Non-Palindromes in Base 2 by Reverse-and-Add Process/Mistake
Jump to navigation
Jump to search
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): $43$