# Automorphic number

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In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base ${\displaystyle b}$ whose square "ends" in the same digits as the number itself.

## Definition and properties

Given a number base ${\displaystyle b}$, a natural number or ${\displaystyle b}$-adic number ${\displaystyle n}$ with ${\displaystyle k}$ digits is an automorphic number if there exists a ${\displaystyle b}$-adic number solution of the equation ${\displaystyle n^{2}=x}$ such that ${\displaystyle n\equiv x{\bmod {b}}^{k}}$.

For example, with ${\displaystyle b=10}$, there are four 10-adic solutions to ${\displaystyle x^{2}=x}$, the last 10 digits of which are

${\displaystyle \ldots 0000000000}$
${\displaystyle \ldots 0000000001}$
${\displaystyle \ldots 8212890625}$ (sequence A018247 in the OEIS)
${\displaystyle \ldots 1787109376}$ (sequence A018248 in the OEIS)

Thus, the automorphic numbers in base 10 are 0, 1, 5, 6, 25, 76, 376, 625, 9376, ... (sequence A003226 in the OEIS).

In any given base ${\displaystyle b}$ there are ${\displaystyle 2^{\omega (b)}}$ ${\displaystyle b}$-adic solutions to ${\displaystyle x^{2}=x}$ uniquely determined by their last digit, where the prime omega function ${\displaystyle \omega (b)}$ is the number of distinct prime factors in ${\displaystyle b}$. This is because in the ring of integers modulo ${\displaystyle b}$, there are ${\displaystyle 2^{\omega (b)}}$ solutions to ${\displaystyle x^{2}=x}$. An element ${\displaystyle x}$ in ${\displaystyle \mathbb {Z} /b\mathbb {Z} }$ satisfies the equation ${\displaystyle x^{2}=x}$ if and only if ${\displaystyle x\equiv 0{\bmod {p}}^{v_{p}(b)}}$ or ${\displaystyle x\equiv 1{\bmod {p}}^{v_{p}(b)}}$ for all ${\displaystyle p|x}$ Since there are two possible values in ${\displaystyle \lbrace 0,1\rbrace }$, and there are ${\displaystyle \omega (b)}$ such ${\displaystyle p|x}$, there are ${\displaystyle 2^{\omega (b)}}$ solutions to ${\displaystyle x^{2}=x}$.

${\displaystyle x}$ and ${\displaystyle 1-x}$ are both ${\displaystyle b}$-adic solutions to ${\displaystyle x^{2}=x}$.

As 0 is always a zero divisor, 0 and 1 are automorphic numbers in every base. These solutions are called trivial automorphic numbers. If ${\displaystyle b}$ is a prime power, then the ring of ${\displaystyle b}$-adic numbers has no zero divisors other than 0, so the only solutions to ${\displaystyle n^{2}=n}$ are 0 and 1. As a result, nontrivial automorphic numbers, those other than 0 and 1, only exist when the base ${\displaystyle b}$ has at least two distinct prime factors.

### Nontrivial ${\displaystyle b}$-adic solutions to ${\displaystyle x^{2}=x}$

All ${\displaystyle b}$-adic numbers are represented in base ${\displaystyle b}$, using A−Z to represent digit values 10 to 35.

${\displaystyle b}$ Prime factors of ${\displaystyle b}$ Nontrivial solutions in ${\displaystyle \mathbb {Z} /b\mathbb {Z} }$ to ${\displaystyle x^{2}=x}$ Nontrivial ${\displaystyle b}$-adic solutions to ${\displaystyle x^{2}=x}$
6 2, 3 3, 4

