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Inmathematics, anautomorphic number (sometimes referred to as acircular number) is anatural number in a givennumber base${\displaystyle b}$ whosesquare "ends" in the same digits as the number itself.

Given a number base${\displaystyle b}$, a natural number${\displaystyle n}$ with${\displaystyle k}$ digits is anautomorphic number if${\displaystyle n}$ is afixed point of thepolynomial function${\displaystyle f(x)=x^2}$ over${\displaystyle \mathbb{Z}/b^k\mathbb{Z}}$, thering ofintegers modulo${\displaystyle b^k}$. As theinverse limit of${\displaystyle \mathbb{Z}/b^k\mathbb{Z}}$ is${\displaystyle {\mathbb{Z}}_b}$, the ring of${\displaystyle b}$-adic integers, automorphic numbers are used to find the numerical representations of the fixed points of${\displaystyle f(x)=x^2}$ over${\displaystyle {\mathbb{Z}}_b}$.

For example, with${\displaystyle b=10}$, there are four 10-adic fixed points of${\displaystyle f(x)=x^2}$, the last 10 digits of which are:

${\displaystyle \dots 0000000000}$

${\displaystyle \dots 0000000001}$

${\displaystyle \dots 8212890625}$ (sequenceA018247 in theOEIS)

${\displaystyle \dots 1787109376}$ (sequenceA018248 in theOEIS)

Thus, the automorphic numbers inbase 10 are 0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376, 59918212890625, ... (sequenceA003226 in theOEIS).

A fixed point of${\displaystyle f(x)}$ is azero of the function${\displaystyle g(x)=f(x)-x}$. In the ring of integers modulo${\displaystyle b}$, there are${\displaystyle 2^{\omega (b)}}$ zeroes to${\displaystyle g(x)=x^2-x}$, where theprime omega function${\displaystyle \omega (b)}$ is the number of distinctprime factors in${\displaystyle b}$. An element${\displaystyle x}$ in${\displaystyle \mathbb{Z}/b\mathbb{Z}}$ is a zero of${\displaystyle g(x)=x^2-x}$if and only if${\displaystyle x\equiv 0{modp}^{v_p(b)}}$ or${\displaystyle x\equiv 1{modp}^{v_p(b)}}$ for all${\displaystyle p|b}$. Since there are two possible values in${\displaystyle \{0,1\}}$, and there are${\displaystyle \omega (b)}$ such${\displaystyle p|b}$, there are${\displaystyle 2^{\omega (b)}}$ zeroes of${\displaystyle g(x)=x^2-x}$, and thus there are${\displaystyle 2^{\omega (b)}}$ fixed points of${\displaystyle f(x)=x^2}$. According toHensel's lemma, if there are${\displaystyle k}$ zeroes or fixed points of a polynomial function modulo${\displaystyle b}$, then there are${\displaystyle k}$ corresponding zeroes or fixed points of the same function modulo any power of${\displaystyle b}$, and this remains true in theinverse limit. Thus, in any given base${\displaystyle b}$ there are${\displaystyle 2^{\omega (b)}}$${\displaystyle b}$-adic fixed points of${\displaystyle f(x)=x^2}$.

As 0 is always azero-divisor, 0 and 1 are always fixed points of${\displaystyle f(x)=x^2}$, and 0 and 1 are automorphic numbers in every base. These solutions are calledtrivial automorphic numbers. If${\displaystyle b}$ is aprime power, then the ring of${\displaystyle b}$-adic numbers has no zero-divisors other than 0, so the only fixed points of${\displaystyle f(x)=x^2}$ 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.

