Long Period Prime/Examples/19
Jump to navigation
Jump to search
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): $052,631,578,947,368,421$