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        riddles >> putnam exam (pure math) >> Parity with powers and the greatest integer
        
(Message started by: ecoist on Aug 14^th, 2006, 5:21pm)

Title: Parity with powers and the greatest integer
Post by ecoist on Aug 14^th, 2006, 5:21pm

(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)?

Title: Re: Parity with powers and the greatest integer
Post by towr on Aug 15^th, 2006, 12:45am

Sounds familiar, I'd go with yes.

[hide]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.[/hide]

Title: Re: Parity with powers and the greatest integer
Post by Eigenray on Aug 17^th, 2006, 1:16pm

[link=http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1138301013]Yes[/link], in fact uncountably many such.