# Non-Palindromes in Base 2 by Reverse-and-Add Process

## Theorem

Let the number $22$ be expressed in binary: $10110_2$.

When the reverse-and-add process is performed on it repeatedly, it never becomes a palindromic number.

## Proof

\(\ds 10110_2 + 01101_2\) | \(=\) | \(\ds 100011_2\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 100011_2 + 110001_2\) | \(=\) | \(\ds 1010100_2\) |

It remains to be shown that a binary number of this form does not become a palindromic number.

Let $d_n$ denote $n$ repetitions of a binary digit $d$ in a number.

Thus:

- $10111010000$

can be expressed as:

- $101_3010_4$

**Beware** that the subscript, from here on in, does not denote the number base.

It is to be shown that the reverse-and-add process applied to:

- $101_n010_{n + 1}$

leads after $4$ iterations to:

- $101_{n + 1}010_{n + 2}$

Thus:

\(\ds 101_n010_{n + 1} + 0_{n + 1}101_k01\) | \(=\) | \(\ds 110_n101_{n - 1}01\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 110_n101_{n - 1}01 + 101_{n - 1}010_n11\) | \(=\) | \(\ds 101_{n + 1}010_{n + 1}\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds 101_{n + 1}010_{n + 1} + 0_{n + 1}101_{n + 1}01\) | \(=\) | \(\ds 110_{n - 1}10001_{n - 1}01\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds 110_{n - 1}10001_{n - 1}01 + 101_{n - 1}00010_{n - 1}11\) | \(=\) | \(\ds 101_{n + 1}010_{n + 2}\) |

As neither $101_n010_{n + 1}$ nor $101_{n + 1}010_{n + 2}$ are palindromic numbers, nor are any of the intermediate results, the result follows.

$\blacksquare$

## Historical Note

This result, according to David Wells in his $1986$ book *Curious and Interesting Numbers*, was given by Roland Sprague, but the source for this has not been identified.

Independently of Sprague's work, Glyn Johns and James Wiegold, in the context of a children's mathematics club in Cardiff, Wales, noted the reverse-and-add behaviour of $1010100_2$.

Their subsequent report in Volume $78$, issue $483$ of *The Mathematical Gazette* was used as the basis of the analysis given here in $\mathsf{Pr} \infty \mathsf{fWiki}$, but it needs to be pointed out that there are a number of mistakes in that source.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $196$ - Nov. 1994: Glyn Johns and James Wiegold:
*The Palindrome Problem in Base 2*(*The Mathematical Gazette***Vol. 78**,*no. 483*: pp. 312 – 314) www.jstor.org/stable/3620206 - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $196$