Automorphic number

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

## Contents

* 1 Definition and properties
* 1.1 Automorphic numbers in base b
* 2 Extensions
* 2.1 a-automorphic numbers
* 2.2 Trimorphic numbers
* 3 Programming example
* 4 See also
* 5 References
* 6 External links

## Definition and properties

Given a number base [math]\displaystyle{ b }[/math], a natural number [math]\displaystyle{ n }[/math] with [math]\displaystyle{ k }[/math] digits is an automorphic number if [math]\displaystyle{ n }[/math] is a fixed point of the polynomial function [math]\displaystyle{ f(x) = x^2 }[/math] over [math]\displaystyle{ \mathbb{Z}/b^k\mathbb{Z} }[/math], the ring of integers modulo [math]\displaystyle{ b^k }[/math]. As the inverse limit of [math]\displaystyle{ \mathbb{Z}/b^k\mathbb{Z} }[/math] is [math]\displaystyle{ \mathbb{Z}_b }[/math], the ring of [math]\displaystyle{ b }[/math]-adic integers, automorphic numbers are used to find the numerical representations of the fixed points of [math]\displaystyle{ f(x) = x^2 }[/math] over [math]\displaystyle{ \mathbb{Z}_b }[/math].

For example, with [math]\displaystyle{ b = 10 }[/math], there are four 10-adic fixed points of [math]\displaystyle{ f(x) = x^2 }[/math], the last 10 digits of which are one of these

[math]\displaystyle{ \ldots 0000000000 }[/math]
[math]\displaystyle{ \ldots 0000000001 }[/math]
[math]\displaystyle{ \ldots 8212890625 }[/math] (sequence A018247 in the OEIS)
[math]\displaystyle{ \ldots 1787109376 }[/math] (sequence A018248 in the OEIS)

Thus, the automorphic numbers in base 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, ... (sequence A003226 in the OEIS).

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

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

### Automorphic numbers in base b

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

[math]\displaystyle{ b }[/math] | Prime factors of [math]\displaystyle{ b }[/math] | Fixed points in [math]\displaystyle{ \mathbb{Z}/b\mathbb{Z} }[/math] of [math]\displaystyle{ f(x) = x^2 }[/math] | [math]\displaystyle{ b }[/math]-adic fixed points of [math]\displaystyle{ f(x) = x^2 }[/math] | Automorphic numbers in base [math]\displaystyle{ b }[/math]

|
|
|
|

6 | 2, 3 | 0, 1, 3, 4 |

[math]\displaystyle{ \ldots 0000000000 }[/math]

[math]\displaystyle{ \ldots 0000000001 }[/math]

[math]\displaystyle{ \ldots 2221350213 }[/math]

[math]\displaystyle{ \ldots 3334205344 }[/math]

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 |

[math]\displaystyle{ \ldots 0000000000 }[/math]

[math]\displaystyle{ \ldots 0000000001 }[/math]

[math]\displaystyle{ \ldots 8212890625 }[/math]

[math]\displaystyle{ \ldots 1787109376 }[/math]

| 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 |

[math]\displaystyle{ \ldots 0000000000 }[/math]

[math]\displaystyle{ \ldots 0000000001 }[/math]

[math]\displaystyle{ \ldots 21\text{B}61\text{B}3854 }[/math]

[math]\displaystyle{ \ldots 9\text{A}05\text{A}08369 }[/math]

| 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 |

[math]\displaystyle{ \ldots 0000000000 }[/math]

[math]\displaystyle{ \ldots 0000000001 }[/math]

[math]\displaystyle{ \ldots 7337\text{A}\text{A}0\text{C}37 }[/math]

[math]\displaystyle{ \ldots 6\text{A}\text{A}633\text{D}1\text{A}8 }[/math]

| 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 |

[math]\displaystyle{ \ldots 0000000000 }[/math]

[math]\displaystyle{ \ldots 0000000001 }[/math]

