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]