12876,2447) was prime and that this wasn't a known Leyland prime. As a result of having now examined all 75545875 smaller Leyland numbers, I can say that 955 of the then-known Leyland primes were in fact indexable!

I had made a bit of a fuss over the three Leyland primes I discovered last October. So far this month, chronologically after L(12876,2447), I found L(13307,3442), L(13227,2200), L(13371,3068), and L(13051,2448). It seems that I'm picking low-hanging, perhaps mistakenly ignored fruit. Norbert Schneider is also looking for Leyland primes not far from where I'm currently searching, so I have competition! Regardless, I will continue my effort to close the gaps between known Leyland primes — extending their indices as far as I am able.