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?
[ meaning "if s in S , then f(s) is a subset of S " ] in case anyone is still interested ... i completed a depth-first search through 50 generations. the largest number arising within 50 generations has 110 decimal digits: 3457746548572498971631711458544819351439214287725631661- 1563654569796733285181233292965618589219724572135111824 i haven't had the chance to code breadth-first search or a hybrid approach. mike
----- Original Message ----- From: "Michael Reid" <reid@math.ucf.edu> To: <math-fun@mailman.xmission.com> Sent: Saturday, January 28, 2006 7:49 PM Subject: Re: [math-fun] Digital silliness
in case anyone is still interested ... i completed a depth-first search through 50 generations. the largest number arising within 50 generations has 110 decimal digits:
3457746548572498971631711458544819351439214287725631661- 1563654569796733285181233292965618589219724572135111824
i haven't had the chance to code breadth-first search or a hybrid approach.
mike
I'm guessing any approach will fail. In order to find and verify the largest number, I cannot see how we can avoid examining every number in the set. If this set includes even one 110-digit number, it is reasonable to think it contains some modest fraction of, say, the 50-digit numbers. Even a modest fraction (say 10^-10) of the 50-digit numbers would not be tractable to process by any algorithm. It might be worth a try to run this algorithm in small numerical bases to get a feel for the size and largest element of the base-10 set in order to judge its tractability.
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David Wilson -
Michael Reid