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Prime division & repetition period

  1. Jun 1, 2010 #1
    Something odd i noticed while playing around with primes.

    We have the set of prime numbers P and a p ∈ P.
    Define a function f:Q → N that will give the period of the repetition in the decimal expansion of some number r ∈ Q.

    1) ∀ p ∈ P: ∃ n ∈ N: ∀ q ∈ P, q < p: f(q/p) = n.
    So n is independant of q.

    So define a function g:N → N: ∀ p ∈ P, ∀ q ∈ P, q < p: g(p) = f(q/p).

    2) ∀ p ∈ P: ((p - 1) / g(p)) ∈ N.

    You'll always get a unit fraction 1/2, 1/3, 1/4... never something like 5/7.
    I was wondering why?
     
  2. jcsd
  3. Jun 2, 2010 #2
    Because the number system we use is base 10 :eek:)
     
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