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Re: factors of n^p- (n-1)^p

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  • grostoon
    In fact, one can extend this forms of numbers to guassian integers as well as quatratic integers. That is, numbers of the form ... integers or |z|=|a +
    Message 1 of 8 , Nov 21, 2005
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      In fact, one can extend this forms of numbers to guassian integers as
      well as quatratic integers. That is, numbers of the form
      |u^p - v^p| with |u - v| = 1 and |z|=|a + b*i|=a^2 + b^2 for guassian
      integers or |z|=|a + b*sqrt(N)|=|a^2 - N*b^2| for quadratic integers
      (for some integer N that is not a perfect square of course).

      Examples:

      A/ |(1 + i)^p - 1| which are called Guassian Mersenne Norms. See
      http://primes.utm.edu/top20/page.php?id=41,

      B/ |(2 + i)^p - (1 + i)^p| which is prime for p=2, 3, 5, 7, 11, 17,
      19, 23, ...

      C/ |(3 + sqrt(3))^p - 1| = 6^p + 1 - lucasV(6, 6, p) which is prime
      for p=2, 3, 11, ..., 151537, 170371 (last 2 exponents are Mike's
      records),

      D/ |(5 + sqrt(3))^p - 3^p| which is prime for p=2, 7, 13, ...


      J-L.


      --- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...>
      wrote:
      >
      >
      >
      > grostoon <grostoon@y...> wrote: Hi Robert,
      >
      > --- In primenumbers@yahoogroups.com, "Robert" <rw.smith@b...> wrote:
      > >
      > > Factors of n^p-(n-1)^p are all of the form F=2kP+1, where k is an
      > > integer and P is a prime.
      > >
      >
      > Yes, and if p is a Sophie Germain prime such that p=3 mod 4 and
      > (2p+1) divides n-2 then 2p+1 divides n^p-(n-1)^p, which generalizes
      > some well known properties of mersenne's numbers factors.
      >
      > BAck in 1995 when I was trying to find a "super formula" to
      generate primes, I looked at these numbers together with Carol and
      Kynea Numbers. I called these numbers Mersenne Class Numbers. I
      think that the properties are more or less the same as Mersenne.
      I've gone out on a limb to say that people searching for Mersenne
      back in the days were aware of the Mersenne Class Numbers, but
      thought that the smallest one (Mersenne Numbers) were more
      interesting. I do hope that someday we can develop a Mersenne-like
      test will be available for these numbers so that we can prove them
      prime.
      >
      > Although I spent more time on these Classes of numbers (in fact,
      my mathematics Senior Seminar was 80% about these numbers). I chose
      to move on with Carol and Kynea Numbers simply because they were
      yielded more primes in a given range than these classes of numbers.
      You can see how all of these numbers resembles Mersenne Numbers:
      > Carol Number ---> (2^p-1)^2-2; where p = 2,3,4,...
      > Kynea Number ---> (2^p+1)^2-2; where p = 0,1,2,3,4,...
      > Mersenne Number-> (2^p-1); where p is prime
      > Mersenne Class--> n^p-(n-1)^p; where p is prime
      >
      > You should take a look at
      > http://www.primenumbers.net/prptop/prptop.php for a lot of PRP of
      > this form. My current records are 9^170099-8^170099 (162316 digits)
      > and 8^173687-7^173687 (156855 digits) and Mike's record is
      currently
      > 3^256199-2^256199 (122238 digits).
      >
      > Impressive records!!!!!
      >
      > Recently, I found the smallest exponents such that the two "hard
      > cases" 48^p-47^p and 61^p-60^p are PRP. They are 58543 and 54517.
      >
      > This is interesting!!! I knew that 48 was tough, but I didn't
      realize that it would go this far.
      > Now, I'm working on the next hard case 134^p-133^p which I expect
      > will give one of the biggest PRP.
      >
      > Good luck on finding a PRP with that n value. It would be good
      to see a record PRP in this form...
      >
      > Regards,
      >
      > Jean-Louis.
      >
      >
      >
      >
      >
      >
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