Automorphic number

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

For example, with $$b = 10$$, there are four 10-adic fixed points of $$f(x) = x^2$$, the last 10 digits of which are:
 * $$\ldots 0000000000$$
 * $$\ldots 0000000001$$
 * $$\ldots 8212890625$$
 * $$\ldots 1787109376$$

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, ... .

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

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

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 $$f(x) = ax^2$$

For example, with $$b = 10$$ and $$a = 2$$, as there are two fixed points for $$f(x) = 2x^2$$ in $$\mathbb{Z}/10\mathbb{Z}$$ ($$x = 0$$ and $$x = 8$$), according to Hensel's lemma there are two 10-adic fixed points for $$f(x) = 2x^2$$,
 * $$\ldots 0000000000$$
 * $$\ldots 0893554688$$

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

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, ...