Index has reached #1102.

[URL="http://factordb.com/index.php?id=1100000000841025098"]L(15156,3209)[/URL]
[URL="http://factordb.com/index.php?id=1100000000841061408"]L(15277,3042)[/URL]

[URL="http://chesswanks.com/num/a094133.txt"]Index[/URL] has reached #1113 L(19783,492).

[URL="http://factordb.com/index.php?id=1100000000841348804"]L(16112,2349)[/URL]

Please check :) [url]http://www.primefan.ru/xyyxf/primes.html#primes[/url]

[QUOTE=XYYXF;436669]Please check :) [url]http://www.primefan.ru/xyyxf/primes.html#primes[/url][/QUOTE]

I can verify that a sorted version of your (x,y) list matches mine and that my name appears the appropriate number of times. A sorted version of your digits list mismatched mine in one instance: The number of digits in L(12547,4354) is ten more than what you have.

Fixed that.

Thank you Hans!

[URL="http://factordb.com/index.php?id=1100000000841447182"]L(13693,9262)[/URL]
[URL="http://factordb.com/index.php?id=1100000000841486701"]L(13683,9154)[/URL]
[URL="http://factordb.com/index.php?id=1100000000841494689"]L(14287,6138)[/URL]
[URL="http://factordb.com/index.php?id=1100000000842138953"]L(14898,5909)[/URL]

[URL="http://factordb.com/index.php?id=1100000000844098896"]L(13367,11418)[/URL]
[URL="http://factordb.com/index.php?id=1100000000844999830"]L(15108,3917)[/URL]
[URL="http://factordb.com/index.php?id=1100000000845093399"]L(14428,7849)[/URL]
[URL="http://factordb.com/index.php?id=1100000000845531539"]L(16534,2505)[/URL]

Updated: [url]http://www.primefan.ru/xyyxf/primes.html#0[/url]

[URL="http://factordb.com/index.php?id=1100000000846986895"]L(15711,3752)[/URL]
[URL="http://factordb.com/index.php?id=1100000000847615733"]L(13835,7366)[/URL]
[URL="http://factordb.com/index.php?id=1100000000848347621"]L(13105,12132)[/URL]
[URL="http://factordb.com/index.php?id=1100000000848354657"]L(13392,9919)[/URL]

[URL="http://factordb.com/index.php?id=1100000000848405372"]L(14147,5892)[/URL] = #1114
[URL="http://factordb.com/index.php?id=1100000000848557203"]L(14990,4011)[/URL]

[URL="http://chesswanks.com/num/a094133.txt"]Index[/URL] has reached #1119 L(20992,359).

[URL="http://factordb.com/index.php?id=1100000000848641620"]L(13691,12352)[/URL]
[URL="http://factordb.com/index.php?id=1100000000848667062"]L(15319,9862)[/URL]
[URL="http://factordb.com/index.php?id=1100000000848827743"]L(14051,8766)[/URL]
[URL="http://factordb.com/index.php?id=1100000000848853409"]L(14609,6406)[/URL]

I reached x=23,000 and found 2 new PRP:
677^21702+21702^677, 61430 digits,
747^22198+22198^747, 63782 digits.

Updated: [url]http://www.primefan.ru/xyyxf/primes.html#0[/url]

[URL="http://factordb.com/index.php?id=1100000000848971616"]L(15340,4221)[/URL]
[URL="http://factordb.com/index.php?id=1100000000849607618"]L(14345,8152)[/URL]
[URL="http://factordb.com/index.php?id=1100000000850270346"]L(16170,2141)[/URL]
[URL="http://factordb.com/index.php?id=1100000000850381812"]L(15504,3955)[/URL]
[URL="http://factordb.com/index.php?id=1100000000850381910"]L(13507,13074)[/URL]
[URL="http://factordb.com/index.php?id=1100000000850717342"]L(15110,3621)[/URL]

[URL="http://factordb.com/index.php?id=1100000000850777464"]L(13086,12769)[/URL]
[URL="http://factordb.com/index.php?id=1100000000850941215"]L(13981,8988)[/URL]
[URL="http://factordb.com/index.php?id=1100000000851611944"]L(14531,6912)[/URL]
[URL="http://factordb.com/index.php?id=1100000000851656234"]L(14754,5987)[/URL]

[URL="http://factordb.com/index.php?id=1100000000851767024"]L(13636,12623)[/URL]
[URL="http://factordb.com/index.php?id=1100000000851917986"]L(13214,11595)[/URL]
[URL="http://factordb.com/index.php?id=1100000000851948227"]L(13725,8128)[/URL] = #1120
[URL="http://factordb.com/index.php?id=1100000000852385550"]L(13918,7343)[/URL]

[URL="http://chesswanks.com/num/a094133.txt"]Index[/URL] has reached #1137 L(16636,1845).

