Long Period Prime/Examples/19

Theorem

The prime number $19$ is a long period prime:

$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$

This sequence is A021023 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

From Reciprocal of $19$:

$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$

Counting the digits, it is seen that this has a period of recurrence of $18$.

Hence the result.

$\blacksquare$


Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $19$
  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $052,631,578,947,368,421$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $19$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $052,631,578,947,368,421$
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