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Topic: Polynomial (Read 823 times) |
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cool_joh
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Polynomial
« on: Aug 18th, 2008, 12:50am » |
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Find all monic integer polynomials p(x) of degree two for which there exists an integer polynomial q(x) such that p(x)q(x) is a polynomial having all
coeffcients ±1.
What if p(x)q(x) is allowed to have zero coefficients?
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Hippo
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Re: Polynomial
« Reply #1 on: Aug 18th, 2008, 2:06am » |
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Equivalently: find a,b such that the sequence A0=0,A1=1,An+2=aAn+1+bAn±1 has only finite number of nonzero elements (for appropriate choice of ± signs).
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| « Last Edit: Aug 18th, 2008, 4:26pm by Hippo » |
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Hippo
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Re: Polynomial
« Reply #2 on: Aug 18th, 2008, 2:57am » |
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So when An+1=An+2=0 we get bAn=±1 therefore b=±1 and An=±1.
Hmmm, my choice of variabes p,q may be rather confusing with independent polynomial variables p(x),q(x) in the original question  ... I've converted it to a,b.
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| « Last Edit: Aug 18th, 2008, 4:27pm by Hippo » |
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Eigenray
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Re: Polynomial
« Reply #3 on: Aug 20th, 2008, 4:39am » |
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This one finally just hit me: the roots of p(x) can't be too big.
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Hippo
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Re: Polynomial
« Reply #4 on: Aug 20th, 2008, 1:12pm » |
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on Aug 20th, 2008, 4:39am, Eigenray wrote:| This one finally just hit me: the roots of p(x) can't be too big. |
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Yes, this is much nicer solution  (at least my one was not finished).
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| « Last Edit: Aug 20th, 2008, 1:13pm by Hippo » |
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