[URL="http://factordb.com/index.php?id=1100000000852429001"]L(14427,7396)[/URL]
[URL="http://factordb.com/index.php?id=1100000000852480741"]L(14569,6846)[/URL]
[URL="http://factordb.com/index.php?id=1100000000852495177"]L(14872,5229)[/URL]
[URL="http://factordb.com/index.php?id=1100000000852504740"]L(13582,13165)[/URL]

I reached x=24,000 and found 6 new PRPs:
610^23017+23017^610, 64110 digits,
595^23148+23148^595, 64225 digits,
611^23586+23586^611, 65712 digits,
467^23596+23596^467, 62986 digits,
572^23601+23601^572, 65078 digits,
500^23793+23793^500, 64217 digits.

[URL="http://factordb.com/index.php?id=1100000000852678562"]L(14210,6849)[/URL]
[URL="http://factordb.com/index.php?id=1100000000852809145"]L(14031,7670)[/URL]
[URL="http://factordb.com/index.php?id=1100000000852811060"]L(14289,6860)[/URL]
[URL="http://factordb.com/index.php?id=1100000000852947007"]L(15307,3958)[/URL]

[URL="http://factordb.com/index.php?id=1100000000853430470"]L(14751,5006)[/URL]
[URL="http://factordb.com/index.php?id=1100000000854007165"]L(15023,4468)[/URL]
[URL="http://factordb.com/index.php?id=1100000000854315586"]L(16366,12925)[/URL]
[URL="http://factordb.com/index.php?id=1100000000854793067"]L(13860,8387)[/URL] = #1138

[URL="http://chesswanks.com/num/a094133.txt"]Index[/URL] has reached #1179 L(17691,1508). I'm diverting my fastest processors to another long-term project. The gaps on which I will still be working are up a bit from the current index high so I don't see myself contributing another index update anytime soon.

Updated the page (I hope correctly): [url]http://www.primefan.ru/xyyxf/primes.html#0[/url]

The range from 13001 to 15000 is now available for decimal length >= 56230 digits, i.e. most of the work here is already done :)

[QUOTE=XYYXF;440581]Updated the page (I hope correctly): [url]http://www.primefan.ru/xyyxf/primes.html#0[/url][/QUOTE]

The (x,y) pairs and their decimal digit lengths are correct, as are the contributions attributed to me.

Thanks for the check :)

[URL="http://factordb.com/index.php?id=1100000000855034913"]L(17458,7153)[/URL]
[URL="http://factordb.com/index.php?id=1100000000855998846"]L(15852,6061)[/URL]
[URL="http://factordb.com/index.php?id=1100000000856099082"]L(18407,4474)[/URL]
[URL="http://factordb.com/index.php?id=1100000000860276661"]L(18738,3955)[/URL]

I reached x=25,000 and found 2 new PRPs:
452^24729+24729^452, 65659 digits,
695^24772+24772^695, 70402 digits.

I reached x=26,000 and found 1 new PRP:
771^25718+25718^771, 74250 digits.

[QUOTE=NorbSchneider;445768]I reached x=26,000 and found 1 new PRP:
771^25718+25718^771, 74250 digits.[/QUOTE]Well done!

I reached x=27,000 and found 1 new PRP:
730^26513+26513^730, 75916 digits.

I reached x=27,500 and found 2 new PRPs:
713^27118+27118^713, 77371 digits,
565^27318+27318^565, 75181 digits.

