- --- In primenumbers@yahoogroups.com, "Mike Oakes" <mikeoakes2@...> wrote:
>

I have today uploaded a file containing the results of my search for all primes of this form for 2<=b<=1000, 2<=p<10000, done in the years 2000-2008.

> --- In primenumbers@yahoogroups.com, "j_chrtn" <j_chrtn@> wrote:

> >

> > --- In primenumbers@yahoogroups.com, "Maximilian Hasler" <maximilian.hasler@> wrote:

> > >

> > > Dear prime number fans,

> > > is there anything available about possible finiteness of primes of the form (x+1)^p-x^p ?

> > > Specifically, some curios reasons led me to look at 7^p-6^p.

> > > It seems that 1399 and 2027 are the largest known p for which this is prime (Sloane's A062573). According to my calculations, the next such p must be larger than 17900.

> > > Also, 2027 is (so far) the only such p of the form n^2+2, n>1.

> > >

> >

> > Hi Maximilian,

> >

> > p=1399 and 2027 are not the current records for base 7. My personal

> > records are p=69371 and p=86689 for 7^p-6^p. And the largest PRP I have found of this form is currently 8^336419-7^336419.

> >

> > Take a look at Henry Lifchitz's PRP records page

> > www.primenumbers.net/prptop/prptop.php for much more primes/PRP of this form.

> >

> > I believe that (1) for any integer n >= 1, there are infinitely many

> > primes p such that (n+1)^p-n^p is prime and that (2) for any prime p,

> > there are infinitely many integers n such that (n+1)^p-n^p is prime as

> > well.

> > But, unfortunately, proving (or disproving) (1) and (2) is far from being trivial I'm afraid.

> >

> > And now, just for fun, a litle challenge for you: find a prime p such that 138^p-137^p is prime or PRP.

> > Good luck ;-)

> >

> > JL

>

> I have done quite a lot of work on this form, initially summarised in my May 2001 post to the NMBRTHRY list:

> http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0105&L=NMBRTHRY&P=R359&I=-3

>

> Since then, Jean-Louis in particular has devoted seemingly enormous numbers of cpu cycles to extending the list of known PRPs.

It is the file "b^p-(b-1)^p.txt", within the "Prime Tables" folder in the Files area of this site.

-Mike Oakes - --- In primenumbers@yahoogroups.com,

"Mike Oakes" <mikeoakes2@...> wrote:

> I have today uploaded a file containing the results of my search

Here is a simple link to Mike's interesting table:

> for all primes of this form for 2<=b<=1000, 2<=p<10000, done in

> the years 2000-2008.

http://tinyurl.com/d3nf9w

David - --- In primenumbers@yahoogroups.com, "David Broadhurst" wrote:
> --- In primenumbers@yahoogroups.com, "Mike Oakes" wrote:

(Thanks, Mike !)

> > I have today uploaded a file containing the results of my search

> > for all primes of this form for 2<=b<=1000, 2<=p<10000, done in

> > the years 2000-2008.

> Here is a simple link to Mike's interesting table:

Thanks, David!

> http://tinyurl.com/d3nf9w

(Remark: These tiny urls are nice, but can be quite annoying when they point to a website that re-arranged its directory structure - with tinyurls ultra-efficient and discreet forwarding system you sometimes can't get the slightest info about where/what you had been pointed to and try to find it "by hand"...Â [this happened to me a few days ago - but I forgot where & what is was about...])

OTOH:

p=2 : A006254 Numbers n such that 2n-1 is prime.

p=3 : A002504 numbers such that 1+3x(x-1) is (a "cuban") prime.

p=5 : A121617 Nexus numbers of order 5 (or A022521[n-1] = n^5 - (n-1)^5) are primes.

p=7 : A121619 Nexus numbers of order 7 (A022523[n-1] = n^7 - (n-1)^7) are primes

p >= 11 seems not yet there, so Mike could enter his numbers into OEIS, starting there !

Maximilian, per proxy SOSR

(Society for On-line Sequence Recovery)