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Topic: n-ugly numbers and perfect squares (Read 874 times) |
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Aryabhatta
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n-ugly numbers and perfect squares
« on: Jun 16th, 2007, 9:36am » |
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A number is called n-ugly if it is only divisible by primes among the first n primes.
You are given an arbitrary set U of n+1 n-ugly numbers.
Show that there is some non-empty subset of U with the property that the product of its elements is a perfect square.
Show that there is some set V of n n-ugly numbers without this property.
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towr
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Re: n-ugly numbers and perfect squares
« Reply #1 on: Jun 16th, 2007, 10:05am » |
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You could make a bitstring for each number, where for each the ith prime the ith bit is set if the highest power of that prime dividing the number is odd. Now the problem is akin to the claim there is a non-empty subset of these n+1 bitstrings for which the XOR to 0.
I'm not sure what you mean with the last part. Is V still a subset of U?
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Aryabhatta
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Re: n-ugly numbers and perfect squares
« Reply #2 on: Jun 16th, 2007, 10:18am » |
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No, U and V are unrelated.
(I admit, the part about V is trivial, it was only put there for completeness)
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| « Last Edit: Jun 17th, 2007, 11:23am by Aryabhatta » |
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rmsgrey
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Re: n-ugly numbers and perfect squares
« Reply #3 on: Jun 18th, 2007, 7:32am » |
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Well, since it's trivial:
V={pi}
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Barukh
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Re: n-ugly numbers and perfect squares
« Reply #4 on: Jun 18th, 2007, 11:07pm » |
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Is the U-part solved?
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towr
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Re: n-ugly numbers and perfect squares
« Reply #5 on: Jun 19th, 2007, 12:45am » |
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on Jun 18th, 2007, 11:07pm, Barukh wrote:Not in this thread.
I just restated the problem in other terms for a different perspective. I haven't given it much more thought yet.
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| « Last Edit: Jun 19th, 2007, 12:52am by towr » |
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Eigenray
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Re: n-ugly numbers and perfect squares
« Reply #6 on: Jun 20th, 2007, 9:22pm » |
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on Jun 16th, 2007, 10:05am, towr wrote:| You could make a bitstring for each number, where for each the ith prime the ith bit is set if the highest power of that prime dividing the number is odd. |
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And then you could take all those bits and put them in an array.
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Barukh
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Re: n-ugly numbers and perfect squares
« Reply #7 on: Jun 20th, 2007, 11:59pm » |
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Aha! Linear dependency?
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Aryabhatta
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Re: n-ugly numbers and perfect squares
« Reply #8 on: Jun 21st, 2007, 10:11am » |
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Correct! Barukh. You got it!
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