After a small hiatus, on December 10 I started again to look for Leyland primes. I have now finished with the Leyland numbers in the gap between L(19021,1576) <60821 digits> and L(29007,128) <61124 digits> and have found therein 6 new PRPs:

L(14708,13707) <60847>
L(15088,10831) <60876>
L(14858,12651) <60950>
L(16069,6258) <61005>
L(14788,13355) <61011>
L(16697,4570) <61110>

Thanks for them. The page is updated: [url]http://www.primefan.ru/xyyxf/primes.html#0[/url]

I have now finished with the Leyland numbers in the two gaps between L(19909,1456) <62976 digits> and L(19546,1741) <63345 digits> and have found therein 5 new PRPs:

L(17549,3936) <63090>
L(15346,12941) <63103>
L(15103,15078) <63106>
L(16976,5253) <63158>
L(16458,7031) <63315>

I have now finished with the Leyland numbers in the two gaps between L(19732,1265) <61211 digits> and L(18879,1850) <61681 digits> and have found therein 6 new PRPs:

L(17567,3094) <61318>
L(14837,13810) <61429>
L(14821,14166) <61526>
L(15369,10226) <61626>
L(18185,2454) <61645>
L(15937,7408) <61672>

I have now finished with the Leyland numbers in the two gaps between L(25742,291) <63426 digits> and L(26336,267) <63905 digits> and have found therein 7 new PRPs:

L(17683,3882) <63466>
L(15852,10157) <63516>
L(16996,5483) <63549>
L(19085,2164) <63654>
L(18855,2402) <63741>
L(18468,2857) <63824>
L(15441,13706) <63879>

I reached x=30,000 and found 2 new PRPs:
474^27863+27863^474, 74556 digits,
536^29847+29847^536, 81458 digits.

I have now finished with the Leyland numbers in the gap between L(40495,114) <83295 digits> and L(35917,214) <83702 digits> and have found therein 6 new PRPs:

L(20850,9971) <83374>
L(20519,11572) <83378>
L(21368,8085) <83500>
L(26870,1293) <83609>
L(19522,19283) <83656>
L(20091,14596) <83664>

I reached x=31,000 and found 3 new PRPs:
734^30453+30453^734, 87270 digits,
423^30634+30634^423, 80456 digits,
758^30693+30693^758, 88386 digits.

I have now finished with the Leyland numbers in the gap between L(35917,214) <83702 digits> and L(39070,143) <84209 digits> and have found therein 7 new PRPs:

L(23543,3610) <83755>
L(20625,11522) <83770>
L(19551,19268) <83773>
L(21457,8200) <83979>
L(21234,9067) <84033>
L(20996,10059) <84038>
L(21485,8224) <84116>

I reached x=32,000 and found 1 new PRP:
496^31671+31671^496, 85369 digits.

I'm looking for the date of discovery of Anatoly Selevich's L(8656,2929) <30008 digits>. This number went on to be proven prime, which may have been why it wasn't in PRPtop when I added it on his behalf in Aug. 2015. For the record, I have eight other Leyland primes with more than 10000 decimal digits for which I don't have a discovery date:

<10041> L(3571,648) Paul Leyland
<10073> L(2930,2739) Greg Childers
<10094> L(3265,1234) Leonid Muraviov
<13740> L(5140,471) Paul Leyland
<15071> L(4405,2638) Greg Childers
<16868> L(5182,1799) Paul Leyland
<17283> L(5154,2255) Paul Leyland
<18195> L(5155,3384) Paul Leyland

If anyone can reference a discovery date for any of these, I'd be very grateful.

There are currently 1000 Leyland primes listed in PRPtop. Since my currently known number is 1302, we therefore have 302 missing: 128 (from Paul Leyland), 54 (Andrey Kulsha), 52 (Greg Childers), 25 (Peter Liaskovsky), 23 (Christ van Willegen), 5 (Alexander Kuzmich), 4 (Rob Binnekamp), 3 (Leonid Muraviov), 2 (each, from Mark Rodenkirch, Henri Lifchitz, Göran Hemdal, & Sander Hoogendoorn). My (Hans Havermann) PRPtop Leyland prime listing includes 6 that are actually from "firejuggler".

There are 1250 Leyland primes in Andrey Kulsha's [URL="http://www.primefan.ru/xyyxf/primes.html"]Jan. 2017 list[/URL]. The missing 52 are accounted for by discoveries from me (44) and Norbert Schneider (8).

My ongoing Leyland prime indexing effort is [URL="http://chesswanks.com/num/a094133.txt"]here[/URL].

[QUOTE=pxp;483924]I'm looking for the date of discovery of Anatoly Selevich's L(8656,2929) <30008 digits>. This number went on to be proven prime, which may have been why it wasn't in PRPtop when I added it on his behalf in Aug. 2015. For the record, I have eight other Leyland primes with more than 10000 decimal digits for which I don't have a discovery date:

<10041> L(3571,648) Paul Leyland
<10073> L(2930,2739) Greg Childers
<10094> L(3265,1234) Leonid Muraviov
<13740> L(5140,471) Paul Leyland
<15071> L(4405,2638) Greg Childers
<16868> L(5182,1799) Paul Leyland
<17283> L(5154,2255) Paul Leyland
<18195> L(5155,3384) Paul Leyland

If anyone can reference a discovery date for any of these, I'd be very grateful.[/QUOTE]It's just about possible, though unlikely, I may be able to find the information for the ones I discovered. It will require digging through decade-old material from when I worked for MSRC. All will be between (roughly) 2000 and 2005.

Paul

I believe that I can construct reasonable discovery dates for all but Anatoly Selevich's L(8656,2929) by using Paul's [URL="http://www.leyland.vispa.com/numth/primes/xyyx.htm"]ranges-being-searched table[/URL]. Obviously the date will fall between when-reserved and when-completed in roughly the same proportion as x falls between xmin and xmax. That leaves L(8656,2929) as being discovered after Oct. 2006, since this number is not yet in Paul's primes-and-strong-pseudoprimes list.

I've put Selevich's Leyland prime output from Aug 2007 through Jan 2008 (as dated in PRPtop) [URL="http://chesswanks.com/num/SelevitchLeylandPrimeDiscoveries.txt"]here[/URL]. It seems that (8656,2929) should fall somewhere in this date-range but it isn't obvious to me what's going on. He was likely searching different (x,y)-ranges on different processors.

The odds of finding a d-digit Leyland prime where d ~ 100000 are (empirically) about 1 in 75. In other words, one should expect to find about 40 Leyland primes with digit-size ranging from 100000 to 103000. What are the odds when d ~ 400000 ?

[QUOTE=pxp;488860]The odds of finding a d-digit Leyland prime where d ~ 100000 are (empirically) about 1 in 75. In other words, one should expect to find about 40 Leyland primes with digit-size ranging from 100000 to 103000. What are the odds when d ~ 400000 ?[/QUOTE]

My estimate is about 1 in 100. So about 30 Leyland primes with digit-size ranging from 400000 to 403000. Seems like a lot.

Norbert Schneider recently PRP'd L(47012,297). At 116250 decimal digits, this is now the 6th largest-known Leyland prime. Moreover, this gives Norbert five of the top ten. Congratulations!

I search in the interval 13,000<=x<=15,000, for new PRPs
and also doublecheck the known PRPs.
Currenttly I reached x=13,800, so far no new PRPs and the
known PRPs are confirmed.

Hans, what is your next range after the 100,000 digits is finished?

The long compute
 

[QUOTE=pxp;488860]The odds of finding a d-digit Leyland prime where d ~ 100000 are (empirically) about 1 in 75. In other words, one should expect to find about 40 Leyland primes with digit-size ranging from 100000 to 103000.[/QUOTE]

I have 51.

Beginning on 26 June 2017, I am today done with my search of Leyland numbers between L(40210,287) <98832> and L(40945,328) <103013>. I found 67 new primes. The (x,y) values of the 25046458 Leyland numbers in the gap were precomputed, sorted by size, and packaged into bundles of 88670. Each bundle was assigned to an available core on one of my Macs.

The computations were done in Mathematica (versions 8 or 9) where each Leyland number was checked for GCD(x,y)==1 before applying PrimeQ. A given bundle would take from four to six weeks to check. There's no point in listing the new primes. Refer to my [URL="http://chesswanks.com/num/a094133.txt"]Leyland prime indexing effort[/URL] which is always up-to-date.

Norbert asked what I'm doing next. I'm back to extending the prime indices which have languished at L(17691,1508) <56230>, #1179, for these almost-two years. I've already started.

[QUOTE=pxp;514156]I have 51.

Beginning on 26 June 2017, I am today done with my search of Leyland numbers between L(40210,287) <98832> and L(40945,328) <103013>. I found 67 new primes. The (x,y) values of the 25046458 Leyland numbers in the gap were precomputed, sorted by size, and packaged into bundles of 88670. Each bundle was assigned to an available core on one of my Macs.

