## Re: primes of the form (x+1)^p-x^p

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• ... I have today uploaded a file containing the results of my search for all primes of this form for 2
Message 1 of 22 , May 7, 2009
--- In primenumbers@yahoogroups.com, "Mike Oakes" <mikeoakes2@...> wrote:
>
> --- 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.

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.

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

-Mike Oakes
• ... Here is a simple link to Mike s interesting table: http://tinyurl.com/d3nf9w David
Message 2 of 22 , May 7, 2009
"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.

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

http://tinyurl.com/d3nf9w

David
• ... (Thanks, Mike !) ... Thanks, David! (Remark: These tiny urls are nice, but can be quite annoying when they point to a website that re-arranged its
Message 3 of 22 , May 7, 2009
> --- In primenumbers@yahoogroups.com, "Mike Oakes" 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.

(Thanks, Mike !)

> Here is a simple link to Mike's interesting table:
> http://tinyurl.com/d3nf9w

Thanks, David!
(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)
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