Superautomorphic Numbers

1. An automorphic number is generally understood to be an integer the final digits of whose square (and therefore of any integral power) form the original number, i.e. a is automorphic if a 2 ≡ a (mod 10 n ), where n is the number of digits of a . There are several references to these in books on re...

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Bibliographic Details
Published in:Mathematical gazette 1991-10, Vol.75 (473), p.277-289
Main Author: Pargeter, A. Robert
Format: Article
Language:English
Subjects:
Inflorescences
Integers
Mathematical tables
Mathematical theorems
Numbers
Online Access:Get full text
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Summary:1. An automorphic number is generally understood to be an integer the final digits of whose square (and therefore of any integral power) form the original number, i.e. a is automorphic if a 2 ≡ a (mod 10 n ), where n is the number of digits of a . There are several references to these in books on recreational mathematics, some using also bases other than 10, but otherwise generally confined to definitions and examples. Investigating some elementary properties that I had not seen treated in print led me to submit an article to the Gazette, which was published in October 1989 (no. 465, p 212). This caught the attention of Dr R. J. Cook, of Frome, who kindly sent me the results of some experiments it had led him to make on applying the idea to indices > 2. These results were surprising and at first puzzling, but a lengthy subsequent correspondence has led to the emergence of a remarkably complicated and quite fascinating picture, which this article is an attempt to summarise.
ISSN:0025-5572
2056-6328
DOI:10.2307/3619485