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 [076923] (length=6)

numerator = 76923

denominator = 999999 (6 digits)

computing reduction,

76923/999999 = 1/13

is the answer.

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 10^{e}≡1 (mod p)

(i.e. the order of 10 under mod p.j

In case of 13,

10^{1}≡10 (mod 13)

10^{2}≡ 9 (mod 13)

10^{3}≡12 (mod 13)

10^{4}≡ 3 (mod 13)

10^{5}≡ 4 (mod 13)

10^{6}≡ 1 (mod 13)

so, e=6. At the above example,

111111 = (10^{6}-1)/9 = (10^{e}-1)/9

so, in general,

p | (10^{e}-1)/9

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

e | p-1

so,

p | (10^{p-1}-1)/9

This is equal to Fermat's theorem.

The number 11...11=(10^{n}-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,

(b^{n}-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.jpHisanori Mishima