The computations were done in Mathematica (versions 8 or 9) where each Leyland number was checked for GCD(x,y)==1 before applying PrimeQ. A given bundle would take from four to six weeks to check. There's no point in listing the new primes. Refer to my [URL="http://chesswanks.com/num/a094133.txt"]Leyland prime indexing effort[/URL] which is always up-to-date.

Norbert asked what I'm doing next. I'm back to extending the prime indices which have languished at L(17691,1508) <56230>, #1179, for these almost-two years. I've already started.[/QUOTE]

I'm confused. Did you do the PRP testing with Mathematica?

Yes. PrimeQ is Mathematica's PRP test.

I looked this up. Others might find it useful. The subject has come up more than once in the past.

[QUOTE]
The Rabin-Miller strong pseudoprime test is a particularly efficient test. The Wolfram Language implements the multiple Rabin-Miller test in bases 2 and 3 combined with a Lucas pseudoprime test as the primality test used by the function PrimeQ[n]. Like many such algorithms, it is a probabilistic test using pseudoprimes. In order to guarantee primality, a much slower deterministic algorithm must be used. However, no numbers are actually known that pass advanced probabilistic tests (such as Rabin-Miller) yet are actually composite.
[/QUOTE]

[url]http://mathworld.wolfram.com/PrimalityTest.html[/url]

[QUOTE=pxp;514156]I'm back to extending the prime indices which have languished at L(17691,1508) <56230>, #1179, for these almost-two years.[/QUOTE]

I just looked it up. The index reached #1179 on 23 Aug 2016 already!

[QUOTE=pxp;514172]The index reached #1179 on 23 Aug 2016 already![/QUOTE]

I have examined all Leyland numbers in the gap between L(17691,1508) <56230>, #1179, and L(17605,1908) <57755> and found 19 new primes. That makes L(17605,1908) #1199.

[QUOTE=pxp;515491]That makes L(17605,1908) #1199.[/QUOTE]

I have examined all Leyland numbers in the three gaps between L(17605,1908) <57755>, #1199, and L(26530,163) <58690> and found 12 new primes. That makes L(26530,163) #1214.

[QUOTE=pxp;516540]That makes L(26530,163) #1214.[/QUOTE]

I have examined all Leyland numbers in the gap between L(26530,163) <58690>, #1214, and L(125330,3) <59798> and found 8 new primes. That makes L(125330,3) #1223 and advances the index to L(28468,129), #1226.

[QUOTE=pxp;518589]That makes L(125330,3) #1223 and advances the index to L(28468,129), #1226.[/QUOTE]

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.

[QUOTE=pxp;519088]That makes L(19021,1576) #1236 and advances the index to L(19898,1263), #1254.[/QUOTE]

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 [URL="http://chesswanks.com/num/a094133.txt"]Leyland prime indexing effort[/URL] 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 [I]out there[/I] may not reflect its latest changes. For that reason, I am now keeping a [URL="https://drive.google.com/file/d/1QhQQOhZU0mmgt3vFXk1OtugNn5UUPwNH/view"]backup copy on my Google Drive[/URL] that will hopefully be accessible a little longer than the file on my computer. Just in case anyone is interested.

[QUOTE=pxp;520920]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 [URL="http://chesswanks.com/num/a094133.txt"]Leyland prime indexing effort[/URL] 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 [I]out there[/I] may not reflect its latest changes. For that reason, I am now keeping a [URL="https://drive.google.com/file/d/1QhQQOhZU0mmgt3vFXk1OtugNn5UUPwNH/view"]backup copy on my Google Drive[/URL] that will hopefully be accessible a little longer than the file on my computer. Just in case anyone is interested.[/QUOTE]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?

[QUOTE=xilman;520923]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?[/QUOTE]

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 [URL="http://www.primefan.ru/xyyxf/primes.html"]Andrey Kulsha's list[/URL] augmented by [URL="https://www.mersenneforum.org/showthread.php?t=19348&page=2"]RichD's 27 subsequent additions[/URL] and I might do that. But that takes us only to November 2017 and constitutes a blog post at most.

