000RM.dvi

18.5 The period length of a prime 523

18.5 The period length of a prime ...........

The period length of a prime numberpmeans the length of the shortest
repeating block of digits in the decimal expansion of^1 p.

Suppose^1 phas period lengthλ:

1

p

=0.a 1 ···aλ.

Then moving the decimal placesnplaces to the right, we have

10 λ

p

=a 1 ···aλ.a 1 ···aλ,

and
10 λ
p

−

1

p

=a 1 ···aλ

is an integer. This means thatpdivides 10 λ− 1. Clearly,pcannot be 2
or 5. It is known that ifp=2, 5 , then 10 p−^1 − 1 is divisible byp, and
any numberλfor which 10 λ− 1 is divisible bypdividesp− 1.

Theorem 18.1.Ifp=2, 5 , the period length ofpis thesmallestdivisor
λofp− 1 such thatpdivides 10 λ− 1.

Theorem 18.2.Letp> 5 be a prime. The period length ofpis the
smallest divisornofp− 1 such thatpdividesRn.

Proof.Note that 10 n−1=9n =9×Rn.Ifp=3, thenpdivides
Rn.

We say thatpis a primitive prime divisor the repunitn. Thus, for
a primep> 5 , the period length ofpis the numbernfor whichpis
primitive prime divisor. A table of primitive prime divisors of repunits
is given in an Appendix.