Leyland Primes (x^y+y^x primes)

[pxp]

Originally Posted by pxp

That makes L(125330,3) #1223 and advances the index to L(28468,129), #1226.

I have examined all Leyland numbers in the gap between L(28468,129) <60085>, #1226, and L(19021,1576) <60821> and found 9 new primes. That makes L(19021,1576) #1236 and advances the index to L(19898,1263), #1254.

[pxp]

Originally Posted by pxp

That makes L(19021,1576) #1236 and advances the index to L(19898,1263), #1254.

I have examined all Leyland numbers in the gap between L(19898,1263) <61712>, #1254, and L(19909,1456) <62976> and found 11 new primes. That makes L(19909,1456) #1266 and advances the index to L(26336,267), #1283.

I count 160 unindexed primes in my list. I'm going to be adding some more cores to the project next month, so I'm looking forward to seeing what the number will be in six months.

My Leyland prime indexing effort is of course just a file on my computer. I'm getting old and there will come a time when that domain no longer reaches that document and any then-existing versions out there may not reflect its latest changes. For that reason, I am now keeping a backup copy on my Google Drive that will hopefully be accessible a little longer than the file on my computer. Just in case anyone is interested.

[xilman]

Originally Posted by pxp

I have examined all Leyland numbers in the gap between L(19898,1263) <61712>, #1254, and L(19909,1456) <62976> and found 11 new primes. That makes L(19909,1456) #1266 and advances the index to L(26336,267), #1283.

I count 160 unindexed primes in my list. I'm going to be adding some more cores to the project next month, so I'm looking forward to seeing what the number will be in six months.

My Leyland prime indexing effort is of course just a file on my computer. I'm getting old and there will come a time when that domain no longer reaches that document and any then-existing versions out there may not reflect its latest changes. For that reason, I am now keeping a backup copy on my Google Drive that will hopefully be accessible a little longer than the file on my computer. Just in case anyone is interested.

Good work.

May I suggest that you add an indication whether primality is known to have been proven or whether the number has so-far only passed a PRP test?

[pxp]

Originally Posted by xilman

May I suggest that you add an indication whether primality is known to have been proven or whether the number has so-far only passed a PRP test?

Proof of primality is for me a can of worms as I have no personal experience with (or understanding of) such proofs. I could use factordb.com as my authority but I believe that this will miss some of the larger supposedly proven Leyland primes.

It might be enough to collate the "proven" primes in Andrey Kulsha's list augmented by RichD's 27 subsequent additions and I might do that. But that takes us only to November 2017 and constitutes a blog post at most.

[xilman]

Originally Posted by pxp

Proof of primality is for me a can of worms as I have no personal experience with (or understanding of) such proofs. I could use factordb.com as my authority but I believe that this will miss some of the larger supposedly proven Leyland primes.

It might be enough to collate the "proven" primes in Andrey Kulsha's list augmented by RichD's 27 subsequent additions and I might do that. But that takes us only to November 2017 and constitutes a blog post at most.

Either approach would work fine. The default state is unproven, so missing any through inadequacies of your source material is, at the very least, completely harmless.

[pxp]

I've created a Proven Leyland Primes document of the Leyland primes up to L(8656,2929), #715. In the final column I've indicated factordb if factordb.com has the number as P. I've indicated Kulsha if factordb.com has the number as PRP but Andrey Kulsha's list suggests it is proven. I count 257 of the former (which excludes index #1) and 31 of the latter, for a total of 288 proven primes.

I believe Kulsha's list has 260 entries for proven primes and RichD had 27 additions for a total of 287. The missing prime appears to be L(3028,483).

[xilman]

Originally Posted by pxp

I've created a Proven Leyland Primes document of the Leyland primes up to L(8656,2929), #715. In the final column I've indicated factordb if factordb.com has the number as P. I've indicated Kulsha if factordb.com has the number as PRP but Andrey Kulsha's list suggests it is proven. I count 257 of the former (which excludes index #1) and 31 of the latter, for a total of 288 proven primes.

I believe Kulsha's list has 260 entries for proven primes and RichD had 27 additions for a total of 287. The missing prime appears to be L(3028,483).

Thanks!

Paul

[pxp]

Originally Posted by pxp

That makes L(19909,1456) #1266 and advances the index to L(26336,267), #1283.

I have examined all Leyland numbers in the four gaps between L(26336,267) <63905>, #1283, and L(32907,92) <64623> and found 9 new primes. That makes L(32907,92) #1296.

