Can someone prove that \(\displaystyle{\sum_{{{i}={1}}}^{{{89}}}}{{\sin}^{{{2}{n}}}{\left({\frac{{\pi}}{{{180}}}}{i}\right)}}\) is a

Riley Quinn

Riley Quinn

Answered question


Can someone prove that i=189sin2n(π180i) is a dyadic rational for all positive integers n, or find counterexample?

Answer & Explanation



Beginner2022-03-17Added 4 answers

That last sum (over j) is a sum of roots of unity - specifically, if d=GCD(mn,k), the last sum is the sum of the kd roots of unity d times. So the sum is equal to 0, except when d=k, in which case all the terms are 1 and the sum is k. This implies that the summand of the sum over m is an integer, so the entire expression is an integer divided by 4n.
In the case that n<k, we get the simple expression
For the original problem I posed, we have
and we have proven that the last sum is a dyadic ratioanl, so j=189sin2n(πi180) is a dyadic rational as well.

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