Mathematical_Recreations-Kraitchik-2e

Numerical Pastimes 77
8. AUTOMORPHIC NUMBERS
A number is called automorphic in the scale of notation
with base B if all of its powers end in the same digits in this
scale of notation.* If n is the number of digits appearing at
the end of every power of the automorphic number N, we
have N'" == N (modulo Bn), for x > O. But this will be satisfied
if N2 == N (modulo Bn).
Suppose first that B is a prime.
Then the congruence
N2 == N (modulo Bn) has only the solutions N == 0 and
N == 1 (modulo Bn).
N ow suppose B composite, say B = 6 = 2·3. The congruence
N2 == N (modulo 6n ) may be replaced by the simultaneous
congruences, N2 == N (modulo 2") and N2 == N
(modulo 3"). These have the solutions, N == 0 or 1 (modulo 2")
and N == 0 or 1 (modulo 3"), from which we find four solutions
modulo 6". Of these, two are trivial, namely N == 0
and N == 1 (modulo 6"). The other two are given by N == 0
(modulo 2") and 1 (modulo 3"), or N == 1 (modulo 2") and
o (modulo 3"). Thus the numbers (in the scale with base 6)
which end in 4, 44, 344, and so on or in 3, 13, 213, and so
on, are automorphic.
In the usual scale of notation, with base 10 =
2·5, we find
similarly the trivial solutions, N == 0 and N == 1 (modulo
10"), and the true automorphic numbers given by N == 0
(modulo 2") and 1 (modulo 5"), or N == 1 (modulo 2") and 0
(modulo 5"). Thus numbers ending in 6, 76, 376, ... , or in 5,
25, 625, "., are automorphic.
* The terminology as regards cyclic numbers and automorphic numbers
is not as yet fixed by usage. The lexicographers of Webster's New International
Dictionary, Second Edition, describe as cyclic numbers or circular
numbers those numbers which are here called automorphic; they
do not define the term automorphic as it applies to numbers. Of necessity,
we must therefore define the terms and use them according to our definitions.