...0213

...5344

10 2, 5 5, 6

...0625

...9376

12 2, 3 4, 9

...3854

...8369

14 2, 7 7, 8

...0C37

...D1A8

15 3, 5 6, 10

...DA86

...146A

18 2, 3 9, 10

...1249

...GFDA

20 2, 5 5, 16

...B6B5

...8D8G

21 3, 7 7, 15

...H7G7

...3D4F

22 2, 11 11, 12

...185B

...KDGC

24 2, 3 9, 16

...D0L9

...AN2G

26 2, 13 13, 14
28 2, 7 15, 16
30 2, 3, 5 6, 10, 15, 16, 21, 25

...B2J6

...H13A

...1Q7F

...S3MG

...CSQL

...IRAP

33 3, 11 12, 22
34 2, 17 17, 18
35 5, 7 15, 21
36 2, 3 9, 28

...DN29

...MCXS

### ${\displaystyle a}$-automorphic numbers

Given a number base ${\displaystyle b}$, a natural number or ${\displaystyle b}$-adic number ${\displaystyle n}$ with ${\displaystyle k}$ digits is an ${\displaystyle a}$-automorphic number if there exists a ${\displaystyle b}$-adic number solution of the equation ${\displaystyle ax^{2}=x}$ such that ${\displaystyle n\equiv x{\bmod {b}}^{k}}$. 0 is trivially an ${\displaystyle a}$-automorphic number for all such ${\displaystyle a}$.

For example, with ${\displaystyle b=10}$, there are two 10-adic solutions to ${\displaystyle 2x^{2}=x}$,

${\displaystyle \ldots 0000000000}$
${\displaystyle \ldots 0893554688}$

so the 2-automorphic numbers in base 10 are 0, 8, 88, 688, 4688...

${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle b}$-adic solutions to the equation ${\displaystyle ax^{2}=x}$ OEIS sequences
0 10 ${\displaystyle \ldots 0000000000}$ A000004
1 10

${\displaystyle \ldots 0000000000}$

${\displaystyle \ldots 0000000001}$

${\displaystyle \ldots 8212890625}$

${\displaystyle \ldots 1787109376}$

A000004, A000012, A007185, A016090
2 10

${\displaystyle \ldots 0000000000}$

${\displaystyle \ldots 0893554688}$

A000004, A030984
3 10

${\displaystyle \ldots 0000000000}$

${\displaystyle \ldots 7262369792}$

${\displaystyle \ldots 9404296875}$

${\displaystyle \ldots 6666666667}$

A000004, A030985, A030986, A067275
4 10

${\displaystyle \ldots 0000000000}$

${\displaystyle \ldots 0446777344}$

A000004, A030987
5 10

${\displaystyle \ldots 0000000000}$

${\displaystyle \ldots 3642578125}$

A000004, A030988
6 10

${\displaystyle \ldots 0000000000}$

${\displaystyle \ldots 3631184896}$

A000004, A030989
7 10

${\displaystyle \ldots 0000000000}$

${\displaystyle \ldots 7142857143}$

${\displaystyle \ldots 4548984375}$

${\displaystyle \ldots 1683872768}$

A000004, A030990, A030991, A030992
8 10

${\displaystyle \ldots 0000000000}$

${\displaystyle \ldots 0223388672}$

A000004, A030993
9 10

${\displaystyle \ldots 0000000000}$

${\displaystyle \ldots 5754123264}$

${\displaystyle \ldots 3134765625}$

${\displaystyle \ldots 8888888889}$

A000004, A030994, A030995, A059482

## m-morphic numbers

Given a number base ${\displaystyle b}$, a natural number or ${\displaystyle b}$-adic number ${\displaystyle n}$ with ${\displaystyle k}$ digits is an ${\displaystyle m}$-morphic number (also known as a ${\displaystyle m-1}$-spherical number) if there exists a ${\displaystyle b}$-adic number solution of the equation ${\displaystyle n^{m}=x}$ such that ${\displaystyle n\equiv x{\bmod {b}}^{k}}$. All automorphic numbers are ${\displaystyle m}$-morphic, i.e. if the square of a number ends in the same digits as the number, then so do all its higher powers. The terms circular and spherical were formerly used for the slightly different case of a number whose powers all have the same last digit as the number itself.[1]

### Trimorphic numbers

For ${\displaystyle m=3}$ the numbers are known as trimorphic numbers.

For base ${\displaystyle b=10}$, the trimorphic numbers are:

0, 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, ... (sequence A033819 in the OEIS)

For base ${\displaystyle b=12}$, the trimorphic numbers are:

0, 1, 3, 4, 5, 7, 8, 9, B, 15, 47, 53, 54, 5B, 61, 68, 69, 75, A7, B3, BB, 115, 253, 368, 369, 4A7, 5BB, 601, 715, 853, 854, 969, AA7, BBB, 14A7, 2369, 3853, 3854, 4715, 5BBB, 6001, 74A7, 8368, 8369, 9853, A715, BBBB, ...

## References

1. ^ "spherical number". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. (Subscription or UK public library membership required.)