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Prime factors of (c^n-b^n)/(c-b)

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  • someoneltd
    Hi, All! Is the next statement true? Let n is prime, b and c are integer, c b 0, GCD(c,b)=1,GCD(c-b,n)=1. Then all prime factors of the number (c^n-b^n)/(c-b)
    Message 1 of 3 , Jan 22, 2006
      Hi, All!

      Is the next statement true?

      Let n is prime, b and c are integer, c>b>0, GCD(c,b)=1,GCD(c-b,n)=1.
      Then
      all prime factors of the number (c^n-b^n)/(c-b) have the form 2kn+1,
      where
      k>0 is integer.

      Where can I find it?

      Thanks.
      --
      Best regards. Someone.

      http://www.someoneltd.boom.ru/ http://home.tula.net/frazez/
    • Kermit Rose
      From: someoneltd Date: 01/22/06 11:41:50 To: primenumbers@yahoogroups.com Subject: [PrimeNumbers] Prime factors of (c^n-b^n)/(c-b) Hi, All! Is the next
      Message 2 of 3 , Jan 22, 2006
        From: someoneltd
        Date: 01/22/06 11:41:50
        To: primenumbers@yahoogroups.com
        Subject: [PrimeNumbers] Prime factors of (c^n-b^n)/(c-b)

        Hi, All!

        Is the next statement true?

        Let n is prime, b and c are integer, c>b>0, GCD(c,b)=1,GCD(c-b,n)=1.
        Then
        all prime factors of the number (c^n-b^n)/(c-b) have the form 2kn+1,
        where
        k>0 is integer.

        Where can I find it?

        Thanks.


        Kermit says:


        p divides (c^n - b^n)/(c-b)

        p divides (c^n - b^n)


        C^n - b^n = 0 mod p

        c^n = b^n mod p

        (c/b)^n = 1 mod p

        n divides (p-1)

        kn = p-1

        kn+1 = p

        p = kn + 1

        [Non-text portions of this message have been removed]
      • mikeoakes2@aol.com
        ... You statement as written is false. Counterexample: c = 4, b = 3, n = 2. For n = 2, you can t have the 2 in 2kn+1 . For n 2 (and so odd) your statement
        Message 3 of 3 , Jan 23, 2006
          In an email dated Sun, 22 1 2006 4:38:44 pm GMT, "someoneltd" <umax@...> writes:

          >Is the next statement true?
          >
          >Let n is prime, b and c are integer, c>b>0, GCD(c,b)=1,GCD(c-b,n)=1.
          >Then
          >all prime factors of the number (c^n-b^n)/(c-b) have the form 2kn+1,
          >where
          >k>0 is integer.


          You statement as written is false.
          Counterexample: c = 4, b = 3, n = 2.

          For n = 2, you can't have the "2" in "2kn+1".
          For n > 2 (and so odd) your statement is true.

          -Mike Oakes
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