## 3. How to find the fraction from the repeating sequence

### How to find the fraction from the repeating sequence

It is very easy.

1/9   = 0.1111111 ...
1/99  = 0.0101010 ...
1/999 = 0.0010010 ...
...

so,

• let numerator be the repeating sequence
• let denominator be consecutive "9" which length is the length of repeating sequence
• reduction

For example, in case  (length=6)

numerator   = 76923
denominator = 999999 (6 digits)

computing reduction,

76923/999999 = 1/13

Now, multiply both sides of equation by the denominator, then,

76923 * 13 = 999999 = 9 * 111111

so, we find a non-trivial factor 13 except trivial factors, 3, 11 and 111.

The length e of repeating sequence of 1/p is given by the following formula,

e is the smallest number where 10e≡1  (mod p)
(i.e. the order of 10 under mod p.Åj

In case of 13,

101≡10 (mod 13)
102≡ 9 (mod 13)
103≡12 (mod 13)
104≡ 3 (mod 13)
105≡ 4 (mod 13)
106≡ 1 (mod 13)

so, e=6. At the above example,

111111 = (106-1)/9 = (10e-1)/9

so, in general,

p | (10e-1)/9

and because the length of repeating sequence is at most p-1,

e | p-1

so,

p | (10p-1-1)/9

This is equal to Fermat's theorem.

### Repunit

The number 11...11=(10n-1)/9  (n times iteration of "1") is called as repunit, denoting by Rp. For example, 111111 = R6 (above case).
Especially, when the length is prime, it does not have trivial factors, so,

• is it prime or not
• factorization of composite case

are the main concerns.
Harvey Dubner researched (2007) primes up to 200000 and found when

p = 2, 19, 23, 317, 1031, 49081, 86453, 109297

then repunits are also prime.
(last three cases, p=49081, 86453, 109297, then Rp are probable prime.)
And Maksym Voznyy found that R270343 is a probable prime (July 15, 2007).

More generally,

(bn-1)/(b-1)

are also under reseach. Which is called as general repunit, base-b repunit.

"10,n-" in The Cunningham Project is the current results of factorization of repunit.
The following are under computation. On March 03, 2008, there are "First Five Holes".

10,241- c229
10,257- c241
10,263- c216
10,269- c233
10,271- c214

so, all the repunits under 239 were completely factorized.

For factors of repunits Rp where p is prime under 100 are here.

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E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima