Summing Complex Fractional Exponents

Algebra Level 5

\[\text{A} = \sum_{r=0}^{2016} \exp \left(\frac{i 2016\pi r^2}{2017}\right), \qquad \text{B} = \sum_{r=0}^{2015} \exp \left(-\frac{i 2017\pi r^2}{2016}\right) \]

Given that \(\text{A}\) and \(\text{B}\) are defined as above, \(\Re \left( \dfrac {\text{A}}{\text{B}} \right)\) can be written as \(\sqrt{\dfrac{p}{q}}\) for coprime positive integers \(p\) and \(q\). Evaluate \(p+q\).


  • \(\exp (x) = e^x\), where \(e \approx 2.71828\) denotes the Euler's number.
  • \( i = \sqrt{-1} \).
  • If \(a\) and \(b\) are real numbers, then \(\Re(a+bi) = a\).

Problem Loading...

Note Loading...

Set Loading...