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4. THE GROUP OF UNITS MOD n, Un (the final battleground)
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Un is the group of integers between 1 and n-1 that have
no factors in common with n. It is a group under
multiplication mod n. E.g. U7 = {1, 2, 3, 4, 5, 6} and
2 x 5 = 10 = 3 (mod 7).
When we do long division to find the decimal for 1/7
(say), we are actually finding the powers of 10 in U7.
.1428571...
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7|1.0000000
7
30 10 = 3 (mod 7)
28
20 100 = 30 = 2 (mod 7)
14
60 1000 = 300 = 20 = 6 (mod 7)
56
40 10,000 = ... = 60 = 4 (mod 7)
35
50 10^5 = ... = 40 = 5 (mod 7)
49
10 10^6 = ... = 50 = 1 (mod 7)
That is, the remainder at every stage of the long division,
3, 2, 6, 4, 5, 1, is the same as the successive powers of 10
mod 7. AND THE LONG DIVISION PROCESS IS JUST REDUCTION OF POWERS
OF 10 MOD 7 (or mod n in general).
So per(1/n) = the order of 10 in Un and we may use all
the power of group theory to attack the periods of
decimals investigation. (The order of an element in a finite group
is the smallest power which gives 1.)
OTHER BASES
Note that "decimals" in other bases are fairly difficult. For
example, the "decimal" in base two for 1/7 involves writing 7 in
base 2 as (111), and doing long division in base 2.
.001...
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111 |1.000...
111
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1
Since we get the remainder of 1 here everything will repeat. That is,
in base 2 the decimal for 1/7 = 1/(111) = .001001001... (base 2).
However, finding the order of 2 in U7 is easy. 2^1 = 2, 2^2 = 4, and
2^3 = 8 = 1 (mod 7). So we see "2 has order 3 in U7" exactly corresponds
to "1/7 has period 3 as a binary decimal".
EVEN MORE QUESTIONS
Find the order of 10 in a bunch of Un's to check it is the
same as per(1/n): pick n's with no 2's or 5's
Explore decimals in other bases. Find a prime, p, which has
period p-1 as a base 2 decimal. Check that the repeating
block is indeed a base 2 carousel number
Form a list of carousel primes and try to see what they
have in common. Predict a new carousel prime and test it.
(This is likely hard. Does that put you off, or inspire you?)