Editing Automorphic number (section)

==Extensions==
Automorphic numbers can be extended to any such polynomial function of [[degree of a polynomial|degree]] <math>n</math> <math display="inline">f(x) = \sum_{i = 0}^{n} a_i x^i</math> with ''b''-adic coefficients <math>a_i</math>. These generalised automorphic numbers form a [[Tree structure|tree]].

===''a''-automorphic numbers===
An <math>a</math>-'''automorphic number''' occurs when the polynomial function is <math>f(x) = ax^2</math> 

For example, with <math>b = 10</math> and <math>a = 2</math>, as there are two fixed points for <math>f(x) = 2x^2</math> in <math>\mathbb{Z}/10\mathbb{Z}</math> (<math>x = 0</math> and <math>x = 8</math>), according to [[Hensel's lemma]] there are two 10-adic fixed points for <math>f(x) = 2x^2</math>, 
: <math>\ldots 0000000000</math>
: <math>\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>f(x) = x^3</math>.<ref>[http://www.numericana.com/answer/p-adic.htm#decimal See Gérard Michon's article at]</ref> 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.<ref>{{cite OED|spherical number}}</ref>

For base <math>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, ... {{OEIS|id=A033819}}

For base <math>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, ...