Definition:Automorphic Number

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Definition

An automorphic number is a positive integer all of whose powers end in that number.


Sequence of Automorphic Numbers

The sequence of automorphic numbers begins as:

\(\displaystyle 1\) \(:\) \(\displaystyle 1^2\) \(\displaystyle = 1\)
\(\displaystyle 5\) \(:\) \(\displaystyle 5^2\) \(\displaystyle = 25\)
\(\displaystyle 6\) \(:\) \(\displaystyle 6^2\) \(\displaystyle = 36\)
\(\displaystyle 25\) \(:\) \(\displaystyle 25^2\) \(\displaystyle = 625\)
\(\displaystyle 76\) \(:\) \(\displaystyle 76^2\) \(\displaystyle = 3776\)
\(\displaystyle 376\) \(:\) \(\displaystyle 376^2\) \(\displaystyle = 141 \, 376\)
\(\displaystyle 625\) \(:\) \(\displaystyle 625^2\) \(\displaystyle = 390 \, 625\)
\(\displaystyle 9 \, 376\) \(:\) \(\displaystyle 9 \, 376^2\) \(\displaystyle = 87 \, 909 \, 376\)


Also known as

Some sources refer to these as curious numbers.

The term automorph for automorphic number can also sometimes be seen.


Also see

  • Results about automorphic numbers can be found here.


Sources