Automorphic Number
A number 
such that 
has its last
digit(s) equal to 
is called 
-automorphic. For
example, 
(Wells 1986, pp. 58-59)
and 
(Wells 1986, p. 68),
so 5 and 6 are 1-automorphic. Similarly, 
and 
, so 8 and 88 are
2-automorphic. de Guerre and Fairbairn (1968) give a history of automorphic numbers.
The first few 1-automorphic numbers are 1, 5, 6, 25, 76, 376, 625, 9376, 90625, ... (OEIS A003226, Wells 1986, p. 130). There
are two 1-automorphic numbers with a given number of digits, one ending in 5 and
one in 6 (except that the 1-digit automorphic numbers include 1), and each of these
contains the previous number with a digit prepended. Using this fact, it is possible
to construct automorphic numbers having more than 
digits (Madachy
1979). The first few 1-automorphic numbers ending with 5 are 5, 25, 625, 0625, 90625,
... (OEIS A007185), and the first few ending
with 6 are 6, 76, 376, 9376, 09376, ... (OEIS A016090).
The 1-automorphic numbers 
ending in 5
are idempotent (mod 
) since
![[a(n)]^2=a(n) (mod 10^n)](/images/equations/AutomorphicNumber/NumberedEquation1.gif) |
(Sloane and Plouffe 1995).
The following table gives the 10-digit 
-automorphic numbers.
 |  -automorphic numbers | Sloane |
| 1 | 0000000001, 8212890625, 1787109376 | A007185, A016090 |
| 2 | 0893554688 | A030984 |
| 3 | 6666666667, 7262369792, 9404296875 | A030985, A030986 |
| 4 | 0446777344 | A030987 |
| 5 | 3642578125 | A030988 |
| 6 | 3631184896 | A030989 |
| 7 | 7142857143, 4548984375, 1683872768 | A030990, A030991, A030992 |
| 8 | 0223388672 | A030993 |
| 9 | 5754123264, 3134765625, 8888888889 | A030994, A030995 |
The infinite 1-automorphic number ending in 5 is given by ...56259918212890625 (OEIS A018247), while the infinite 1-automorphic number ending in 6 is given by ...740081787109376 (OEIS A018248).
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