Loading ...
Sorry, an error occurred while loading the content.

Base 2 Prime Reciprocal Proof?

Expand Messages
  • Jeff Huth
    I have been studying prime reciprocals in binary, but I am still fairly novice to number theory. I would like to understand if there is a proof for the
    Message 1 of 2 , Jan 1, 2008
    • 0 Attachment
      I have been studying prime reciprocals in binary, but I am still fairly novice to number theory.
      I would like to understand if there is a proof for the following statement. A proof probably exists for Base 2, or generalized for other bases. If so, would someone kindly refer me to a proof, so that I may understand it better.
      "In binary notation (Base 2), the reciprocal of a prime number (1/p, where p>2) will repeat with a period length of L=(p-1)/k, where k = is a nonnegative integer (k>=0)."

      For example:
      p=3, 1/3 = 0.01 01 01 01... (k=1)
      p=5, 1/5 = 0.0011 0011 0011... (k=1)
      p=7, 1/7 = 0.001 001 001... (k=2)
      p=11, 1/11 = 0.0001011101 0001011101 0001011101... (k=1)
      p=13, 1/13 = 0.000100111011 000100111011 000100111011... (k=1)
      p=17, 1/17 = 0.00001111 00001111 00001111... (k=2)
      p=23, 1/23 = 0.00001011001 00001011001 00001011001... (k=2)
      p=29, 1/29 = 0.0000100011010011110111001011 0000100011010011110111001011... (k=1)
      p=31, 1/31 = 0.00001 00001 00001... (k=6)

      There are many interesting patterns when you example primes from this angle. I will present more in another message. I would like to learn more about what others have discovered.


      ____________________________________________________________________________________
      Never miss a thing. Make Yahoo your home page.
      http://www.yahoo.com/r/hs

      [Non-text portions of this message have been removed]
    • Phil Carmody
      ... Information on /unique period primes/ should help you. Think about the process of performing the division by hand, and compare that to the concept of the
      Message 2 of 2 , Jan 1, 2008
      • 0 Attachment
        --- Jeff Huth <jeff_huth@...> wrote:
        > I have been studying prime reciprocals in binary, but I am still fairly
        > novice to number theory.
        > I would like to understand if there is a proof for the following statement.
        > A proof probably exists for Base 2, or generalized for other bases. If so,
        > would someone kindly refer me to a proof, so that I may understand it better.
        >
        > "In binary notation (Base 2), the reciprocal of a prime number (1/p, where
        > p>2) will repeat with a period length of L=(p-1)/k, where k = is a
        > nonnegative integer (k>=0)."
        >
        > For example:
        > p=3, 1/3 = 0.01 01 01 01... (k=1)
        > p=5, 1/5 = 0.0011 0011 0011... (k=1)
        > p=7, 1/7 = 0.001 001 001... (k=2)
        > p=11, 1/11 = 0.0001011101 0001011101 0001011101... (k=1)
        > p=13, 1/13 = 0.000100111011 000100111011 000100111011... (k=1)
        > p=17, 1/17 = 0.00001111 00001111 00001111... (k=2)
        > p=23, 1/23 = 0.00001011001 00001011001 00001011001... (k=2)
        > p=29, 1/29 = 0.0000100011010011110111001011 0000100011010011110111001011...
        > (k=1)
        > p=31, 1/31 = 0.00001 00001 00001... (k=6)
        >
        > There are many interesting patterns when you example primes from this angle.
        > I will present more in another message. I would like to learn more about
        > what others have discovered.

        Information on /unique period primes/ should help you.
        Think about the process of performing the division by hand, and compare that to
        the concept of the multiplicative order of 2 modulo p. (And in decimal, the
        order of 10 mod p.)

        Phil


        () ASCII ribbon campaign () Hopeless ribbon campaign
        /\ against HTML mail /\ against gratuitous bloodshed

        [stolen with permission from Daniel B. Cristofani]


        ____________________________________________________________________________________
        Be a better friend, newshound, and
        know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ
      Your message has been successfully submitted and would be delivered to recipients shortly.