ghgonnet (06/21/82)
Many years ago, we met a nutty, self-taught type who was searching for what he called "numeros lindos" (pretty numbers). These numbers have the property that when squared (or any power) the last digits coincide with the initial number. E.g. 109376 ** 2 = 11963109376. How many lindos are there with n digits? How to generate them? Is this problem known in the literature? (In base 10 this problem is not trivial, since 10 is not a prime) Gaston H. Gonnet Waterloo
G:shallit (06/22/82)
The problem is very well-known. These numbers are sometimes called "automorphic" numbers in the literature. Martin Gardner has written about them in his Mathematical Games column in Scientific American (but I can't remember when). If my memory is correct, it's not hard to prove there are exactly 2 n-digit automorphs (in base 10) for every n. Their sum is 1+10^n. Also, the Journal of Recreational Mathematics has many articles on generalization to other bases, powers, etc.
davide (06/23/82)
Oops... In my previous reply to the numeros lindos query, I said there was
only one "pretty number" of n digits. shallit stated in his follow-article
that there were exactly two for every n. This is close, but not completely
correct. There are two for most n.
In my previous reply, I mentioned the series that begins 6, 76, 376, 9376,
109376. I forgot the series that starts 5, 25, 625, 90625, 890625. Note
that for n=1,2,3,6 there are two automorphic numbers in base 10, but for
n=4,5 there are only two. For higher values of n, there are *usually* two.
David Eby
Tektronix, Instrument Division