# Trimorphic Number is not necessarily Automorphic

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## 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): $49$