# 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:

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

## 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.