20210624, 23:40  #12 
Jan 2020
2^{3} Posts 

20210625, 08:19  #13 
"Kebbaj Reda"
May 2018
Casablanca, Morocco
136_{8} Posts 

20210625, 14:16  #14  
Feb 2017
Nowhere
2^{2}·17·73 Posts 
Quote:
I later had a simple idea that greatly improved my script. I only pushed to 10^400, though. But when I saw 10^545, how could I resist pushing further? I can report that, if my improved script is writ right, there are two solutions between 10^545 and 10^600. 

20210625, 18:57  #15 
Jan 2017
2^{3}×3×5 Posts 
With Sage, generating all solutions up to 10^1000 was pretty fast (there are 25 including the example solutions). I tried generating one with more digits, and got a solution above 10^9000.

20210625, 20:54  #16 
Feb 2017
Nowhere
2^{2}×17×73 Posts 
Thanks for giving me a way to check my script! It was easy to adjust it to check up to 10^1000. Lo and behold, 25 solutions!

20210626, 04:41  #17  
Jan 2020
2^{2}×3^{2} Posts 
Quote:
25 solutions up to 10^1000. 34 solutions up to 10^2000. 46 solutions up to 10^3000. Or less? Last 12 candidates only passed a BPSW test, I stopped primality proving after 10^2000 because APRCL was becoming too lengthy. Anyway, my aged YAFU implementation can only prove numbers up to 6021 digits, so your finding above 10^9000 is doubly remarkable to me. I wonder how long did you take to prove it prime, and if you used some code specially written for Leyland numbers. Last fiddled with by 0scar on 20210626 at 04:43 

20210626, 06:45  #18 
Jun 2003
3^{2}·571 Posts 
Are people referring to http://www.primefan.ru/xyyxf/primes.html ?

20210626, 12:14  #19  
Feb 2017
Nowhere
4964_{10} Posts 
Quote:
I note that PariGP had no trouble finding representations of the solutions by the quadratic forms specified. 

20210626, 21:46  #20 
Jan 2017
2^{3}×3×5 Posts 
I didn't "prove" it prime  in fact I explicitly turned proofs off in Sage to speed things up (I'm not sure why they're enabled by default, seems silly to me). The unrealistic chance that a probabilistic primality test could return a wrong result is a much less relevant worry than the script being buggy, other software error on the machine, or even a hardware failure.

20210627, 02:37  #21 
Romulan Interpreter
Jun 2011
Thailand
2^{2}·7·349 Posts 

20210701, 03:25  #22 
Jan 2020
100100_{2} Posts 
Actually, it seems that I spoiled it before. Within this thread, I was the first one to say "Leyland" (should I say "xilman"?)
Wikipedia page about Leyland numbers mentions the largest known Leyland proven primes and references XYYXF's search project. This forum also has many threads about Leyland prp finding / primality proving. 
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