Superautomorphic numbers

Extract

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 a² ≡ a (mod 10ⁿ), 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.