Trimorphic Number is not necessarily Automorphic

Theorem

Let $n \in \Z_{>0}$ be a trimorphic number.

Then it is not necessarily the case that $n$ is also an automorphic number.

Proof

Take as an example $n = 49$.

We have that:

$49^3 = 117 \, 6 \mathbf{49}$

demonstrating that $49$ is trimorphic.

However, we also have that:

$49^2 = 2401$

demonstrating that $49$ is not automorphic.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $49$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $49$