Yes, you have discerned my intended interpretation. It is true that all of the small numbers without digit 0 are in S. However, if you pick a sufficiently long random number with no zeroes and square it, the result will almost certainly include zeroes with about the expected frequency of 1 digit in 10. This means that for sufficiently long n, the blocks of nonzero digits in n^2 should be almost always be shorter than n. This would argue that S is finite. A better experiment might be to try to find the largest possible elements you can in S. ----- Original Message ----- From: "Edwin Clark" <eclark@math.usf.edu> To: "math-fun" <math-fun@mailman.xmission.com> Cc: "Sequence Fans" <seqfan@ext.jussieu.fr> Sent: Wednesday, January 11, 2006 12:40 AM Subject: Re: [math-fun] Digital silliness
On Tue, 10 Jan 2006, David Wilson wrote:
For a number n, let f(n) be the set of numbers gotten by splitting n^2 at the 0 digits. For example
29648^2 = 879003904
so f(29648) = { 4, 39, 879 }
Let S be the smallest set of numbers containing 2 and fixed by f. What is the largest element of S?
I assume you mean by S: The smallest set of positive integers satisfying
1. 2 is in S 2. if n is in S then f(n) is a subset of S
In this case, my (not very extensive) experiments lead me to conjecture that every positive integer which contains no 0 is in S.
--Edwin