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Re: [PrimeNumbers] Base 2 Prime Reciprocal Proof?

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  • 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 1 of 2 , Jan 1, 2008
      --- 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


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