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

(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).

REFERENCES:

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.

Schroeppel, R. Item 59 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 23, Feb. 1972. https://www.inwap.com/pdp10/hbaker/hakmem/number.html#item59.

Sloane, N. J. A. Sequences A003226/M3752, A007185/M3940, A016090, A018247, and A018248 in "The On-Line Encyclopedia of Integer Sequences."

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, England: Penguin Books, pp. 59 and 171, 178, 191-192, 1986.