Circles, the Basel Problem, and the Apparent Brightness of Stars

On Pi Day 2016, I wrote in this post about the remarkable fact, discovered by Euler, that the probability that two randomly chosen integers have no prime factors in common is \frac{6}{\pi^2}.  The proof makes use of the  famous identity \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}, often referred to as the “Basel problem”, which is also due to Euler.  In the 2016 post I presented Euler’s original solution to the Basel problem using the Taylor series expansion for \frac{\sin(x)}{x}.

In honor of Pi Day 2018, I’d like to explain a simple and intuitive solution to the Basel problem due to Johan Wästlund.  (Wästlund’s paper is here; see also this YouTube video, which is where I first heard about this approach – thanks to Francis Su for sharing it on Facebook!)  Wästlund’s approach is motivated by physical considerations (the inverse-square law which governs the apparent brightness of a light source) and uses only basic Euclidean geometry and trigonometry.

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Many of you have undoubtedly heard by now the math puzzle about Cheryl’s birthday which has been sweeping across the internet.  I appeared on CNN on Wednesday to explain the solution — here is a link to the problem and my explanation.  Since that appearance, I’ve received dozens of emails about the problem and/or my explanation of it.   I thought I’d share a few of my thoughts following this flurry of activity. Continue reading

Conference on p-adic methods in number theory

After somewhat of a hiatus, I’m back to blogging again.  The purpose of this post is to advertise the conference “p-adic Methods in Number Theory”, which will be held in Berkeley, CA from May 26-30, 2015.  The conference, which I am helping to organize, is in honor of the mathematical legacy of Robert Coleman.  Please spread the word!  Here is the current version of the conference poster, which will be mailed out soon to a math department near you:

Coleman Conference Poster

Many thanks to Janet Ziebell of the Georgia Tech College of Sciences for her help creating this poster, and to Ken McMurdy for designing the conference website.

Here is a memorial article about Robert which I co-authored with Barry Mazur and Ken Ribet.   I encourage you to read it!  It will be published in the new open access journal Research in the Mathematical Sciences, in a special volume dedicated to Robert.

You can find other interesting links related to Robert Coleman’s life and work here, and in this older blog post of mine.