# Largest Pandigital Square including Zero

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## Theorem

The largest pandigital square (in the sense where pandigital includes the zero) is $9 \, 814 \, 072 \, 356$:

- $9 \, 814 \, 072 \, 356 = 99 \, 066^2$

## Proof

We check all the square of numbers from $99 \, 067$ up to $\floor {9876543210} = 99 \, 380$, with the following constraints:

Since all these squares has $9$ as its leftmost digit, the number cannot end with $3$ or $7$.

The number cannot end with $0$ since its square will end in $00$.

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

These constraints leaves us with the around $60$ candidates:

- $99069, 99072, 99075, 99078, 99081, 99084, 99096, 99099, 99102, 99105, 99108, 99111, 99114, 99126, 99129, 99132, \dots, 99378$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $9,814,072,356$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $9,814,072,356$