# Long Period Prime/Examples/61

Jump to navigation
Jump to search

## 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 $9$.

This page has been identified as a candidate for refactoring.In particular: Extract that second result into a separate pageUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

## 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): $61$