## Base 2 Prime Reciprocal Proof?

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• 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
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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.

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• ... 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
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--- 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.
> what others have discovered.

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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