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Find least $n$ so that a certain polynomial exists.
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freemind
Riemann Hypothesis
Riemann Hypothesis


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#1
Find least $n$ so that a certain polynomial exists.
Moldova National MO 2008, 12 Grade, Problem 5 (Day 2)
Find the least positive integer n so that the polynomial P(X)=\sqrt3\cdot X^{n+1}-X^n-1 has at least one root of modulus 1.
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The fate of equilibrium is to end the eternity...

PostPosted: Sun Mar 02, 2008 4:50 pm
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Birch & Swinnerton Dyer
Birch & Swinnerton Dyer

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#2
Let X = exp(2\pi \phi i), then we get \phi n = \pm \frac 16 \mod 1, \ \phi (n + 1) = \frac {1}{12} \mod 1.
It give n = 10\mod 12 and exactly 2 roots X = exp(\frac {\pm \pi i}{6}).

PostPosted: Mon Mar 03, 2008 6:53 am
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