Re: Fractions: 1/7 and others like it

msb@xxxxxxx (Mark Brader) wrote:

Jeff Kenton:
The fractions 1/7, 2/7 ... 6/7 all are repeating decimals where the same
six digits repeat in the same order. Find all sets of fractions less
than 100 with the same property (i.e., such that all n - 1 proper
fractions with denominator n are composed of the same n - 1 digits
repeating in the same order).

Enjoy.

Spoiler:

40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2

It's all the ones where 1/n has a repeating decimal of period n-1, namely
n = 7, 17, 19, 23, 29, 47, 59, 61, 97. However, this list of numbers was
found by experiment; I don't know a way to predict the period from n.

Albert H. Beiler covers this in the chapter 'Cycling towards Infinity' in his
1964 book 'Recreations in the theory of numbers: The Queen of Mathematics
entertains'.

Paraphrasing:
Consider k and n, where n is prime and k is the smallest integer satisfying
'10^k is congruent to 1 mod n'.

Fermat's theorem tells us that 10^(n-1) is always congruent to 1 mod n.

Those primes for which a smaller k exists than n-1 are the ones which have a
shorter decimal period.

Pre-computers, William Shanks computed k for all primes less than 120000, and in
1873 the 'Proceedings of the Royal Society of London' printed his results up to
the prime 29989. The Archives of the Society have the remainder in manuscript
form.

So for many years people have been interested in this, and apparently nobody has
come up with a shortcut for recognising numbers for which k is equal to n-1,
beyond checking this congruence for all the potential values of k (being the
divisors of n-1), eg for n=97 you need to check the remainders of 10^2, 10^3,
10^4, 10^6, 10^8, 10^12, 10^16, 10^24, 10^32 and 10^48 when divided by 97.

Beiler also notes that where k is even, you can divide the digits into two parts
in which the corresponding pairs always sum to 9, eg for n=7 we get
142 857 and we can pair up 1st and 4th, 2nd and 5th, 3rd and 6th digits to
always get a sum of 9.

If n=17 you get 05882352 94117647 which all pair up etc.

For those primes which have k=n-1, 10 is said to be a primitive root of n.

If you're still reading, you should get this book!
Dover Publications, Inc, SBN 486-21096-0

--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@xxxxxxxxxxx)
.

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