sequences (x+1)^x +/- x^(x-1)

[cino hilliard]

>From: "Pfoertner, Hugo" <Hugo.Pfoertner at muc.mtu.de>
>To: cino hilliard <hillcino368 at hotmail.com>
>CC: Sequence Fanatics <seqfan at ext.jussieu.fr>
>Subject: RE: sequence (x+1)^x + x^x
>Date: Fri, 29 Aug 2003 12:41:51 +0200
>
>-----Original Message-----
>
>Cino Hilliard's message contained (in part):
> > >      print1(3" ");
> > >      forstep(x=2,n,2,
> > >        y=(x+1)^x+x^x;
> > >         if(isprime(y),print1(y" "))
> > >
> > > Maybe someone can further limit the search or find more terms?
> >
>
>To which Jim Nastos replied (in part):
> >   Since primality testing is costly for large numbers,
>[...]
>
>R. Shepherd replied:
>
> > To which I add/clarify that it's my experience that even Pari's
>*probable*-
> > primality testing is costly for sufficiently large numbers
>[...]
>
>I have now tested up to x=6000, for which (x+1)^x+x^x is ~ 3*10^22669
>and found nothing. Too bad, but I think it's time to give up here :-(

Here are a couple more I submitted  that are hard to find many primes.
x   (x+1)^x - x^(x-1)
2   7
5   7151
6   109873
9   956953279
24 3497141354765072424170242943188801

x   (x+1)^x + x^(x-1)
2 11
3 73
8 45143873
9 1043046721
15 1182114430632237601
16 49814113380273715457
219 
9801615952462170929952045821890821396710912031226648911944763748433755762721
84659731111074611842070880901520914942190933192494519220552874378329955576805823
31414743390903536886563255691872388147909719862350046890442775021598737436921851
63667389559331434975763359338546594933739820041944796738755156455137466548509022
81132259357635256208866903996846334447225950922770720253259910043958607024082197
52645173803192242484411714890463677525984144715600817377579530580772435014764054
8359202458575823525352138166782389241


I have found a glitch with the prime prover isprime and the ispseudoprime in 
pari ver 2.2.6  on my
pentium III 1000 mhz 256 meg ram system.
(01:06) gp > isprime(220^219 + 219^218)
%1 = 0
and
(01:06) gp > ispseudoprime(220^219 + 219^218)
%1 = 1

On the p4 2519 mhz 2 gig ram system  we have
(02:01) gp > isprime(220^219 + 219^218)
%6 = 1
a proven prime according to Pari 2.2.6.

This problem also occurred on 3^x - 2^x for x-2381 which is a proven prime 
on the pentium 4
2519 mhz 2 gig ram system. Also on x=2833 and 3613 the p3 gave bad results 
which I did not
prove primality on the p4.

I am running GIMPS in the background on both systems.

Maybe if some other fan has a pIII they could check this. It may be just my 
system or it may be
Pari 2.2.6.

Cino Hilliard

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