[math]\displaystyle{ \ldots 624\text{D}4\text{B}\text{D}\text{A}86 }[/math]

[math]\displaystyle{ \ldots 8\text{C}\text{A}1\text{A}3146\text{A} }[/math]

| 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

## Extensions

Automorphic numbers can be extended to any such polynomial function of degree [math]\displaystyle{ n }[/math] [math]\displaystyle{ f(x) = \sum_{i = 0}^{n} a_i x^i }[/math] with b-adic coefficients [math]\displaystyle{ a_i }[/math]. These generalised automorphic numbers form a tree.

### a-automorphic numbers

An [math]\displaystyle{ a }[/math]-automorphic number occurs when the polynomial function is [math]\displaystyle{ f(x) = ax^2 }[/math]

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

[math]\displaystyle{ \ldots 0000000000 }[/math]
[math]\displaystyle{ \ldots 0893554688 }[/math]

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

### Trimorphic numbers

A trimorphic number or spherical number occurs when the polynomial function is [math]\displaystyle{ f(x) = x^3 }[/math] .[1] All automorphic numbers are trimorphic. 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.[2]

For base [math]\displaystyle{ b = 10 }[/math], 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 [math]\displaystyle{ b = 12 }[/math], 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, ...

## Programming example

def hensels_lemma(polynomial_function, base: int, power: int):
"""Hensel's lemma."""
if power == 0:
return [0]
if power > 0:
roots = hensels_lemma(polynomial_function, base, power - 1)
new_roots = []
for root in roots:
for i in range(0, base):
new_i = i * base ** (power - 1) + root
new_root = polynomial_function(new_i) % pow(base, power)
if new_root == 0:
new_roots.append(new_i)
return new_roots

base = 10
digits = 10

def automorphic_polynomial(x):
return x ** 2 - x

for i in range(1, digits + 1):
print(hensels_lemma(automorphic_polynomial, base, i))

## See also

* Arithmetic dynamics
* Kaprekar number
* P-adic number
* P-adic analysis
* Zero divisor

## References

1. ↑ See Gérard Michon's article at
2. ↑ spherical number (3rd ed.), Oxford University Press, September 2005, http://oed.com/search?searchType=dictionary&q=spherical+number (Subscription or UK public library membership required.)

* examples of 1-automorphic numbers at PlanetMath.org.

## External links

* Weisstein, Eric W.. "Automorphic number". http://mathworld.wolfram.com/AutomorphicNumber.html.
* Weisstein, Eric W.. "Trimorphic Number". http://mathworld.wolfram.com/TrimorphicNumber.html.

* v
* t
* e

Classes of natural numbers

Powers and
related numbers|

* Achilles
* Power of 2
* Power of 10
* Square
* Cube
* Fourth power
* Fifth power
* Sixth power
* Seventh power
* Perfect power
* Powerful
* Prime power

Of the form
a × 2b ± 1|

* Cullen
* Double Mersenne
* Fermat
* Mersenne
* Proth
* Thabit
* Woodall

Other polynomial
numbers|

* Carol
* Hilbert
* Idoneal
* Kynea
* Leyland
* Loeschian
* Lucky numbers of Euler

Recursively defined
numbers|

* Fibonacci
* Jacobsthal
* Leonardo
* Lucas
* Padovan
* Pell
* Perrin

Possessing a
specific set
of other numbers|

* Knödel
* Riesel
* Sierpinski

Expressible via
specific sums|

* Nonhypotenuse
* Polite
* Practical
* Primary pseudoperfect
* Ulam
* Wolstenholme

Figurate numbers|

| 2-dimensional|

| centered|

* Centered triangular
* Centered square
* Centered pentagonal
* Centered hexagonal
* Centered heptagonal
* Centered octagonal
* Centered nonagonal
* Centered decagonal
* Star

|

non-centered|

* Triangular
* Square
* Square triangular
* Pentagonal
* Hexagonal
* Heptagonal
* Octagonal
* Nonagonal
* Decagonal
* Dodecagonal