[kar_bon]

I've created a category for Leyland primes/PRPs in the Wiki here.
The templates takes the number of digits for sorting so not for ascending x-values.
The table of all numbers is sortable by any column inserted a counting column so you can sort by date found or proven.

@pxp:
I'm using the full date of findings from FactorDB, is this correct here? (see last examples) You gave only year/month in your list.
Is there a rule to determine the Leyland # I could use instead of giving as parameter of any number?
The same can be done for x^y - y^x then.

[pxp]

Originally Posted by kar_bon

I'm using the full date of findings from FactorDB, is this correct here? (see last examples) You gave only year/month in your list.
Is there a rule to determine the Leyland # I could use instead of giving as parameter of any number?

The date in my list is not from the FactorDB list but (primarily) the date used in PRPtop which is month/year. This is because I endeavoured to date all Leyland primes > 10000 digits, not just the ones that made it to FactorDB (quite a few did not). Even so, I had issues because not everyone submitted their primes to PRPtop. Where I could, I added missing primes to PRPtop on the person's behalf, but in these cases the PRPtop date would be somewhat later than the actual discovery date (which I gleaned from the mersenneforum thread) so I used the actual discovery dates. For a smallish number of primes I had no discovery date, so I used an estimate based on other similar-sized numbers contributed to PRPtop by the author. For my own contributions where I actually had a precise day of discovery, a couple of last-day-of-the-month finds show up in PRPtop dated the following month because I didn't submit them before midnight (PRPtop time). In those cases I used PRPtop months: close enough! In only one instance — Anatoly Selevich's L(8656,2929) — did I not have sufficient evidence of a probable discovery date. It may have been removed from PRPtop after it was proven prime (I resubmitted it in 2015). I went through Anatoly's submissions to PRPtop and finally settled on November 2007 as the most likely date of that find, although the evidence is far from certain. That is why in my Leyland primes indexing list, the date for L(8656,2929) is the only one preceded by a "~".

The Leyland # of a given L(x,y) is given by its position in OEIS sequence A076980. The easiest way to associate a given L(x,y) with its Leyland # is to simply sort all Leyland numbers up to a specified decimal-digit size. I did this back in 2015 for Leyland numbers up to ~100000 decimal-digit size: >331 million (x,y) pairs. The difficulty is in the storing/sorting of large Leyland numbers that are approximately equal, for example: (240240,2), (120120,4), (80080,8), (60060,16), (48048,32), (40040,64), (34320,128), (30030,256), (24024,1024), (21840,2048), (20020,4096), (18480,8192), and (17160,16384). I subsequently developed a Mathematica procedure for determining the Leyland # of any given (x,y) that does a count based on L(x,y) approximations up to a digit-size slightly smaller than the digit-size of L(x,y) and then adding a count for exact L(x,y) in the final stretch. Still, it took me 13 hours to determine the Leyland # of Serge Batalov's L(328574,15).

Indices for x^y - y^x numbers would (ideally) correspond to their position in A045575. For numbers larger than ~10^218 (the limit of how many terms are listed in the b-file for that sequence) I suppose that's as easy/hard as the corresponding x^y + y^x situation.

[pxp]

Originally Posted by kar_bon

I'm using the full date of findings from FactorDB, is this correct here? (see last examples)

I only now just looked at your numbers. For any prime of which I am the finder, the FactorDB date will be the day of the discovery or possibly the day after. That's because I run my finds through FactorDB before submitting them to PRPtop. Norbert Schneider is the only other person who has been actively looking for new primes since January 2017 (the date of Andrey Kulsha's last update to his list) and I don't know if/when he submits them to FactorDB. But I do know that other discoverers have been inconsistent about feeding their primes to FactorDB if they submitted them at all.

For example, Anatoly Selevich's L(8656,2929) has a FactorDB create-time of "before November 4, 2018" which is a far cry from his discovery date of ~ November 2007. When I went through the first 715 primes in FactorDB last week for my proven-primes list, I noted that for fifteen or twenty (or so) my query was the first query for that number (appearing as U). I just now tried it with 7789^7302+7302^7789 (prime #716) and it came back as U. So there are plenty of unknown/untried primes still that are not yet in FactorDB. The FactorDB creation date therefore strikes me as unindicative of the the prime's discovery date, except (as noted) for people like myself.