Smallest Prime Number whose Period is of Maximum Length
Jump to navigation
Jump to search
Theorem
$7$ is the smallest prime number the period of whose reciprocal, when expressed in decimal notation, is maximum:
- $\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$
Proof
From Maximum Period of Reciprocal of Prime, the maximum period of $\dfrac 1 p$ is $p - 1$.
- $\dfrac 1 2 = 0 \cdotp 5$: not recurring.
- $\dfrac 1 3 = 0 \cdotp \dot 3$: recurring with period $1$.
- $\dfrac 1 5 = 0 \cdotp 2$: not recurring.
- $\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$: recurring with period $6$.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $7$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $7$