Left-Truncated Automorphic Number is Automorphic

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
Let $n$ be an automorphic number of $d$ digits, expressed in base $b$.

By, we have:


 * $n^2 \equiv n \pmod {b^d}$

Let some digits be removed from the left-hand end of $n$, so that only $d'$ digits remain.

This only makes sense when $d' < d$.

Define this new number as $n'$.

Then we have:


 * $n \equiv n' \pmod {b^{d'} }$

Thus we have:

Hence $n'$ is an automorphic number of $d'$ digits in base $b$.

Also see
Similar proofs can give similar results for other similarly defined numbers, e.g. Trimorphic Numbers, Tri-Automorphic Numbers.