[QUOTE=pxp;520944]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 [URL="http://www.primefan.ru/xyyxf/primes.html"]Andrey Kulsha's list[/URL] augmented by [URL="https://www.mersenneforum.org/showthread.php?t=19348&page=2"]RichD's 27 subsequent additions[/URL] and I might do that. But that takes us only to November 2017 and constitutes a blog post at most.[/QUOTE]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.

I've created a [URL="http://chesswanks.com/num/ProvenLeylandPrimes.txt"]Proven Leyland Primes[/URL] document of the Leyland primes up to L(8656,2929), #715. In the final column I've indicated [I]factordb[/I] if factordb.com has the number as P. I've indicated [I]Kulsha[/I] 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).

[QUOTE=pxp;520984]I've created a [URL="http://chesswanks.com/num/ProvenLeylandPrimes.txt"]Proven Leyland Primes[/URL] document of the Leyland primes up to L(8656,2929), #715. In the final column I've indicated [I]factordb[/I] if factordb.com has the number as P. I've indicated [I]Kulsha[/I] 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).[/QUOTE]Thanks!

Paul

[QUOTE=pxp;520920]That makes L(19909,1456) #1266 and advances the index to L(26336,267), #1283.[/QUOTE]

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.

I've created a category for Leyland primes/PRPs in the Wiki [url='https://www.rieselprime.de/ziki/Category:Leyland_prime_P']here[/url].
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.

[QUOTE=kar_bon;521977]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?[/QUOTE]

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 [I]all[/I] 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 [URL="http://chesswanks.com/num/a094133.txt"]Leyland primes indexing[/URL] 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 [URL="http://oeis.org/A076980"]A076980[/URL]. 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 [URL="http://oeis.org/A045575"]A045575[/URL]. 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.

[QUOTE=kar_bon;521977]I'm using the full date of findings from FactorDB, is this correct here? (see last examples)[/QUOTE]

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.

Yes, the "date" "before November 4, 2018" in FactorDB was created after server/database moving by Markus, so not available older dates here.

The list of Leyeland primes contains also some dates like for x=1031-1050 as reserved/complete, that's why I used "2001-06-04" for those 4 primes then. For other dates expressions like "2002-06" is enough here as you gave for >#294 in your list.

Dates from your and Norberts finds should be better than.

Will see how this can be handled. The great advantage of the table is the individual sorting:
I tried first to sort by x-value in the category by the template, but this needed a click to sort the table by digits. Now it's done by sorting by digits and calling the table only. Inserting a new number you don't need to create an index (first column), it's done in the table call.
Also:
Clicking on "Digits" column first (upwards sorting) and than on "Prover" the smallest unproven number are listed in descending digit order: could be helpful to prove some smaller numbers.

ToDo's:
pages with information about those numbers, links, stats, current work/reservation, minus-side.

[url='https://www.rieselprime.de/ziki/Leyland_prime_P_2240_87']Here[/url] is a Leyland prime of 4345 digits without a certificate in FactorDB but proven prime in the list.

Same [url='https://www.rieselprime.de/ziki/Leyland_prime_P_1610_993']here[/url]: 4826 digits.

[QUOTE=kar_bon;522086][url='https://www.rieselprime.de/ziki/Leyland_prime_P_2240_87']Here[/url] is a Leyland prime of 4345 digits without a certificate in FactorDB but proven prime in the list.
Same [url='https://www.rieselprime.de/ziki/Leyland_prime_P_1610_993']here[/url]: 4826 digits.[/QUOTE]

Yes. These are the smallest two of thirty-one instances previously noted here. You can easily find all 31 by searching for the word 'Kulsha' in my [URL="http://chesswanks.com/num/ProvenLeylandPrimes.txt"]proven Leyland primes[/URL] list.

I found 3 new PRPs:
634^42803+42803^634, 119938 digits,
736^44335+44335^736, 127104 digits,
9946^17491+17491^9946, 69923 digits.

Those three new ones are in the Wiki, too.

Some notes:

- I'm using a date of discovery for old Leyland primes according to the "When completed" listed [url='http://www.leyland.vispa.com/numth/primes/xyyx.htm']here[/url]. So listing/sorting in the table is available for those, too.

- Primes with missing certificate in FactorDB are marked with a remark in the page and listed as orange in column "Prover" in the table. These certs should be inserted later.

- I've found some certs. in FactorDB but not yet listed in pxp's list. I used the date and info of program from FactorDB.

