Re: [math-fun] Knuth's Boolean Synthesis problem; digit silliness
Rich asks: << Do all such sans-0-digit strings appear somewhere in the square of a sans-0-digit number? Statistically, it seems unlikely that most such strings appear in the square of a *shorter* s0d number. This would mean that, if all s0d numbers occur in the closure of {2}, most of them are reached through a path that involves shrinking down from a larger number. Second question: Do all the s0d numbers generate the same set? Or, Do all s0d numbers generate 2?
Thinking of a digit string s as defining s/10^p in [0,1), it's clear that by going out enough places the distance between successive squares (k+1)^2/10^2r - k^2/10^2r = (2k+1)/10^2r can be made small enough so that there exist r = r(p) sufficiently large and k = k(p,r) such that .s < k^2/10^2r < .s + 1/10^p, ensuring that s occurs as a substring of k^2. --Da
[replying to Dan's remark below] Sure. But can you make an s0d string that occurs as a substring in the square of an s0d number? If you have a construction, we could likely fix up the surrounding 0s. Possible construction idea: If N is an s0d number, then the square of N000...001 will have 2N in the middle. So N000...005 will have 10N in the middle, which is equivalent to N when the 0s are stripped away. We can ditch the blob of 000...00 in the root by complementing, using 999N'999...995 as the squaree, with N' being the 9s complement of N. But the proposed root will fail as an s0d number if N contains a 9. Maybe N can be split into L+M and a root designed that adds L and M together, while L' and M' are s0d. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Daniel Asimov Sent: Thu 1/12/2006 10:33 AM To: math-fun Subject: Re: [math-fun] Knuth's Boolean Synthesis problem; digit silliness Rich asks: << Do all such sans-0-digit strings appear somewhere in the square of a sans-0-digit number? Statistically, it seems unlikely that most such strings appear in the square of a *shorter* s0d number. This would mean that, if all s0d numbers occur in the closure of {2}, most of them are reached through a path that involves shrinking down from a larger number. Second question: Do all the s0d numbers generate the same set? Or, Do all s0d numbers generate 2?
Thinking of a digit string s as defining s/10^p in [0,1), it's clear that by going out enough places the distance between successive squares (k+1)^2/10^2r - k^2/10^2r = (2k+1)/10^2r can be made small enough so that there exist r = r(p) sufficiently large and k = k(p,r) such that .s < k^2/10^2r < .s + 1/10^p, ensuring that s occurs as a substring of k^2. --Da _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
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Daniel Asimov -
Schroeppel, Richard