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: 57 Thanks: 5 
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: 7,635 Thanks: 2620 Math Focus: Mainly analysis and algebra 
I think it's just a case of counting the conjugate pairs.


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