y^x-x^y primes

[rogue]

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

[Batalov]

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.

[NorbSchneider]

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.

[Batalov]

Originally Posted by NorbSchneider

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.

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.

[xilman]

Originally Posted by Batalov

Re: y^x-x^y :

FWIW, I'm building tables of factorizations of numbers of this form. Not yet published but likely to be RSN.

Paul

[Batalov]

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

[rogue]

Completed thru 11600. Here are the new PRPs:

3945^11504-11504^3945
20^11507-11507^20 	
3169^11552-11552^3169
4340^11553-11553^4340
9170^11589-11589^9170

[Batalov]

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^y/g-y^x/g = 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 prp₁₈₄₆₂₉.
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).

[rogue]

Originally Posted by Batalov

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^y/g-y^x/g = 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.

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

[NorbSchneider]

The y^x-x^y PRPs page 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:

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

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

PRPs with 100,000+ digits:

        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

[rogue]

Completed thru 11700:

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