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Oldie but a goodie...

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  • Jack Brennen
    Quick little prime chestnut for you: Find all primes of the form 4^n+n^4, where n is a positive integer.
    Message 1 of 3 , Jul 1 11:56 AM
      Quick little prime chestnut for you:

      Find all primes of the form 4^n+n^4, where n is a positive integer.
    • David Broadhurst
      ... Let f(n) = 4^n + n^4. If n is even, then so is f(n). If n is odd, then f(n) = (A(n) + B(n))*(A(n) - B(n)) with A(n) = 2^n + n^2 B(n) = 2^((n+1)/2)*n Hence
      Message 2 of 3 , Jul 1 1:57 PM
        --- In primenumbers@yahoogroups.com,
        Jack Brennen <jfb@...> wrote:

        > Find all primes of the form 4^n+n^4,
        > where n is a positive integer.

        Let f(n) = 4^n + n^4.
        If n is even, then so is f(n).
        If n is odd, then
        f(n) = (A(n) + B(n))*(A(n) - B(n))
        with
        A(n) = 2^n + n^2
        B(n) = 2^((n+1)/2)*n

        Hence only f(1) = 5 is prime.

        Not also that 5|f(n) if n is odd and coprime to 5.

        David (per proxy Léon François Antoine)
      • Phil Carmody
        ... Ah - memories of http://www.leyland.vispa.com/numth/primes/xyyx.htm Phil
        Message 3 of 3 , Jul 4 10:31 AM
          --- On Wed, 7/1/09, Jack Brennen <jfb@...> wrote:
          > Quick little prime chestnut for you:
          >
          > Find all primes of the form 4^n+n^4, where n is a positive integer.

          Ah - memories of http://www.leyland.vispa.com/numth/primes/xyyx.htm

          Phil
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