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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).