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20029Re: primes of the form (x+1)^p-x^p

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  • Mark Underwood
    Apr 9, 2009
      --- In primenumbers@yahoogroups.com, "Maximilian Hasler" <maximilian.hasler@...> wrote:
      >
      > > Format: (n, number of m less than n such that 2^m+3^n or 2^n+3^m is prime)
      > >
      > > (1,1) (2,2) (3,2) (4,3) (5,3) (6,3) (9,3) (11,3) (25,2)
      > > (33,3) (34,3) (54,1) (69,2) (70,3) (97,3) (103,3) (115,3)
      > > (...)
      > > (951,3) (958,3) (966,1) (970,3)
      >
      > Very cool. To notch it up on the coolness factor one could submit this sequence to OEIS.
      >
      > Modulo correcting "less than" to "not exceeding" (or specifying that m may be zero, or correcting a(1)=0 and others in the above).
      >
      > Actually there are at least 6 sequences:
      > MU = { n | MU1(n)<=3 } (or why not <= 2 or even <= 1 ?)
      > MU1(n) = # { m <= n | 2^n+3^m or 2^m+3^n is prime } (or "<" ?)
      > MU2(n) = min { m | 2^n+3^m or 2^m+3^n is prime }
      > MU3(n) = max { m <= n | 2^n+3^m or 2^m+3^n is prime } (or "<" ?)
      > MU4(n) = min { m | 2^n+3^m is prime }
      > MU5(n) = min { m | 2^m+3^n is prime }
      >
      >
      > I checked that they are not in OEIS except for MU5 :
      >
      > A123359 Least m such that 3^n+2^m is prime.
      >
      > But maybe better double-check...
      >
      > Maximilian
      >

      Thank you Maximilian,

      The OESIS thing would only be appealing to me if the minimum m that caused

      2^n+3^m or 2^m+3^n to be prime was always found to be less than n for all n.

      Then I might consider adding the sequence

      1,54,129,135,170,175,185,274,280,335,460,489,547,611, 614,...

      These are the n such that only one m less than n makes 2^n+3^m or 2^m+3^n prime.

      So far, so good. Up to n=1420 there is at least one m < n that that makes a prime.

      Mark
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