Thursday, September 22, 2016
Entry 7
Ramanujan found the nice trigonometric relation, 3√2cos(2π7)+3√2cos(4π7)+3√2cos(6π7)=−3√−5+3⋅71/3=−0.904… Equivalently, let xi be the three roots of the cubic x3+x2−2x−1=0. Then 3∑k=1x1/3k=−3√−5+3⋅71/3=−0.904… The natural question to ask is, can this be generalized to quintics and cos(2π11)? It turns out it can. Noam Elkies found that if we let yi be the five roots of the quintic y5+6y4−y3−32y2+16y−1=0, then 5∑k=1y1/5k=−5√−274+5(−21⋅111/5+13⋅112/5+13⋅113/5)=−0.093… The yi can also be expressed as cosines yk=−(z2k−1)2−(zk−1),andzk=2cos(2πk11) Similarly, one can find a 7th deg relation using cos(2π29) and so on.
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Can you explain in more detail how these interesting identities can be derived?
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