3-dimensional|

| centered|

* Centered tetrahedral
* Centered cube
* Centered octahedral
* Centered dodecahedral
* Centered icosahedral

|

non-centered|

* Tetrahedral
* Cube
* Octahedral
* Dodecahedral
* Icosahedral
* Stella octangula

pyramidal|

* Square pyramidal
* Pentagonal pyramidal
* Hexagonal pyramidal
* Heptagonal pyramidal

4-dimensional|

| centered|

* Centered pentachoric
* Squared triangular

|

non-centered|

* Pentatope
* 4-cube

Base-dependent
numbers|

| Arithmetic functions and dynamics|

* Digit sum
* Digital root
* Multiplicative
* Dudeney
* Factorion
* Happy
* Kaprekar
* Kaprekar's constant
* Keith
* Lychrel
* Meertens
* Narcissistic
* Perfect digit-to-digit invariant
* Perfect digital invariant
* Persistence
* Additive
* Multiplicative
* Self
* Sum-product

|

P-adic numbers related|

* Automorphic

Divisor related|

* Equidigital
* Extravagant
* Frugal
* Harshad
* Polydivisible
* Smith

Digit-composition related|

* Palindromic
* Pandigital
* Repdigit
* Repunit
* Self-descriptive
* Smarandache–Wellin
* Strictly non-palindromic
* Undulating

Digit-permutation related|

* Cyclic
* Digit-reassembly
* Parasitic
* Primeval
* Transposable

Other|

* Friedman
* Vampire

Combinatorial
numbers|

* Bell
* Cake
* Catalan
* Dedekind
* Delannoy
* Euler
* Fuss–Catalan
* Lazy caterer's sequence
* Lobb
* Motzkin
* Narayana
* Ordered Bell
* Schröder
* Schröder–Hipparchus

Pseudoprimes|

* Carmichael number
* Catalan pseudoprime
* Elliptic pseudoprime
* Euler pseudoprime
* Euler–Jacobi pseudoprime
* Fermat pseudoprime
* Frobenius pseudoprime
* Lucas pseudoprime
* Somer–Lucas pseudoprime
* Strong pseudoprime

Arithmetic functions|

| By properties of σ(n)|

* Abundant
* Arithmetic
* Colossally abundant
* Descartes
* Hemiperfect
* Highly abundant
* Highly composite
* Hyperperfect
* Multiply perfect
* Perfect
* Practical number
* Primitive abundant
* Quasiperfect
* Refactorable
* Sublime
* Superabundant
* Superior highly composite
* Superperfect
* Untouchable

|

By properties of Ω(n)|

* Almost prime
* Semiprime

By properties of φ(n)|

* Highly cototient
* Highly totient
* Noncototient
* Nontotient
* Perfect totient
* Sparsely totient

By properties of s(n)|

* Almost perfect
* Amicable
* Betrothed
* Deficient
* Perfect
* Semiperfect
* Sociable

Other|

* Euclid
* Fortunate

Dividing a quotient|

* Wieferich
* Wall–Sun–Sun
* Wolstenholme prime
* Wilson

Other prime factor
or divisor related
numbers|

* Blum
* Erdős–Nicolas
* Erdős–Woods
* Friendly
* Giuga
* Harmonic divisor
* Lucas–Carmichael
* Pronic
* Regular
* Rough
* Smooth
* Sociable
* Sphenic
* Størmer
* Super-Poulet
* Zeisel

Binary numbers|

* Evil
* Odious
* Pernicious

Generated
via a sieve|

* Lucky
* Prime

Sorting related|

* Pancake number
* Sorting number

Natural language related|

* Aronson's sequence
* Ban

Graphemics related|

* Strobogrammatic

* Mathematics portal

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