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.

$\blacksquare$


Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $90,625$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $90,625$
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