Definition:Automorphic Number

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:

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

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.