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

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

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

        http://tinyurl.com/d3nf9w

        David
      • Maximilian Hasler
        ... (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
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          --- In primenumbers@yahoogroups.com, "David Broadhurst" wrote:
          > --- 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"... [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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