Smallest Prime Number whose Period is of Maximum Length

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): $7$