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 December 6th, 2015, 03:52 PM #1 Member   Joined: Jun 2015 From: Ohio Posts: 99 Thanks: 19 Putnam Exam Question Most of the questions from this year's Putnam exam completely stumped me, but here was one I was very curious about. Compute $\displaystyle \log_2 \prod_{a=1}^{2015} \prod_{b=1}^{2015} (1 + e^{2 \pi i ab/2015})$ I had no idea how to do this. I thought I did at first, but then I remembered typical exponent laws don't work when you're working with complex numbers so I couldn't turn the $\displaystyle e^{2 \pi i}$ into 1 (since there were other things in the exponent). Anybody know how to do this?
 December 9th, 2015, 09:14 AM #2 Member   Joined: Oct 2013 Posts: 60 Thanks: 6 Hard for calculating a double prod up to 2015. Even for a computer build in 2015! We know: 2015 = 5*13*31 replace 2015 by 5 and calculate 9 replace 2015 by 13 and calculate 25 replace 2015 by 31 and calculate 61 finally, calculate the answer 9*25*61 = 13725 math is fun!
 December 9th, 2015, 10:03 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra I think it's just a case of counting the conjugate pairs.

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