- I've inserted some discovery dates for numbers found during doublechecking from [url='http://www.primefan.ru/xyyxf/news.html#0']here[/url].

- Categories for proven and PRPs numbers available now.

More to come.

[QUOTE=kar_bon;522194] I've found some certs. in FactorDB but not yet listed in pxp's list.[/QUOTE]

I'm not sure what you are saying here. There were only two types of proven Leyland primes in [URL="http://chesswanks.com/num/ProvenLeylandPrimes.txt"]my list[/URL]: Those that were shown as P in FactorDB (257, indicated by the word 'factordb') and those that were shown as PRP in FactorDB but were indicated as proven by Kulsha (31, indicated by the word 'Kulsha'). If you are saying that there are certificates in the former, that's certainly to be expected. I wasn't concerned about certificates in creating my list but only in distinguishing proven primes from PRPs.

Reserving the range x = 20001-30000, y = 801-1000.

[QUOTE=pxp;522216]I'm not sure what you are saying here.[/QUOTE]

I'm trying to combine several data files but found no prover/date in any.

Example: L(2448,535) no dates for <#295 in you a094133.txt and no prover/proven date at all, so I used the data from FactorDB.

[QUOTE=kar_bon;522230]I'm trying to combine several data files but found no prover/date in any. Example: L(2448,535) no dates for <#295 in you a094133.txt and no prover/proven date at all, so I used the data from FactorDB.[/QUOTE]

The point of my [URL="http://chesswanks.com/num/a094133.txt"]a094133.txt[/URL] document is to have in one place an up-to-date list of [I]all[/I] known Leyland primes and to track my continuing effort to index them by size. I've added the Leyland#, decimal-digit size, discoverer, and the discovery date for indices >294 (PRPtop did not accept PRPs smaller than 10000 decimal digits and I've culled my discovery dates primarily from them, except where they were missing or clearly incorrect).

I was never very interested in the proven/PRP distinction and created the [URL="http://chesswanks.com/num/ProvenLeylandPrimes.txt"]proven Leyland primes[/URL] list only as a courtesy to [I]xilman[/I] who asked me for something like it on July 6. [URL="http://www.primefan.ru/xyyxf/primes.html"]Andrey Kulsha's list[/URL] has prover/proven-date information on 260 of the known proven Leyland primes, including 31 that are still PRP in FactorDB. For these 31 you will not be able to get prover/proven-date from FactorDB. For the remaining 229 you can get the information from either Kulsha's list or from FactorDB but I would guess that the FactorDB prover/proven-date may or may not agree with Kulsha's. To grab a random example: L(3100,11) is proven prime by Jonathan A. Zylstra on 30 December 2003 according to Kulsha. It is proven prime by Edwin Hall on 15 March 2017 according to FactorDB. Clearly Kulsha's information is likely to be preferred.

There are 28 proven Leyland primes that are [I]not[/I] in Kulsha's list and for these one necessarily has to rely on FactorDB alone. But these are easy. 27 of them are from RichD (the last three posts [URL="https://www.mersenneforum.org/showthread.php?t=19348&page=2"]here[/URL]) and one is from [URL="http://factordb.com/index.php?id=1100000000936497498"]Anonymous[/URL].

[QUOTE=Dylan14;522221]Reserving the range x = 20001-30000, y = 801-1000.[/QUOTE]

Hey Dylan. Welcome aboard.

Just so you know, Andrey Kulsha has been missing (I'm guessing that he died) since his last update in January 2017, so there is really no-one here to track the reservations. Since January 2017 there have only been two users that are actively searching for new Leyland primes: Norbert Schneider who has added 33 and myself who has added 177. Those 210 combined with the 1250 in Kulsha's list give us the 1460 known Leyland primes to date.

While Norbert still conducts his searches by using the (x,y)-range system, I do not. My searches are strictly by L(x,y) decimal-digit size and my current search interval is L(32907,92) <64623 decimal digits> to L(29934,157) <65733>.

Be sure to post any of your finds to either PRPtop or here (preferably both) so that I can add them to my own list of known Leyland primes.

Another new PRP:
746^44541+44541^746, 127955 digits.

