y^x-x^y primes
 

...I also search PRPs of the form y^x-x^y.

I made a webpage to these PRPs, similar to Andrey's page
to the y^x+x^y PRPs. You can find the page at [URL="http://primfakt.atw.hu"]primfakt.atw.hu[/URL],
y^x-x^y PRPs exists much more than y^x+x^y PRPs.
For example to x= 5000 894 y^x-x^y and 426 y^x+x^y PRPs,
x=10000 1530 y^x-x^y and 787 y^x+x^y PRPs.
I have all the y^x-x^y PRPs to x=10800, and a few for higher x values.
Andrey, Hans or someone else, are you interesting to join me
searching the y^x-x^y PRPs?

I remember than the numbers of the form x^y-y^x was factorized by Torbjörn Alm some years ago. That's his posting to ggnfs yahoogroup dated 23rd of October, 2005:[quote]I have been running a little factoring job on numbers
X^Y-Y^X and I have factored almost all numbers up to X=80.
I wonder if there is any interest in the result and where to
publish it.

The factoring has been done spep by step with
trial div up to 10^7, P-1, P+1 and rho followed by MPQS and ECM
in order to remove small factors. Some numbers has been factored
algebraic at this point.
Finally SNFS + some GNFS to break the composites. I have a database in
Paradox to keep it and a Delphi prgoram to administer it.
I have kept all GNFS/SNFS result files as well as the log files.

If there is an interest , I will gladly made it available as
a text file or as a PDF document.

Torbjörn Alm[/quote]Some of the results is still available in the Files section of the group:
[url]https://groups.yahoo.com/neo/groups/ggnfs/files[/url]

I found 2 new PRPs:
7406^12879+12879^7406, 49837 digits,
8335^12882+12882^8335, 50510 digits.

Andrey, the file factortable_xy-yx_1_100.txt in [url]https://groups.yahoo.com/[/url]
neo/groups/ggnfs/files group contains all factorization for x < 101 as
in your results.txt. For 100 < x < 151 as in your results2.txt have I
nothing found to the y^x-x^y numbers.

To the y^x-x^y PRPs have I nothing new found in the above group,
you know y^x-x^y PRPs, thats are not on my webpage?

Nothing more than Henri Lifchitz's prptop + factordb.com.

[QUOTE=pxp;429459]New PRP: L(15215,4762). I'm still working on eight gaps but that's down from as many as eighteen, having diverted processes to another project: [URL="http://chesswanks.com/seq/a037053.txt"]smallest prime containing a given number of zeros[/URL]. So I don't see myself doing y^x-x^y, Norbert. If anything, after I've had my fill of this diversion, I'll get back to advancing my indexing full on.[/QUOTE]

y^x-x^y sounds like an interesting project. I have a couple of week left for the project I am currently working on. I could probably start working on that soon after.

Mark, I also search PRPs of the form y^x-x^y.
At [url]http://primfakt.atw.hu/[/url] can you see, which ranges are completed and which
are available for searching.

I started sieving for x=11301 to x=12400. Unfortunately I only have one computer that I can run my sieving code on and that is the one with the slowest GPU. Now if I could only talk my wife into letting me get a new 27" iMac...

Down to 650,000 (from nearly 3,000,000) and only sieved to 131,519.

Sieving is done. I have about 400,000 candidates to test. I'm guessing about 40 days of PRP testing, but only after I suspect what I am currently doing.

FYI, I'm making a small change to the PRPNet server code so that server stats use y^x-x^y for the - form and x^y+y^x for the + form.

Here are a few PRPs for the minus form: It is complete to x=11400. Still crunching away.

7980^11317-11317^7980
5577^11320-11320^5577
3638^11327-11327^3638
765^11336-11336^765
3415^11342-11342^3415
2181^11344-11344^2181
7707^11344-11344^7707
2684^11355-11355^2684
2779^11364-11364^2779
7287^11366-11366^7287
7813^11372-11372^7813
243^11384-11384^243

I reached x=12,970 and found 1 new PRP:
10821^12968+12968^10821, 52317 digits.

Mark the [URL="http://primfakt.atw.hu/"]y^x-x^y PRPs page[/URL] is updated, the new PRPs from you and me
are on the page now.

Hans, the Leyland# to a given (x,y) pair determine you also with a database
and a Mathematica program, thank for sharing this. I try to write a program
in C# to determine the "Leyland#" to the y^x-x^y PRPs.

I guess I missed one at 11400. Here is thru 11500 (really):

10933^11400-11400^10933
9704^11403-11403^9704
5737^11406-11406^5737
2908^11421-11421^2908
9411^11422-11422^9411
315^11434-11434^315
10043^11466-11466^10043
7338^11467-11467^7338
331^11484-11484^331
4933^11490-11490^4933

Re: y^x-x^y :
whoever worked the 10<=y<=40 interval missed some primes (or reported the finished ranges misleadingly to 70,000).
11^44772-44772^11 is a "new" PRP.

