“Automorphic Number”的概念、定义、习题解题方法及翻译-数学辞典

A number

such that

has its last digits equal to

is called

-automorphic. For example,

and

are 1-automorphic and

and

are 2-automorphic. deGuerre 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 andone in 6 (except that the 1-digit automorphic numbers include 1), and each of these contains the previous number with adigit 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 fewending with 6 are 6, 76, 376, 9376, 09376, ... (Sloane's A016090). The 1-automorphic numbers

ending in 5 are Idempotent (mod

) since

	-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, --

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. ``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. SequencesA016090,A003226/M3752, andA007185/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.