Long Period Prime/Examples/23

Theorem

The prime number $23$ is a long period prime:

$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$

This sequence is A021027 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

From Reciprocal of $23$:

$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$

Counting the digits, it is seen that this has a period of recurrence of $22$.

Hence the result.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $23$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0,434,782,608,695,652,173,913$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $23$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0,434,782,608,695,652,173,913$