Smallest Pandigital Square

Theorem
The smallest pandigital square is $1 \, 026 \, 753 \, 849$:


 * $1 \, 026 \, 753 \, 849 = 32 \, 043^2$

Proof
We check all the squares of numbers from $\ceiling {\sqrt {1 \, 023 \, 456 \, 789} } = 31 \, 992$ up to $32 \, 042$, with the following constraints:

Since all these squares has $10$ as its two leftmost digits, the number cannot end with $0$, $1$ or $9$.

A pandigital number is divisible by $9$, so our number must be divisible by $3$.

These constraints leaves us with the following $12$ candidates:

By inspection, none of these numbers are pandigital.