The y^x-x^y PRPs page is updated, the new PRPs are on the page now.

Serge, the interval 30,101<=x<=70,000, 2<=y<=40 are not finished yet.
I search this interval and are at x=31,200. The info Available for y > 40
are to the others searchers. Thanks for findig a new PRP.

[QUOTE=NorbSchneider;432327]Serge, the interval 30,101<=x<=70,000, 2<=y<=40 are not finished yet.
I search this interval and are at x=31,200. The info Available for y > 40
are to the others searchers. Thanks for findig a new PRP.[/QUOTE]
I see. It is perhaps best to add the explicit reservation there for any existing work in progress.

I ran some single y values from 70k up, but for some y, tried to "double-check" the 30k-70k range (because the whole sub-range is obviously faster than even the first several k above 70k) and was surprised.

[QUOTE=Batalov;432285]Re: y^x-x^y : [/QUOTE]
FWIW, I'm building tables of factorizations of numbers of this form. Not yet published but likely to be RSN.

Paul

14^119741-119741^14 is a 137,239-digit PRP

Completed thru 11600. Here are the new PRPs:

[code]
3945^11504-11504^3945
20^11507-11507^20
3169^11552-11552^3169
4340^11553-11553^4340
9170^11589-11589^9170
[/code]

I wanted to suggest a small change to the new sieve (and if possible / if it is still maintained, to Multisieve). I have not looked at the source - perhaps it is already implemented in it; if so, kindly disregard.

For both + and - forms, any (x,y) pair where gcd(x,y)=g>1 should be removed as soon as the sieve is started; these have an algebraic factorization (with the rare exception if x[SUP]y/g[/SUP]-y[SUP]x/g[/SUP] = 1 - this is only important for tiny (x,y)). For example, if y=15, then all x :: 3|x or 5|x should be removed. In case of Multisieve, I know that this is not happening. Of course, if g=2, it is happening trivially by the parity argument. But for y=14, all x :: 7|x should be removed. For y=11, all x :: 11|x should be removed, etc.
__________________

P.S. Also 14^161089-161089^14 is a prp[SUB]184629[/SUB].
I started quite a while ago by running the quasi-near-repdigit 10^x-x^10 series (and I am well above x>500000 ...w/o any new primes), then I observed that for y=16, no primes are possible (as well as for y=27 or 36, algebraically), and then sieved and ran 10<=y<="16" for x<=200000 (still running; x<=70000 was expected to be a sanity double-check, as discussed above; didn't expect to find anything there).

[QUOTE=Batalov;432957]I wanted to suggest a small change to the new sieve (and if possible / if it is still maintained, to Multisieve). I have not looked at the source - perhaps it is already implemented in it; if so, kindly disregard.

For both + and - forms, any (x,y) pair where gcd(x,y)=g>1 should be removed as soon as the sieve is started; these have an algebraic factorization (with the rare exception if x[SUP]y/g[/SUP]-y[SUP]x/g[/SUP] = 1 - this is only important for tiny (x,y)). For example, if y=15, then all x :: 3|x or 5|x should be removed. In case of Multisieve, I know that this is not happening. Of course, if g=2, it is happening trivially by the parity argument. But for y=14, all x :: 7|x should be removed. For y=11, all x :: 11|x should be removed, etc.[/QUOTE]

I am not going to fix MultiSieve. xyyxsievecl does have a gcd() check.

The [URL="http://primfakt.atw.hu/"]y^x-x^y PRPs page[/URL] is updated, the new PRPs are on the page now.

Mark, you have found the 100 wide interval with the fewest PRPs.

100 wide intervalls with the fewest PRPs:
[CODE]
x=11501-11600 5 PRPs

x=6101-6200 6 PRPs
x=9201-9300 6 PRPs

x=5301-5400 7 PRPs
x=10701-10800 7 PRPs

x=5701-5800 8 PRPs
x=6001-6100 8 PRPs
x=6301-6400 8 PRPs
x=7301-7400 8 PRPs
[/CODE]

Serge, you have found the top 2 largest y^x-x^y PRPs!

