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Automorphic number

Number whose square ends in the same digits

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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)=x2 over Z/bkZ, the ring of integers modulo bk. As the inverse limit of Z/bkZ is Zb, the ring of b-adic integers, automorphic numbers are used to find the numerical representations of the fixed points of f(x)=x2 over Zb.

For example, with b=10, there are four 10-adic fixed points of f(x)=x2, the last 10 digits of which are:

…0000000000
…0000000001
…8212890625 (sequence A018247 in the OEIS)
…1787109376 (sequence A018248 in the OEIS)

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).

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 2ω(b) zeroes to g(x)=x2−x, where the prime omega function ω(b) is the number of distinct prime factors in b. An element x in Z/bZ is a zero of g(x)=x2−x if and only if x≡0modpvp(b) or x≡1modpvp(b) for all p|b. Since there are two possible values in {0,1}, and there are ω(b) such p|b, there are 2ω(b) zeroes of g(x)=x2−x, and thus there are 2ω(b) fixed points of f(x)=x2. 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 2ω(b) b-adic fixed points of f(x)=x2.

As 0 is always a zero-divisor, 0 and 1 are always fixed points of f(x)=x2, 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)=x2 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.

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