I did not know — until I read Gamma (page 168) — that there actually is a formula for generating 
th prime number by directly plugging in . The catch is that it’s a mess:
![p_n = 1 + \displaystyle \sum_{m=1}^{2^n} \left[ n^{1/n} \left( \sum_{i=1}^m \cos^2 \left( \pi \frac{(i-1)!+1}{i} \right) \right)^{-1/n} \right]](https://s0.wp.com/latex.php?latex=p_n+%3D+1+%2B+%5Cdisplaystyle+%5Csum_%7Bm%3D1%7D%5E%7B2%5En%7D+%5Cleft%5B+n%5E%7B1%2Fn%7D+%5Cleft%28+%5Csum_%7Bi%3D1%7D%5Em+%5Ccos%5E2+%5Cleft%28+%5Cpi+%5Cfrac%7B%28i-1%29%21%2B1%7D%7Bi%7D+%5Cright%29+%5Cright%29%5E%7B-1%2Fn%7D+%5Cright%5D&bg=f3f3f3&fg=888888&s=0 "p_n = 1 + \displaystyle \sum_{m=1}^{2^n} \left[ n^{1/n} \left( \sum_{i=1}^m \cos^2 \left( \pi \frac{(i-1)!+1}{i} \right) \right)^{-1/n} \right]")
,
where the outer brackets ![[~ ]](https://s0.wp.com/latex.php?latex=%5B%7E+%5D&bg=f3f3f3&fg=888888&s=0 "[~ ]")
represent the floor function.
This mathematical curiosity has no practical value, as determining the 10th prime number would require computing 
different terms!
When I researching for my series of posts on conditional convergence, especially examples related to the constant 
, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.
Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.
In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.
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Posted by John Quintanilla on October 15, 2016
https://meangreenmath.com/2016/10/15/what-i-learned-from-reading-gamma-exploring-eulers-constant-by-julian-havil-part-15/
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