PRPs with 100,000+ digits:
[CODE]
number length discoverer
14^161089-161089^14 184629 Serge Batalov
14^119741-119741^14 137239 Serge Batalov
29504^30069-30069^29504 134405 Norbert Schneider
28118^30097-30097^28118 133902 Norbert Schneider
18565^30002-30002^18565 128070 Norbert Schneider
14120^30009-30009^14120 124533 Norbert Schneider
[/CODE]

Completed thru 11700:

[code]
408^11603-11603^408
3513^11606-11606^3513
11018^11621-11621^11018
1747^11624-11624^1747
4568^11641-11641^4568
8491^11652-11652^8491
9518^11653-11653^9518
270^11671-11671^270
608^11693-11693^608
[/code]

Completed thru 11,800:

[code]
2254^11721-11721^2254
6619^11730-11730^6619
9955^11742-11742^9955
2164^11751-11751^2164
7397^11772-11772^7397
11702^11773-11773^11702
6780^11773-11773^6780
5624^11777-11777^5624
9049^11778-11778^9049 [/code]

The [URL="http://primfakt.atw.hu/"]y^x-x^y PRPs page[/URL] is updated, the new PRPs are on the page now.

[QUOTE]
then sieved and ran 10<=y<="16" for x<=200000 (still running;
[/QUOTE]

Serge, the search to x=200,000 is still running?
What is with the number x<=200,000, 2<=y<=9, you search also
this number?

10<=y<=17 for x<=200000 is done.
[QUOTE=NorbSchneider;434785]What is with the number x<=200,000, 2<=y<=9, you search also this number?[/QUOTE]
No, I have not looked at x<=200,000, 2<=y<=9 because it appears to be finished by others: Donovan Johnson, Henri Lifchitz, maybe some others.

Completed thru 11,900:

[code]
3002^11819-11819^3002
1379^11822-11822^1379
2150^11823-11823^2150
3137^11826-11826^3137
3017^11838-11838^3017
3373^11850-11850^3373
11195^11852-11852^11195
11161^11858-11858^11161
3617^11862-11862^3617
32^11879-11879^32
5520^11879-11879^5520
5046^11887-11887^5046
463^11888-11888^463
[/code]

Completed thru 12,000

[code]
10755^11902-11902^10755
11141^11910-11910^11141
11391^11920-11920^11391
9578^11927-11927^9578
10442^11933-11933^10442
5639^11954-11954^5639
4220^11963-11963^4220
2582^11971-11971^2582
5715^11974-11974^5715
704^11977-11977^704
1578^11983-11983^1578
[/code]

Completed thru 12,100:

[code] 3723^12010-12010^3723
8678^12011-12011^8678
9021^12022-12022^9021
4231^12032-12032^4231
6262^12033-12033^6262
2984^12057-12057^2984
11687^12066-12066^11687
9741^12068-12068^9741
9090^12071-12071^9090
458^12085-12085^458
6659^12092-12092^6659[/code]

Completed thru 12,200:

[code] 2622^12101-12101^2622
2^12103-12103^2
5870^12127-12127^5870
5889^12128-12128^5889
455^12132-12132^455
4820^12141-12141^4820
2106^12143-12143^2106
5922^12155-12155^5922
1175^12162-12162^1175
2495^12168-12168^2495
6867^12170-12170^6867
11163^12172-12172^11163
10626^12175-12175^10626
8132^12183-12183^8132
7590^12193-12193^7590 [/code]

I found the following new PRPs from x=10990, to x=11055

[CODE]
10568^10993-10993^10568
10502^11005-11005^10502
3552^11009-11009^3552
7178^11011-11011^7178
7471^11028-11028^7471
8918^11049-11049^8918
7704^11051-11051^7704
8684^11055-11055^8684
[/CODE]

The [URL="http://primfakt.atw.hu/"]y^x-x^y PRPs page[/URL] is updated, the new PRPs are on the page now.

My reserved range is finally complete. Here are the final primes:

[code] 368^12203-12203^368
10982^12203-12203^10982
271^12210-12210^271
4272^12211-12211^4272
2048^12225-12225^2048
1164^12229-12229^1164
10377^12230-12230^10377
5562^12269-12269^5562
5768^12269-12269^5768
2618^12275-12275^2618
3419^12308-12308^3419
6934^12309-12309^6934
12209^12330-12330^12209
3498^12337-12337^3498
8202^12341-12341^8202
2768^12355-12355^2768
5800^12369-12369^5800
602^12371-12371^602
1962^12373-12373^1962
11971^12374-12374^11971
695^12386-12386^695
9746^12387-12387^9746
145^12392-12392^145
[/code]

I found the following new PRPs from x=11056, to x=11150

[CODE]
9384^11059-11059^9384
1584^11083-11083^1584
774^11089-11089^774
1470^11089-11089^1470
5245^11094-11094^5245
5175^11096-11096^5175
9407^11102-11102^9407
4243^11126-11126^4243
10600^11127-11127^10600
3099^11132-11132^3099
7448^11147-11147^7448
[/CODE]

The [URL="http://primfakt.atw.hu/"]y^x-x^y PRPs page[/URL] is updated, the new PRPs are on the page now.

