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.
digits is an automorphic number if
is a fixed point of the polynomial function
, the ring of integers modulo
-adic integers, automorphic numbers are used to find the numerical representations of the fixed points of
, the last 10 digits of which are: 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).
In the ring of integers modulo
, where the prime omega function
is the number of distinct prime factors in
zeroes or fixed points of a polynomial function modulo
corresponding zeroes or fixed points of the same function modulo any power of
, and this remains true in the inverse limit.
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.
is a prime power, then the ring of
-adic numbers has no zero-divisors other than 0, so the only fixed points of
As a result, nontrivial automorphic numbers, those other than 0 and 1, only exist when the base
has at least two distinct prime factors.
-adic numbers are represented in base
, using A−Z to represent digit values 10 to 35.
Automorphic numbers can be extended to any such polynomial function of degree
These generalised automorphic numbers form a tree.
-automorphic number occurs when the polynomial function is
), 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... 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.
, the trimorphic numbers are: For base