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

digits is an automorphic number is a fixed point of the polynomial function, the ring of integers modulo. As the inverse limit, the ring of b-adic.

, 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, ... .

is a zero, where the prime omega function is the number of distinct prime factors if and only if. According to Hensel's lemma, and this remains true in the inverse limit.

As 0 is always a zero-divisor, and 0 and 1 are automorphic numbers in every base. These solutions are called trivial automorphic numbers is a prime power are 0 and 1. As a result, nontrivial automorphic numbers has at least two distinct prime factors.

Automorphic numbers 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 with b. These generalised automorphic numbers form a tree.

a-automorphic numbers

-automorphic number

), according to Hensel's lemma,

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

Trimorphic numbers

A trimorphic number or spherical number. 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:

1. 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, ...

, the trimorphic numbers are:

1. 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

See also

• Arithmetic dynamics

• Kaprekar number

• p-adic number

• p-adic analysis

• Zero-divisor

References

External links