Long Period Prime/Examples/61

Theorem

The prime number $61$ is a long period prime:

$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$

It also contains an equal number ($6$) of each of the digits from $0$ to $9$.

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Proof

From Reciprocal of $61$:

$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$

Counting the digits, it is seen that this has a period of recurrence of $60$.

Inspecting the expansion and counting the digits, we find that each one appears exactly $6$ times.

Hence the result.

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $61$