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20009Re: primes of the form (x+1)^p-x^p

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  • Mike Oakes
    Apr 6, 2009
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      --- 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.

      My own record (and favourite PRP of all, being so "Mersenne-like") is
      3^336353-2^336353
      at 160482 digits.

      -Mike Oakes
      [PS I replied on this website about 8 hours ago, but the message seems to have vanished into the ether.]
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