Re: [math-fun] Digital silliness
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?
perhaps you mean "if s in S , then f(s) is a subset of S ". it is not obvious to me that S is finite. it is easy to exhibit arbitrarily large integers that do not have 0 as a decimal digit, and whose squares also do not have any 0's . of course this does not show that your S is infinite, but raises that issue as a possibility. is there reason to believe (i.e. an heuristic argument) that S is finite? or better yet, a proof? mike
There are many ways to vary this problem, most notably by changing the base b, the set of initial numbers A, and the set of splitting digits, which I will call D. Case 1: b = 10, A = {2}, D = {1}. In this case S is infinite, containing an infinitude of numbers of the form 6*10^k. Case 2: b = 10, A = {2}, D = {0,1,5}, In this case S is finite, with largest element 6347862777922. Case 3: b = 10, A = {2}, D = {0}. Finiteness of S unknown. I wrote a program to generate elements for S for b = 10 and a given A and D, and print a list of largest elements encountered. This program quickly exposed the digital patterns in Case 1 that could be exploited to prove that S was infinite, as happened for several other cases with infinite S. For Case 2, and other cases with finite S, there were obviously no digital patterns that could be used to prove the infiniteness of S, and the elements printed by the program appeared patternless. When I ran the program on Case 3, no obvious digital patterns were exposed. This argues strongly, but not conclusively, that S is finite. ----- Original Message ----- From: "Michael Reid" <reid@math.ucf.edu> To: <math-fun@mailman.xmission.com> Sent: Tuesday, January 10, 2006 5:41 PM Subject: Re: [math-fun] Digital silliness
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?
perhaps you mean "if s in S , then f(s) is a subset of S ".
it is not obvious to me that S is finite. it is easy to exhibit arbitrarily large integers that do not have 0 as a decimal digit, and whose squares also do not have any 0's . of course this does not show that your S is infinite, but raises that issue as a possibility.
is there reason to believe (i.e. an heuristic argument) that S is finite? or better yet, a proof?
mike
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