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

## Source Work

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.