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

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## Source Work

1997: David Wells: *Curious and Interesting Numbers* (2nd ed.):

- The Dictionary
- $43$

## Mistake

*In base $2$, $43 = 101011$. This base $2$ number never becomes a palindrome by the reverse-and-add process.*

This is false:

\(\ds 101011_2 + 110101_2\) | \(=\) | \(\ds 1100000_2\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 1100000_2 + 0000011_2\) | \(=\) | \(\ds 1100011_2\) |

The mistake does not originate with David Wells. It comes from the original article in *The Mathematical Gazette* from which he lifted the result without checking it:

*Finally, remember that we started with the base $2$ number $101011$, which is $43$ in base $10$. In base $19$, $43$ palindromises in just one step: $43 + 34 = 77$. Palindromising is a property of the expansion, not the number.*

For a start, the article in question actually begins with the number $1010100_2$, which is $84_{10}$. However, its reverse-and-add sequence results in:

- $84 \to 132 \to 363$

so the argument continues to hold water.

Secondly, what is that "base $19$" doing there? Surely a misprint for base $10$, which works perfectly well.

## Sources

- 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): $43$