Automorphic Number

A number $k$ such that $nk^2$ has its last digits equal to $k$ is called $n$-automorphic. For example, $1\cdot{\underline{5}}^2 = 2{\underline{5}}$ and $1\cdot{\underline{6}}^2 = 3{\underline{6}}$ are 1-automorphic and $2\cdot{\underline{8}}^2=12{\underline{8}}$ and $2\cdot{\underline{88}}^2=154{\underline{88}}$ 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, ... (Sloane's 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 25,000 digits (Madachy 1979). The first few 1-automorphic numbers ending with 5 are 5, 25, 625, 0625, 90625, ... (Sloane's A007185), and the first few ending with 6 are 6, 76, 376, 9376, 09376, ... (Sloane's A016090). The 1-automorphic numbers $a(n)$ ending in 5 are Idempotent (mod $10^n$) since

$$[a(n)]^2\equiv a(n)\ \left({{\rm mod\ } {10^n}}\right)$$

(Sloane and Plouffe 1995).

The following table gives the 10-digit $n$-automorphic numbers.

$n$	$n$-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, --

See also Idempotent, Narcissistic Number, Number Pyramid, Trimorphic Number

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 59 in HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Fairbairn, R. A. ``More on Automorphic Numbers.'' J. Recr. Math. 2, 170-174, 1969.

Fairbairn, R. A. Erratum to ``More on Automorphic Numbers.'' J. Recr. Math. 2, 245, 1969.

de Guerre, V. and Fairbairn, R. A. ``Automorphic Numbers.'' J. Recr. Math. 1, 173-179, 1968.

Hunter, J. A. H. ``Two Very Special Numbers.'' Fib. Quart. 2, 230, 1964.

Hunter, J. A. H. ``Some Polyautomorphic Numbers.'' J. Recr. Math. 5, 27, 1972.

Kraitchik, M. ``Automorphic Numbers.'' §3.8 in Mathematical Recreations. New York: W. W. Norton, pp. 77-78, 1942.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 34-54 and 175-176, 1979.

Sloane, N. J. A. Sequences A016090, A003226/M3752, and A007185/M3940, in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex: Penguin Books, pp. 171, 178, 191-192, 1986.