I've made some changes/approvements in the Wiki for Leyland primes/PRPs:

The latest PRP can be found [url='https://www.rieselprime.de/ziki/Leyland_prime_P_44541_746']here[/url]
- added a history entry to give the post# with date
- the "date found" from FactorDB
- not yet Leyland# filled

There's also a "Proved" history entry possible: see [url='https://www.rieselprime.de/ziki/Leyland_prime_P_2284_1985']here[/url]
- "Prove date" and "Program" from FactorDB
- History date from forum post with link

There're also different categories for proven primes and PRP. Because of sorting by digits, the smallest unproven number can be found as first entry [url='https://www.rieselprime.de/ziki/Category:Leyland_prime_P_PRP']here[/url] = Leyland prime P 3147 214 = 7334 digits.
This can help to identify and prove smaller PRPs for others.
In the [url='https://www.rieselprime.de/ziki/Leyland_prime_P_table']table view[/url] the numbers which marked "proven" but no proof is available in FactorDB are marked.

Further there's a page which creates a [url='https://www.rieselprime.de/ziki/Leyland_prime_P_csv']CSV format[/url] to copy/paste for further use.

I'm now going to insert more/all known Leyland numbers in the wiki with data I've found:
- proven dates/names from the other thread (most from RichD) and found dates from here
- dates from old primes from Leyland page as "When reserved"/"When completed".

I don't know if this was shown somewhere but found nothing, so here's the current distribution of known Leyland primes/PRP's for x<40000 and y<25000:

I recognize those 100000-digit numbers on the right. :) I played in Mathematica with a similar graph last night with the idea of superimposing curves of 60000-, 80000-, and 100000-digit x^y+y^x but I couldn't even figure out how to generate those curves. :/

[QUOTE=pxp;522711]I played in Mathematica with a similar graph last night with the idea of superimposing curves of 60000-, 80000-, and 100000-digit x^y+y^x but I couldn't even figure out how to generate those curves.[/QUOTE]

I finally kludged something together (ContourPlot was the Mathematica function that I was missing). I'm not sure if the waves on the end of the <60000> curve are real or an artifact.

[QUOTE=pxp;522753]I finally kludged something together (ContourPlot was the Mathematica function that I was missing). I'm not sure if the waves on the end of the <60000> curve are real or an artifact.[/QUOTE]An interesting plot. I think the waves are very likely an artefact.

Above about, say, 8000,the distribution of the points looks to me very much like a uniformly random sample of the triangle. Presumably it is not, or the distribution would look random at the lower regions as well.

It may be interesting to apply a scaling to the Y values, such that the populated area becomes square and then to investigate the hypothesis that the (x,y) co-ordinates are drawn independently from a uniform random distribution. If the likelihood is significantly different try to discover a distribution which better matches the observations.

Any takers? I'm not sure my statistics ability is (yet) up to the task.

(added in edit: Henry,this seems like an area where you have some expertise.)

[QUOTE=xilman;522757]I think the waves are very likely an artefact...[/QUOTE]

Indeed. I've fixed that, fixed the aspect ratio to make the x=y line bisect the axes, made the points smaller, and added two greenish curves to indicate the interval that I am currently exploring (I should be done in ten days).

[QUOTE=pxp;521648]That makes L(32907,92) #1296.[/QUOTE]

I have examined all Leyland numbers in the nine gaps between L(32907,92) <64623>, #1296, and L(29934,157) <65733> and found 22 new primes. That makes L(29934,157) #1327.

A look ahead
 

I have written a blog article regarding my ambitious [URL="https://gladhoboexpress.blogspot.com/2019/08/a-look-ahead.html"]Leyland-prime search schedule[/URL] for the next two years.

Another new PRP:
7257^17528+17528^7257, 67672 digits.

[QUOTE=pxp;523458]That makes L(29934,157) #1327.[/QUOTE]

I have examined all Leyland numbers in the gap between L(29934,157) <65733>, #1327, and L(40182,47) <67189> and found 20 new primes. That makes L(40182,47) #1348 and advances the index to L(31870,131), #1354.

Another new PRP:
12511^17556+17556^12511, 71933 digits.

Another new PRP:
13952^17559+17559^13952, 72776 digits.

Another new PRP:
11699^17560+17560^11699, 71437 digits.

Another new PRP:
16696^16957+16957^16696, 71603 digits.

Another new PRP:
15813^16916+16916^15813, 71031 digits.

Another new PRP:
14742^16915+16915^14742, 70512 digits.