y^x-x^y primes

[pxp]

Originally Posted by Andrew Usher

pxp: I see that you have a table of these primes now. As for the proven primes, my search just done the same way showed that the only larger ones are the six just above 10,000 digits (search complete to over 11,300 and also checked those with other discoverers larger than that). I hope your procedure at least checks only x,y coprime and of opposite parity - in fact, that is the simplest form of sieving here.

As to completeness limits, for the first type Leylands one of the sticky threads in this forum gave 150,000 digits, which seems to be correct. For this, I am not sure as Schneider appears to search by size of y only.

Mathematica's PrimeQ isn't totally dumb. It starts with some divisibility tests using small primes. Still, Mathematica is no match for pfgw. That 150000-digits limit for Leyland-plus numbers is due to me. Starting from scratch in 2015 (using Mathematica, exactly as I am doing in my current Leyland-minus table), I switched to xyyxsieve and pfgw in mid-2020 and completed testing the ~687 million Leyland numbers less than 150000 digits the following year.

[Andrew Usher]

Thanks for the clarification. We'll have to see where you want to go with this.

As for primality proof, does anyone have the ability to automatically update a list whenever factordb changes a number from PRP to P? That would seem to be the main obstacle to using them for the purpose.

[pxp]

In 2018 I concocted this conjecture for the Leyland-plus numbers: For d > 11, 10^(d-1)+(d-1)^10 is the smallest (base ten) d-digit term. There are no known primes. Today I noted this similar conjecture for the Leyland-minus numbers: For d > 11, 10^d-d^10 is the largest (base ten) d-digit term. There are two known primes: d = 273 and 399.

[Cybertronic]

I have automatically matched all the Leyland numbers I know with factordb.com. Status PRP or Prime is now entered.

https://pzktupel.de/Primetables/TableLeyland2.php

regards

[pxp]

Originally Posted by Cybertronic

I have automatically matched all the Leyland numbers I know with factordb.com. Status PRP or Prime is now entered.

Good stuff! Concurrently, I have removed the "proven" P column from my list as it would always have lagged behind your values. I've also removed the descending rank numbers column as these would eventually have to be recalibrated.

[Andrew Usher]

I guess my last query is answered in the affirmative - good work. And I see mathwiz has already asked in the other Leyland thread if they would automatically update.

pxp's conjecture (how far has it been verified?) is likely true for high enough d, as d^10/10^d -> 0 quickly there.

[Luminescence]

I am currently proving PRPs below 10k digits and uploading certificates.

I am also the "culprit" behind the proven primes at around 10k digits.
Didn't occur to me to post about it... ooops

I'm gonna post updates from time to time on my progress.

[Cybertronic]

Originally Posted by Luminescence

I am currently proving PRPs below 10k digits and uploading certificates.

I am also the "culprit" behind the proven primes at around 10k digits.
Didn't occur to me to post about it... ooops

I'm gonna post updates from time to time on my progress.

Found 3 new hits on factordb:

171^2600-2600^171
117^2752-2752^117
37^3704-3704^37
now proven prime.

Page was updated

[Cybertronic]

txt-file have now status....
3 new P's found at factordb

https://pzktupel.de/Primetables/TableLeyland2.php

same for Leyland x^y+y^x
https://pzktupel.de/Primetables/TableLeyland1.php

[Cybertronic]

up to date status: PRP / Prime now displayed

[pxp]

Originally Posted by pxp

I have finally been motivated to write a program that sorts (x,y) pairs by the absolute magnitude of x^y-y^x, x<y, for magnitudes < 10^60000. There are 133090654 such pairs. To be more precise, my "Leyland number" for these pairs correspond to the indices of OEIS A045575. For primes of this form, (2,5), (3,4), (2,7), (2,9), (3,10), (2,17), ... these indices are 3, 4, 6, 10, 24, 28, ...

Of course, once I had this list, it became easy to write a (Mathematica) program that sequentially stepped through it, so as to PrimeQ (Mathematica's PRP determinator) each pair. I started that program yesterday and have, so far, rediscovered the smallest 423 x^y-y^x primes. Without sieving or other speedups this will get nowhere fast, but I will let it run (in the background) a few months regardless.

I've now checked the first 15 million Leyland pairs using this program as well as the Leyland pairs from 15 to 20 million on another machine. I will stop there for now. The chart I had been using to document my progress has had a total revamp. I'm now listing all current primes < 10^60000 with a gap after #1078 to indicate the 20 million Leyland pairs that I have checked. The Leyland# column is now the second column; the number of decimal digits, the third. I will not add the discoverer or date of discovery.