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${\displaystyle b}$	Fixed points in${\displaystyle \mathbb{Z}/b\mathbb{Z}}$ of${\displaystyle f(x)=x^2}$	${\displaystyle b}$-adic fixed points of${\displaystyle f(x)=x^2}$	Automorphic numbers in base${\displaystyle b}$
6	2, 3	0, 1, 3, 4	${\displaystyle \dots 0000000000}$ ${\displaystyle \dots 0000000001}$ ${\displaystyle \dots 2221350213}$ ${\displaystyle \dots 3334205344}$	0, 1, 3, 4, 13, 44, 213, 344, 5344, 50213, 205344, 350213, 1350213, 4205344, 21350213, 34205344, 221350213, 334205344, 2221350213, 3334205344, ...
10	2, 5	0, 1, 5, 6	${\displaystyle \dots 0000000000}$ ${\displaystyle \dots 0000000001}$ ${\displaystyle \dots 8212890625}$ ${\displaystyle \dots 1787109376}$	0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, ...
12	2, 3	0, 1, 4, 9	${\displaystyle \dots 0000000000}$ ${\displaystyle \dots 0000000001}$ ${\displaystyle \dots 21\text{B}61\text{B}3854}$ ${\displaystyle \dots 9\text{A}05\text{A}08369}$	0, 1, 4, 9, 54, 69, 369, 854, 3854, 8369, B3854, 1B3854, A08369, 5A08369, 61B3854, B61B3854, 1B61B3854, A05A08369, 21B61B3854, 9A05A08369, ...
14	2, 7	0, 1, 7, 8	${\displaystyle \dots 0000000000}$ ${\displaystyle \dots 0000000001}$ ${\displaystyle \dots 7337\text{A}\text{A}0\text{C}37}$ ${\displaystyle \dots 6\text{A}\text{A}633\text{D}1\text{A}8}$	0, 1, 7, 8, 37, A8, 1A8, C37, D1A8, 3D1A8, A0C37, 33D1A8, AA0C37, 633D1A8, 7AA0C37, 37AA0C37, A633D1A8, 337AA0C37, AA633D1A8, 6AA633D1A8, 7337AA0C37, ...
15	3, 5	0, 1, 6, 10	${\displaystyle \dots 0000000000}$ ${\displaystyle \dots 0000000001}$ ${\displaystyle \dots 624\text{D}4\text{B}\text{D}\text{A}86}$ ${\displaystyle \dots 8\text{C}\text{A}1\text{A}3146\text{A}}$	0, 1, 6, A, 6A, 86, 46A, A86, 146A, DA86, 3146A, BDA86, 4BDA86, A3146A, 1A3146A, D4BDA86, 4D4BDA86, A1A3146A, 24D4BDA86, CA1A3146A, 624D4BDA86, 8CA1A3146A, ...
18	2, 3	0, 1, 9, 10	...000000 ...000001 ...4E1249 ...D3GFDA
20	2, 5	0, 1, 5, 16	...000000 ...000001 ...1AB6B5 ...I98D8G
21	3, 7	0, 1, 7, 15	...000000 ...000001 ...86H7G7 ...CE3D4F
22	2, 11	0, 1, 11, 12	...000000 ...000001 ...8D185B ...D8KDGC
24	2, 3	0, 1, 9, 16	...000000 ...000001 ...E4D0L9 ...9JAN2G
26	2, 13	0, 1, 13, 14	...0000 ...0001 ...1G6D ...O9JE
28	2, 7	0, 1, 8, 21	...0000 ...0001 ...AAQ8 ...HH1L
30	2, 3, 5	0, 1, 6, 10, 15, 16, 21, 25	...0000 ...0001 ...B2J6 ...H13A ...1Q7F ...S3MG ...CSQL ...IRAP
33	3, 11	0, 1, 12, 22	...0000 ...0001 ...1KPM ...VC7C
34	2, 17	0, 1, 17, 18	...0000 ...0001 ...248H ...VTPI
35	5, 7	0, 1, 15, 21	...0000 ...0001 ...5MXL ...TC1F
36	2, 3	0, 1, 9, 28	...0000 ...0001 ...DN29 ...MCXS

Automorphic numbers can be extended to any such polynomial function ofdegree${\displaystyle n}$$f(x)=\sum _{i=0}^n{a}_i{x}^i$ withb-adic coefficients${\displaystyle a_i}$. These generalised automorphic numbers form atree.

An${\displaystyle a}$-automorphic number occurs when the polynomial function is${\displaystyle f(x)=a{x}^2}$

For example, with${\displaystyle b=10}$ and${\displaystyle a=2}$, as there are two fixed points for${\displaystyle f(x)=2x^2}$ in${\displaystyle \mathbb{Z}/10\mathbb{Z}}$ (${\displaystyle x=0}$ and${\displaystyle x=8}$), according toHensel's lemma there are two 10-adic fixed points for${\displaystyle f(x)=2x^2}$,

${\displaystyle \dots 0000000000}$

${\displaystyle \dots 0893554688}$

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

Atrimorphic number orspherical number occurs when the polynomial function is${\displaystyle f(x)=x^3}$.^[1] All automorphic numbers are trimorphic. The termscircular andspherical were formerly used for the slightly different case of a number whose powers all have the same last digit as the number itself.^[2]

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, ... (sequenceA033819 in theOEIS)

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, ...

defhensels_lemma(polynomial_function,base:int,power:int)->list[int]:"""Hensel's lemma."""ifpower==0:return[0]ifpower>0:roots=hensels_lemma(polynomial_function,base,power-1)new_roots=[]forrootinroots:foriinrange(0,base):new_i=i*base**(power-1)+rootnew_root=polynomial_function(new_i)%pow(base,power)ifnew_root==0:new_roots.append(new_i)returnnew_rootsbase=10digits=10defautomorphic_polynomial(x:int)->int:returnx**2-xforiinrange(1,digits+1):print(hensels_lemma(automorphic_polynomial,base,i))

• Arithmetic dynamics
• Kaprekar number
• p-adic number
• p-adic analysis
• Zero-divisor

• examples of 1-automorphic numbers atPlanetMath.

• Weisstein, Eric W."Automorphic number".MathWorld.
• Weisstein, Eric W."Trimorphic Number".MathWorld.

• Arithmetic dynamics
• Base-dependent integer sequences
• Mathematical analysis
• Modular arithmetic
• Number theory
• P-adic numbers
• Ring theory

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