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n^n+ (n+1)^(n+1)

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  • Mark Underwood
    I ve started looking at at n^a + (n+1)^b, where a,b
    Message 1 of 48 , Mar 1, 2005
      I've started looking at at n^a + (n+1)^b, where a,b <= n+1 .

      It generates a reasonable number of primes. I'll be examining for any
      patterns in prime generation and will let you know of any earth
      shattering findings.

      But one thing right off that bat which I find somewhat intriguing is
      this special case:

      n^n + (n+1)^(n+1)

      Examples:

      1^1 + 2^2 = 5 (prime)
      2^2 + 3^3 = 31 (prime)
      3^3 + 4^4 = 283 (prime)

      Seems like it could produce a good amount of primes at this rate. But
      in what appears to be an odds defying feat, no more primes turn up
      for any n up to 500. Perhaps there are no more, for *some* reason.


      Mark
    • djbroadhurst
      ... This is now a conjecture-free OEIS sequence: http://www.research.att.com/~njas/sequences/A165284 David
      Message 48 of 48 , Sep 14, 2009
        --- In primenumbers@yahoogroups.com,
        "djbroadhurst" <d.broadhurst@> wrote:

        > primes p in
        > http://www.research.att.com/~njas/sequences/A068209
        > for which (x+1)^p-x^p-1 and x^x+(x+1)^(x+1)
        > are never simultaneously divisible by p^2.

        This is now a conjecture-free OEIS sequence:
        http://www.research.att.com/~njas/sequences/A165284

        David
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