[QUOTE]
My reserved range is finally complete.
[/QUOTE]

Thank you Mark for searching the x=11301-12400 range, you have found 118 new PRPs.
(One PRP was already known: 2^12103-12103^2)

I found the following new PRPs from x=11151, to x=11300

[CODE]
9253^11168-11168^9253
7493^11178-11178^7493
10413^11182-11182^10413
9560^11187-11187^9560
1364^11201-11201^1364
1784^11221-11221^1784
7742^11229-11229^7742
8101^11232-11232^8101
3479^11240-11240^3479
4448^11243-11243^4448
2606^11267-11267^2606
462^11269-11269^462
11172^11275-11275^11172
563^11280-11280^563
5059^11282-11282^5059
491^11294-11294^491
[/CODE]

From x=12501, to x=12650
[CODE]
6750^12509-12509^6750
9001^12528-12528^9001
7337^12542-12542^7337
8323^12542-12542^8323
11966^12553-12553^11966
6576^12565-12565^6576
6411^12568-12568^6411
2210^12579-12579^2210
3050^12579-12579^3050
4209^12580-12580^4209
4273^12584-12584^4273
5001^12590-12590^5001
9932^12595-12595^9932
5762^12603-12603^5762
6900^12607-12607^6900
9722^12611-12611^9722
1945^12612-12612^1945
228^12631-12631^228
1158^12631-12631^1158
9224^12633-12633^9224
10736^12645-12645^10736
5779^12648-12648^5779
[/CODE]

Other PRPs
[CODE]
97^13088-13088^97
314^13049-13049^314
461^13400-13400^461
588^13231-13231^588
8^60681-60681^8
[/CODE]

I found 2 PRPs on prptop, discovered by Serge Batalov.
42^18385-18385^42
41^23690-23690^41
Serge have you found any other new y^x-x^y PRPs?

The [URL="http://primfakt.atw.hu/"]y^x-x^y PRPs page[/URL] is updated, the new PRPs are on the page now.

[QUOTE=NorbSchneider;450570]I found 2 PRPs on prptop, discovered by Serge Batalov.
42^18385-18385^42
41^23690-23690^41
Serge have you found any other new y^x-x^y PRPs?
[/QUOTE]
No, just these tiny ones.

I found the following new PRPs from x=12651, to x=12800
[CODE]
7760^12677-12677^7760
751^12690-12690^751
5552^12691-12691^5552
12240^12709-12709^12240
1173^12728-12728^1173
2183^12732-12732^2183
8993^12732-12732^8993
11832^12763-12763^11832
5146^12777-12777^5146
9178^12777-12777^9178
444^12791-12791^444
7192^12795-12795^7192
5970^12797-12797^5970
[/CODE]

Other PRPs
[CODE]
920^13221-13221^920
1126^13257-13257^1126
170^13451-13451^170
190^13743-13743^190
200^13401-13401^200
266^13777-13777^266
369^13730-13730^369
466^13527-13527^466
495^13688-13688^495
500^13689-13689^500
774^13691-13691^774
899^13632-13632^899
1062^13447-13447^1062
356^13985-13985^356
470^14023-14023^470
522^14159-14159^522
665^14072-14072^665
1068^13955-13955^1068
38^56123-56123^38
39^67918-67918^39
3^114892-114892^3
[/CODE]

The [URL="http://primfakt.atw.hu/"]y^x-x^y PRPs page[/URL] is updated, the new PRPs are on the page now.

I searched the range x = 15001-20000, y = 41-100 and found the following PRP's:


[CODE]42^18385-1*18385^42
98^15061-1*15061^98
49^15740-1*15740^49
70^15219-1*15219^70
76^18735-1*18735^76[/CODE]


Reserving the range x = 70001-100000, y = 11-20.

[QUOTE=Dylan14;523166]Reserving the range x = 70001-100000, y = 11-20.[/QUOTE]


This range is complete. Only the following was found:


[CODE]18^92125-1*92125^18[/CODE]


With 115642 digits, this should handily enter the PRPtop.

Dylan, congratulations to your new PRP!
18^92125-92125^18 with 115642 digits is the 12th biggest known y^x-x^y PRP.
What is your next range to search?

[QUOTE=NorbSchneider;524572]Dylan, congratulations to your new PRP!
18^92125-92125^18 with 115642 digits is the 12th biggest known y^x-x^y PRP.
What is your next range to search?[/QUOTE]


I'll take x = 70001-100000, y = 21-30 next.

[QUOTE=Dylan14;524758]I'll take x = 70001-100000, y = 21-30 next.[/QUOTE]

This range is complete. The following PRP's were found:

[CODE]24^78283-78283^24
29^80192-80192^29
[/CODE]
Taking x=140001-500000, y = 2-10.

This page, [url]http://primfakt.atw.hu[/url], is not being updated with ranges that have been reserved and tested. That makes is harder for anyone who wants to participate.