Smallest Non-Palindromic Number with Palindromic Square
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Theorem
$26$ is the smallest non-palindromic integer whose square is palindromic.
Proof
Checking the squares of all non-palindromic integers in turn from $10$ upwards, until a palindromic integer is reached:
\(\ds 10^2\) | \(=\) | \(\ds 100\) | ||||||||||||
\(\ds 12^2\) | \(=\) | \(\ds 144\) | ||||||||||||
\(\ds 13^2\) | \(=\) | \(\ds 169\) | ||||||||||||
\(\ds 14^2\) | \(=\) | \(\ds 196\) | ||||||||||||
\(\ds 15^2\) | \(=\) | \(\ds 225\) | ||||||||||||
\(\ds 16^2\) | \(=\) | \(\ds 256\) | ||||||||||||
\(\ds 17^2\) | \(=\) | \(\ds 289\) | ||||||||||||
\(\ds 18^2\) | \(=\) | \(\ds 324\) | ||||||||||||
\(\ds 19^2\) | \(=\) | \(\ds 361\) | ||||||||||||
\(\ds 20^2\) | \(=\) | \(\ds 400\) | ||||||||||||
\(\ds 21^2\) | \(=\) | \(\ds 441\) | ||||||||||||
\(\ds 23^2\) | \(=\) | \(\ds 529\) | ||||||||||||
\(\ds 24^2\) | \(=\) | \(\ds 576\) | ||||||||||||
\(\ds 25^2\) | \(=\) | \(\ds 625\) | ||||||||||||
\(\ds 26^2\) | \(=\) | \(\ds 676\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $26$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $26$