In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an automorphic number (sometimes referred to as a circular number) is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
in a given
number base
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
whose
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
"ends" in the same digits as the number itself.
Definition and properties
Given a number base
, a natural number
with
digits is an automorphic number if
is a
fixed point of the polynomial function
over
, the
ring of
integers modulo . As the
inverse limit of
is
, the ring of
-adic integers, automorphic numbers are used to find the numerical representations of the fixed points of
over
.
For example, with
, there are four 10-adic fixed points of
, the last 10 digits of which are one of these
:
:
:
:
Thus, the automorphic numbers in
base 10
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
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, ... .
A fixed point of
is a
zero of the function . In the
ring of
integers modulo , there are
zeroes to
, where the
prime omega function In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. Thereby \omega(n) (little omega) counts each ''distinct'' prime factor, whereas the related function \Omega(n) (big omega) ...
is the number of distinct prime factors in
. An element
in
is a zero of
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
or
for all
. Since there are two possible values in
, and there are
such
, there are
zeroes of
, and thus there are
fixed points of
. According to
Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to ...
, if there are
zeroes or fixed points of a polynomial function modulo
, then there are
corresponding zeroes or fixed points of the same function modulo any power of
, and this remains true in the
inverse limit. Thus, in any given base
there are
-adic fixed points of
.
As 0 is always a
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
, 0 and 1 are always fixed points of
, and 0 and 1 are automorphic numbers in every base. These solutions are called trivial automorphic numbers. If
is a
prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime powers, while
, and are not.
The sequence of prime powers begins:
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
, then the ring of
-adic numbers has no
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s other than 0, so the only fixed points of
are 0 and 1. As a result, nontrivial automorphic numbers, those other than 0 and 1, only exist when the base
has at least two distinct prime factors.
Automorphic numbers in base ''b''
All
-adic numbers are represented in base
, using A−Z to represent digit values 10 to 35.
Extensions
Automorphic numbers can be extended to any such polynomial function of degree
with b-adic coefficients
. These generalised automorphic numbers form a
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
.
''a''-automorphic numbers
An
-automorphic number occurs when the polynomial function is
For example, with
and
, as there are two fixed points for
in
(
and
), according to
Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to ...
there are two 10-adic fixed points for
,
:
:
so the 2-automorphic numbers in
base 10
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
are 0, 8, 88, 688, 4688...
Trimorphic numbers
A trimorphic number or spherical number occurs when the polynomial function is
. 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.
For base
, 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, ...
For base
, 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 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
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
*
P-adic analysis
In mathematics, ''p''-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers.
The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of l ...
*
Zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
References
*
External links
*
*
{{Classes of natural numbers
Arithmetic dynamics
Base-dependent integer sequences
Mathematical analysis
Modular arithmetic
Number theory
P-adic numbers
Ring theory