# Anning's Theorem

## Theorem

In any base greater than $1$, the fraction:

- $\dfrac {101 \, 010 \, 101} {110 \, 010 \, 011}$

has the property that if the two $1$'s in the center of the numerator and the denominator are replaced by the same odd number of $1$'s, the value of the fraction remains the same.

For example:

- $\dfrac {101 \, 010 \, 101} {110 \, 010 \, 011} = \dfrac {1 \, 010 \, 111 \, 110 \, 101} {1 \, 100 \, 111 \, 110 \, 011} = \dfrac {9091} {9901}$ (in base $10$).

## Proof

Let $b$ be the base in question.

Let $F = \dfrac {101 \, 010 \, 101} {110 \, 010 \, 011}$.

Then:

- $F = \dfrac {1 + b^2 + b^4 + b^6 + b^8} {1 + b + b^4 + b^7 + b^8}$

It is necessary to prove that for all $k \in \Z_{>0}$:

- $F = \dfrac {1 + b^2 + b^4 + b^5 + \cdots + b^{2 k + 2} + b^{2 k + 4} + b^{2 k + 6} } {1 + b + b^4 + b^5 + \cdots + b^{2 k + 2} + b^{2 k + 5} + b^{2 k + 6} }$

This is done by:

- multiplying the numerator of one by the denominator of the other

and then:

- multiplying the denominator of one by the numerator of the other

and checking that they are equal.

Thus we proceed:

\(\displaystyle \) | \(\) | \(\displaystyle 1 + b^2 + b^4 + b^6 + b^8\) | |||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle \times \, \) | \(\displaystyle 1 + b + b^4 + b^5 + \cdots + b^{2 k + 2} + b^{2 k + 5} + b^{2 k + 6}\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 + b + b^4 + b^5 + \cdots + b^{2 k + 2} + b^{2 k + 5} + b^{2 k + 6}\) | |||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle b^2 + b^3 + b^6 + b^7 + \cdots + b^{2 k + 4} + b^{2 k + 7} + b^{2 k + 8}\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle b^4 + b^5 + b^8 + b^9 + \cdots + b^{2 k + 6} + b^{2 k + 9} + b^{2 k + 10}\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle b^6 + b^7 + b^{10} + b^{11} + \cdots + b^{2 k + 8} + b^{2 k + 11} + b^{2 k + 12}\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle b^8 + b^9 + b^{12} + b^{13} + \cdots + b^{2 k + 10} + b^{2 k + 13} + b^{2 k + 14}\) |

and:

\(\displaystyle \) | \(\) | \(\displaystyle 1 + b + b^4 + b^7 + b^8\) | |||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle \times \, \) | \(\displaystyle 1 + b^2 + b^4 + b^5 + \cdots + b^{2 k + 2} + b^{2 k + 4} + b^{2 k + 6}\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 + b^2 + b^4 + b^5 + \cdots + b^{2 k + 2} + b^{2 k + 4} + b^{2 k + 6}\) | |||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle b + b^3 + b^5 + b^6 + \cdots + b^{2 k + 3} + b^{2 k + 5} + b^{2 k + 7}\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle b^4 + b^6 + b^8 + b^9 + \cdots + b^{2 k + 6} + b^{2 k + 8} + b^{2 k + 10}\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle b^7 + b^9 + b^{11} + b^{12} + \cdots + b^{2 k + 9} + b^{2 k + 11} + b^{2 k + 13}\) | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle b^8 + b^{10} + b^{12} + b^{13} + \cdots + b^{2 k + 10} + b^{2 k + 12} + b^{2 k + 14}\) |

Equality can be demonstrated.

$\blacksquare$

## Source of Name

This entry was named for Norman Herbert Anning.

## Historical Note

**Anning's Theorem** appears as a problem in The $1970$ publication *250 Problems in Elementary Number Theory* by Wacław Sierpiński, who cites N. Anning as its source.

Confusingly, later he refers to **P. Anning** in the context of the same result. It can be assumed that this is a mistake.

It is apparent that this result was published in volume $22$ of *Scripta Mathematica*, but it has not been possible to find an online archive to confirm this, or even to determine what the title is of the article it appeared in.

The name **Anning's Theorem** has been coined by $\mathsf{Pr} \infty \mathsf{fWiki}$ as a convenient way to refer to this result, whose description would otherwise be too unwieldy for convenience.

## Sources

- 1956: N. Anning:
*???*(*Scripta Mathematica***Vol. 22**: 227) - 1970: Wacław Sierpiński:
*250 Problems in Elementary Number Theory*: No. $208$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $101,010,101 / 110,010,011$