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Topic: Parity with powers and the greatest integer (Read 453 times) |
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ecoist
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Parity with powers and the greatest integer
« on: Aug 14th, 2006, 5:21pm » |
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(Wish I had thought of this one!)
Is there a positive real number r such that, for all positive integers n, |rn| (equal the greatest integer less or equal rn) has the same parity as n (i.e., are congruent modulo 2)?
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towr
wu::riddles Moderator
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Some people are average, some are just mean.
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Re: Parity with powers and the greatest integer
« Reply #1 on: Aug 15th, 2006, 12:45am » |
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Sounds familiar, I'd go with yes.
If you can find an r and s with -1 < s < 0 and r^n - s^n = 0 (mod 2), and are the solutions to a quadratic you get when trying to find the closed form for an linear integer recurrence equation, then |r^n| alternates between 1 and 0 (mod 2)
There's should be another thread on it somewhere.
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| « Last Edit: Aug 15th, 2006, 12:51am by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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Eigenray
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Re: Parity with powers and the greatest integer
« Reply #2 on: Aug 17th, 2006, 1:16pm » |
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Yes, in fact uncountably many such.
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