Automorphic Numbers with 5 Digits
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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): $90,625$