Automorphic Number - Other Radixes

Other Radixes

Automorphic numbers are radix dependent, and the description above applies to automorphic numbers in base 10. Using other radixes there are different automorphic numbers. 0 and 1 are automorphic numbers in any radix; automorphic numbers other than 0 and 1 only exist when the radix has at least two distinct prime factors.

A single digit number x is automorphic in radix b > x when b divides x2 − x. So 6 is automorphic in a radix which is a divisor of 62 − 6 = 30 that is greater than 6; these divisors are 10, 15 and 30.

In any given radix there are 2p sequences of automorphic numbers where p is the number of distinct prime factors in the radix. For base 10 this gives 22 = 4 sequences, which are 0,1,5 and 6 for 1 digit or 00, 01, 25, 76 for two digits and so on. A prime radix (such as 2,3,4,5,7,8,9,11,13,16,17,...) can only have 0 and 1 (prepended by one or more zeroes) as automorphic numbers. Base 6 is the first radix with non-trivial automorphic numbers and base 15 the first such odd radix. Base 30 is the first radix with three distinct prime factors and has 8 sequences of automorphic numbers. Here some examples of non-trivial 1,2 and 4 digit automorphic numbers in other radixes (using A-Z except I and O to represent digits 10 to 34):

Note that the base 30 numbers expressed in decimal are also automorphic in the last 4 digits.