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Carousel Numbers:
A Lead-in to Number Theory

Gary Klatt

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5.     THE BIG CONNECTING THEOREM
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THEOREM
Assume m/n is a reduced fraction and n' is the result of
dividing all 2's and 5's out of n.

Then the following are equivalent:
  i)   The decimal for m/n eventually repeats every k places

 ii)   The decimal for 1/n' repeats every k places (with no delay)

iii)   1/n' can be written as B/99...9 (k 9's)

 iv)   n' divides 99...9 (k 9's)

  v)   10^k = 1 mod n'  That is, 10^k = 1 in Un'

COROLLARY
Thus the period of m/n (reduced) =
       per(1/n) =
       per(1/n') =
       the length of the smallest Nine number n' divides =
       the order of 10 in Un'

APPLICATION TO CAROUSEL NUMBERS
If p is a prime with 1/p having largest possible period p-1,
then B = 99...9(p-1 9'S)/p is a carousel number.
 
These carousel primes (primes where per(1/p)=p-1) are characterized by
  -    the first Nines number that p divides has length p-1
  -    10 is a generator of the cyclic group Up

A better characterization is needed - we would like an
easily checked property of p to make it a carousel prime.

YET EVEN MORE PROBLEMS
        Prove these results

        Show that every carousel number arises in this way (as a
        Nines number over a carousel prime).

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