„automorphe Zahl“ – Versionsunterschied

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automorphen Zahlen. Das sind Zahlen, deren Quadrat auf dieselbe Ziffer (oder Ziffernfolge) endet wie die Zahl selbst (z. B. 25²=625).

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

Definition and properties

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 $ℤ / b^{k} ℤ$ , the ring of integers modulo $b^{k}$ . As the inverse limit of $ℤ / b^{k} ℤ$ is $ℤ_{b}$ , 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 $ℤ_{b}$ .

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

\dots 0000000000

\dots 0000000001

\dots 8212890625

\{\{OEIS|id=A018247}}

\dots 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 $2^{ω (b)}$ zeroes to $g (x) = x^{2} - x$ , where the prime omega function $ω (b)$ is the number of distinct prime factors in $b$ . An element $x$ in $ℤ / b ℤ$ is a zero of $g (x) = x^{2} - x$ if and only if $x \equiv 0 {mod p}^{v_{p} (b)}$ or $x \equiv 1 {mod p}^{v_{p} (b)}$ for all $p | b$ . Since there are two possible values in ${0, 1}$ , and there are $ω (b)$ such $p | b$ , there are $2^{ω (b)}$ zeroes of $g (x) = x^{2} - x$ , and thus there are $2^{ω (b)}$ 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 $2^{ω (b)}$ $b$ -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.

Automorphic numbers in base b

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

$b$	Prime factors of $b$	Fixed points in $ℤ / b ℤ$ of $f (x) = x^{2}$	$b$ -adic fixed points of $f (x) = x^{2}$	Automorphic numbers in base $b$
6	2, 3	0, 1, 3, 4	$\dots 0000000000$ $\dots 0000000001$ $\dots 2221350213$ $\dots 3334205344$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 8212890625$ $\dots 1787109376$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 21 B 61 B 3854$ $\dots 9 A 05 A 08369$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 7337 A A 0 C 37$ $\dots 6 A A 633 D 1 A 8$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 624 D 4 B D A 86$ $\dots 8 C A 1 A 3146 A$	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

Extensions

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

a-automorphic numbers

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

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

\dots 0000000000

\dots 0893554688

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 $f (x) = x^{3}$ .^[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 $b = 10$ , 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, ...

Programming example

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))

References

↑ See Gérard Michon's article at
↑ \{\{cite OED|spherical number}}

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

External links

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

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

Definition and properties

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 $ℤ / b^{k} ℤ$ , the ring of integers modulo $b^{k}$ . As the inverse limit of $ℤ / b^{k} ℤ$ is $ℤ_{b}$ , 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 $ℤ_{b}$ .

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

\dots 0000000000

\dots 0000000001

\dots 8212890625

\{\{OEIS|id=A018247}}

\dots 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 $2^{ω (b)}$ zeroes to $g (x) = x^{2} - x$ , where the prime omega function $ω (b)$ is the number of distinct prime factors in $b$ . An element $x$ in $ℤ / b ℤ$ is a zero of $g (x) = x^{2} - x$ if and only if $x \equiv 0 {mod p}^{v_{p} (b)}$ or $x \equiv 1 {mod p}^{v_{p} (b)}$ for all $p | b$ . Since there are two possible values in ${0, 1}$ , and there are $ω (b)$ such $p | b$ , there are $2^{ω (b)}$ zeroes of $g (x) = x^{2} - x$ , and thus there are $2^{ω (b)}$ 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 $2^{ω (b)}$ $b$ -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.

Automorphic numbers in base b

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

$b$	Prime factors of $b$	Fixed points in $ℤ / b ℤ$ of $f (x) = x^{2}$	$b$ -adic fixed points of $f (x) = x^{2}$	Automorphic numbers in base $b$
6	2, 3	0, 1, 3, 4	$\dots 0000000000$ $\dots 0000000001$ $\dots 2221350213$ $\dots 3334205344$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 8212890625$ $\dots 1787109376$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 21 B 61 B 3854$ $\dots 9 A 05 A 08369$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 7337 A A 0 C 37$ $\dots 6 A A 633 D 1 A 8$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 624 D 4 B D A 86$ $\dots 8 C A 1 A 3146 A$	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

Extensions

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

a-automorphic numbers

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

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

\dots 0000000000

\dots 0893554688

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 $f (x) = x^{3}$ .^[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 $b = 10$ , 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, ...

Programming example

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))

References

↑ See Gérard Michon's article at
↑ \{\{cite OED|spherical number}}

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

External links

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

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

Definition and properties

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 $ℤ / b^{k} ℤ$ , the ring of integers modulo $b^{k}$ . As the inverse limit of $ℤ / b^{k} ℤ$ is $ℤ_{b}$ , 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 $ℤ_{b}$ .

