Some three months ago, I suggested on Mathematica Stack Exchange that there are 39542 Leyland numbers with exactly one million decimal digits. On my earlier-this-month blog post, I pointed out that I had created a dictionary of 39556982 Leyland-number (x,y) pairs in order of increasing magnitude, starting with the smallest 1000000-digit L(999999,10). The Leyland number L(x,y) = x^y + y^x, x ≥ y > 1. It's easy to cull from this list the first 39542 entries and I present them now here. The final L(190793,174294) is the entry just prior the appearance of the 1000001-digit L(1000000,10).
Tuesday, April 12, 2022
Friday, April 01, 2022
At the end of my February 21 blog post I suggested that I might to try to find (starting in May) a Leyland prime with one million (or more) decimal digits. I am now in a position to assess what this would entail.
Specifically, I would try to see if there are any (probable) primes with 1000000 or more, but fewer than 1000100, digits. There are 3954322 Leyland numbers in this range but by sieving out ones that are divisible by small primes — say, up to 10^11 — only about 61000 should remain. The sieving can be done in three weeks and only then would I start the search. Each remaining candidate needs about five hours to decide if it was composite, which comes to 35 years overall but, fortunately, I can distribute this across 72 Mac-mini cores, so six months. I will likely add some cores to the job but still, five months!
Considering that I have now spent the last ten months charting 300000-digit Leyland primes, it seems doable. There's a possibility that there are no primes in my working range, in which case I would have to commit to the next-larger range. And so on.
I have created a dictionary of Leyland (x,y) pairs from (999999,10) to (1000999,10), sorted by magnitude and preceded by its Leyland-number index (21588818851 to 21628375832). The text file is more than a gigabyte so I don't see much utility in linking to it. Here is a much abridged version showing the initial-, middle-, and final-100 entries: