Left-Truncated Automorphic Number is Automorphic

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Theorem

Let $n$ be an automorphic number, expressed in some conventional number base.

Let any number of digits be removed from the left-hand end of $n$.


Then what remains is also an automorphic number.


Proof


Examples

Left-Truncation of $1 \, 787 \, 109 \, 376$

We have that $1 \, 787 \, 109 \, 376$ is automorphic:

$1 \, 787 \, 109 \, 376^2 = 3 \, 193 \, 759 \, 92 \, \mathbf {1 \, 787 \, 109 \, 376}$


Hence so is $109 \, 376$:

$109 \, 376^2 = 11 \, 963 \, \mathbf {109 \, 376}$


Sources