Long Period Prime/Examples/7
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Theorem
$7$ is the smallest long period prime:
- $\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$
This sequence is A020806 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
From Reciprocal of $7$:
- $\dfrac 1 7 = 0 \cdotp \dot 14285 \, \dot 7$
The reciprocals of $1$, $2$, $4$ and $5$ do not recur:
\(\ds \frac 1 1\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \frac 1 2\) | \(=\) | \(\ds 0 \cdotp 5\) | ||||||||||||
\(\ds \frac 1 4\) | \(=\) | \(\ds 0 \cdotp 25\) | ||||||||||||
\(\ds \frac 1 5\) | \(=\) | \(\ds 0 \cdotp 2\) |
while those of $3$ and $6$ do recur, but with the non-maximum period of $1$:
\(\ds \frac 1 3\) | \(=\) | \(\ds 0 \cdotp \dot 3\) | ||||||||||||
\(\ds \frac 1 6\) | \(=\) | \(\ds 0 \cdotp 1 \dot 6\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $142,857$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $142,857$