Automorphic number

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

Definition and properties

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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:

 
 
  (sequence A018247 in the OEIS)
  (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   is a zero of the function  . In the ring of integers modulo  , there are   zeroes to  , where the prime omega function   is the number of distinct prime factors in  . An element   in   is a zero of   if and only if   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, 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, 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, then the ring of  -adic numbers has no zero-divisors 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

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

  Prime factors of   Fixed points in   of    -adic fixed points of   Automorphic numbers in base  
6 2, 3 0, 1, 3, 4

 

 

 

 

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

 

 

 

 

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

 

 

 

 

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

 

 

 

 

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

 

 

 

 

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

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Automorphic numbers can be extended to any such polynomial function of degree     with b-adic coefficients  . These generalised automorphic numbers form a tree.

a-automorphic numbers

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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 there are two 10-adic fixed points for  ,

 
 

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

Trimorphic numbers

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A trimorphic number or spherical number occurs when the polynomial function is  .[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  , 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  , 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

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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))

See also

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References

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  1. ^ See Gérard Michon's article at
  2. ^ "spherical number". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
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