31 is Smallest Prime whose Reciprocal has Odd Period
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Theorem
$31$ is the smallest prime number to have a decimal expansion of the reciprocal with an odd period greater than $1$:
- $\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$
This sequence is A021035 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
From Reciprocal of $31$:
- $\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$
Counting the digits, it is seen that this has a period of recurrence of $15$, an odd integer.
The prime numbers less than $31$ are $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$.
We investigate the reciprocal of each of these:
\(\ds \dfrac 1 2\) | \(=\) | \(\ds 0 \cdotp 5\) | Reciprocal of $2$: not recurring | |||||||||||
\(\ds \dfrac 1 3\) | \(=\) | \(\ds 0 \cdotp \dot 3\) | Reciprocal of $3$: period $1$ | |||||||||||
\(\ds \dfrac 1 5\) | \(=\) | \(\ds 0 \cdotp 2\) | Reciprocal of $5$: not recurring | |||||||||||
\(\ds \dfrac 1 7\) | \(=\) | \(\ds 0 \cdotp \dot 14285 \dot 7\) | Reciprocal of $7$: period $6$ | |||||||||||
\(\ds \dfrac 1 {11}\) | \(=\) | \(\ds 0 \cdotp \dot 0 \dot 9\) | Reciprocal of $11$: period $2$ | |||||||||||
\(\ds \dfrac 1 {13}\) | \(=\) | \(\ds 0 \cdotp \dot 07692 \dot 3\) | Reciprocal of $13$: period $6$ | |||||||||||
\(\ds \dfrac 1 {17}\) | \(=\) | \(\ds 0 \cdotp \dot 05882 \, 35294 \, 11764 \dot 7\) | Reciprocal of $17$: period $16$ | |||||||||||
\(\ds \dfrac 1 {19}\) | \(=\) | \(\ds 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1\) | Reciprocal of $19$: period $18$ | |||||||||||
\(\ds \dfrac 1 {23}\) | \(=\) | \(\ds 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3\) | Reciprocal of $23$: period $22$ | |||||||||||
\(\ds \dfrac 1 {29}\) | \(=\) | \(\ds 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1\) | Reciprocal of $29$: period $28$ |
and it is seen that none has an odd period greater than $1$.
Hence the result.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $31$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $31$