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RE: [PrimeNumbers] no primes with this forms

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  • Paul Leyland
    ... I would be interested in a proof or disproof of this conjecture. Several years ago I started examining primes of the form x^y + y^x, of which yours is
    Message 1 of 4 , Feb 17, 2001
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      > you tell me two state but by PrimeForm for the first
      > 1000Digits you can find there are an infinite Prime f the form 2^n+n^2
      >
      > can you proof There is an infinite primes with n=3*k

      I would be interested in a proof or disproof of this conjecture. Several
      years ago I started examining primes of the form x^y + y^x, of which yours
      is just a special case. There are clearly an infinite number of such primes
      (take x=1, y=p-1, p prime to see this) but the question of whether there are
      an infinite number with x>y>1 is still open as far as I know.

      It's easy to put some constraints on x and y, for instance they have to be
      of opposite parity and co-prime, but I haven't yet found any way of
      determining the number of primes. I believe there are an infinite number of
      them but it may be, for example, that for any finite value of y there may be
      only a finite number of prime values x^y+y^x with x>y>1

      Two larger examples of primes of this form have appeared in Chris Caldwell's
      tables. The largest of these is 364^405+405^364 with 1038 digits. I have a
      table of other discoveries, but not easily available at the moment, alas.
      If there's general interest I'll dig out the information.


      Paul
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