Automorphic Number

A number k such that nk^2 has its last digit(s) equal to k is called n-automorphic. For example, 1·5__^2=25__ (Wells 1986, pp. 58-59) and 1·6__^2=36__ (Wells 1986, p. 68), so 5 and 6 are 1-automorphic. Similarly, 2·8__^2=128__ and 2·88__^2=15488__, 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 25000 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 a(n) ending in 5 are idempotent (mod 10^n) since

 [a(n)]^2=a(n) (mod 10^n)

(Sloane and Plouffe 1995).

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

nn-automorphic numbersSloane
10000000001, 8212890625, 1787109376A007185, A016090
36666666667, 7262369792, 9404296875A030985, A030986
77142857143, 4548984375, 1683872768A030990, A030991, A030992
95754123264, 3134765625, 8888888889A030994, 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).

See also

Idempotent, Narcissistic Number, Number Pyramid, Trimorphic Number

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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, 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., 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.

Referenced on Wolfram|Alpha

Automorphic Number

Cite this as:

Weisstein, Eric W. "Automorphic Number." From MathWorld--A Wolfram Web Resource.

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