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20003Re: [PrimeNumbers] primes of the form (x+1)^p-x^p

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  • Phil Carmody
    Apr 5, 2009
      --- On Mon, 4/6/09, 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.
      > Are there heuristics in favour of conjecturing finiteness
      > of such primes?
      > Thanks for any hints,

      They are cyclotomic, so have the same kind of rules surrounding admissible factors as Mersennes. (Which are actually quite non-trivial, and the most advanced analysis can not be considered much more than dumb application of the simplest possible rules due to having no reason to think otherwise.) They grow faster, so there's a log ratio between them, but that's just a constant factor.

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