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