Automorphic number#Trimorphic numbers

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

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

: $\backslash ldots\; 0000000000$

: $\backslash ldots\; 0000000001$

: $\backslash ldots\; 8212890625$ {{OEIS|id=A018247}}

: $\backslash ldots\; 1787109376$ {{OEIS|id=A018248}}

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, ... {{OEIS|A003226}}.

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

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

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

Automorphic numbers can be extended to any such polynomial function of degree $n$ $f(x)\; =\; \backslash sum\_\{i\; =\; 0\}^\{n\}\; a\_i\; x^i$ with b-adic coefficients $a\_i$. These generalised automorphic numbers form a tree.

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

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

: $\backslash ldots\; 0000000000$

: $\backslash ldots\; 0893554688$

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

A trimorphic number or spherical number occurs when the polynomial function is $f(x)\; =\; x^3$.[http://www.numericana.com/answer/p-adic.htm#decimal See Gérard Michon's article at] 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.{{cite OED|spherical number}}

For base $b\; =\; 10$, 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, ... {{OEIS|id=A033819}}

For base $b\; =\; 12$, 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, ...

def hensels_lemma(polynomial_function, base: int, power: int) -> list[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: int) -> int:

return x ** 2 - x

for i in range(1, digits + 1):

print(hensels_lemma(automorphic_polynomial, base, i))

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

• {{PlanetMath |urlname=examplesof1automorphicnumbers |title=examples of 1-automorphic numbers}}

• {{MathWorld | urlname=AutomorphicNumber | title=Automorphic number}}
• {{MathWorld | urlname=TrimorphicNumber | title=Trimorphic Number }}

{{Classes of natural numbers}}

Category:Arithmetic dynamics

Category:Base-dependent integer sequences

Category:Mathematical analysis

Category:Modular arithmetic

Category:Number theory

Category:P-adic numbers

Category:Ring theory