Smallest Non-Palindromic Number with Palindromic Square

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): $26$