# Automorphic Numbers with 5 Digits

## Theorem

The only $5$-digit automorphic number which does not begin with a zero is $90 \, 625$.

## Proof

We have:

$90 \, 625^2 = 8 \, 212 \, 8 \mathbf {90 \, 625}$

thus demonstrating it is automorphic.

By Automorphic Numbers in Base 10, the only other possible candidate is $6^{5^4}$.

However:

$6^{5^4} \equiv 09 \, 376 \pmod {10^5}$

begins with a zero.

Hence there are no others.

$\blacksquare$