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

\dots 0000000000

\dots 0000000001

\dots 8212890625

\{\{OEIS|id=A018247}}

\dots 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 $2^{ω (b)}$ zeroes to $g (x) = x^{2} - x$ , where the prime omega function $ω (b)$ is the number of distinct prime factors in $b$ . An element $x$ in $ℤ / b ℤ$ is a zero of $g (x) = x^{2} - x$ if and only if $x \equiv 0 {mod p}^{v_{p} (b)}$ or $x \equiv 1 {mod p}^{v_{p} (b)}$ for all $p | b$ . Since there are two possible values in ${0, 1}$ , and there are $ω (b)$ such $p | b$ , there are $2^{ω (b)}$ zeroes of $g (x) = x^{2} - x$ , and thus there are $2^{ω (b)}$ 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 $2^{ω (b)}$ $b$ -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.

Automorphic numbers in base b

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

$b$	Prime factors of $b$	Fixed points in $ℤ / b ℤ$ of $f (x) = x^{2}$	$b$ -adic fixed points of $f (x) = x^{2}$	Automorphic numbers in base $b$
6	2, 3	0, 1, 3, 4	$\dots 0000000000$ $\dots 0000000001$ $\dots 2221350213$ $\dots 3334205344$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 8212890625$ $\dots 1787109376$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 21 B 61 B 3854$ $\dots 9 A 05 A 08369$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 7337 A A 0 C 37$ $\dots 6 A A 633 D 1 A 8$	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	$\dots 0000000000$ $\dots 0000000001$ $\dots 624 D 4 B D A 86$ $\dots 8 C A 1 A 3146 A$	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

Extensions

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

a-automorphic numbers

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

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

\dots 0000000000

\dots 0893554688

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 $f (x) = x^{3}$ .^[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 $b = 10$ , 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, ...

Programming example

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))

References

↑ See Gérard Michon's article at
↑ \{\{cite OED|spherical number}}

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

External links

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

\{\{Classes of natural numbers}}

[1] See Gérard Michon's article at

[2] \{\{cite OED|spherical number}}

[3] See Gérard Michon's article at

[4] \{\{cite OED|spherical number}}

[5] See Gérard Michon's article at

[6] \{\{cite OED|spherical number}}

[1]

[2]

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[2]

@@ Zeile 269: / Zeile 269: @@
 \{\{Classes of natural numbers}}
-[[Category:Arithmetic dynamics]]
+[[Kategorie:Arithmetic dynamics]]
-[[Category:Base-dependent integer sequences]]
+[[Kategorie:Base-dependent integer sequences]]
-[[Category:Mathematical analysis]]
+[[Kategorie:Mathematical analysis]]
-[[Category:Modular arithmetic]]
+[[Kategorie:Modular arithmetic]]
-[[Category:Number theory]]
+[[Kategorie:Number theory]]
-[[Category:P-adic numbers]]
+[[Kategorie:P-adic numbers]]
 In [[mathematics]], an '''automorphic number''' (sometimes referred to as a '''circular number''') is a [[natural number]] in a given [[number base]] <math>b</math> whose [[square (algebra)|square]] "ends" in the same digits as the number itself.
@@ Zeile 543: / Zeile 543: @@
 \{\{Classes of natural numbers}}
-[[Category:Arithmetic dynamics]]
+[[Kategorie:Arithmetic dynamics]]
-[[Category:Base-dependent integer sequences]]
+[[Kategorie:Base-dependent integer sequences]]
-[[Category:Mathematical analysis]]
+[[Kategorie:Mathematical analysis]]
-[[Category:Modular arithmetic]]
+[[Kategorie:Modular arithmetic]]
-[[Category:Number theory]]
+[[Kategorie:Number theory]]
-[[Category:P-adic numbers]]
+[[Kategorie:P-adic numbers]]
-[[Category:Ring theory]]
+[[Kategorie:Ring theory]]
 In [[mathematics]], an '''automorphic number''' (sometimes referred to as a '''circular number''') is a [[natural number]] in a given [[number base]] <math>b</math> whose [[square (algebra)|square]] "ends" in the same digits as the number itself.

„automorphe Zahl“ – Versionsunterschied

Nikolaos (Diskussion | Beiträge)

Aktuelle Version vom 11. August 2026, 10:43 Uhr

Definition and properties

Automorphic numbers in base b

Extensions

a-automorphic numbers

Trimorphic numbers

Programming example

See also

References

External links

Definition and properties

Automorphic numbers in base b

Extensions

a-automorphic numbers

Trimorphic numbers

Programming example

See also

References

External links

Definition and properties

Automorphic numbers in base b

Extensions

a-automorphic numbers

Trimorphic numbers

Programming example

See also

References

External links