Toot

Written by Shreevatsa R on 2023-12-16 at 16:14

In 1964, someone named C. P. Willans (about whom nothing else is known AFAICT) published an absurd “formula for the nth prime number”, which is literally correct in a trolling kind of way.

[ p_n = 1 + \sum_{i=1}^{2^n} \left\lfloor \left(\frac{n}{\sum_{j=1}^i \left\lfloor\left(\cos \frac{(j-1)! + 1}{j} \pi\right)^2\right\rfloor }\right)^{1/n} \right\rfloor ]

I know this because I once posted an answer ("No") to the question “Is there a known mathematical equation to find the nth prime?”, at https://math.stackexchange.com/questions/1257 and get the occasional comment and downvote over the years, from people interested in this formula.

A couple of months ago I learned of an excellent video on YouTube that explains this formula step-by-step: https://www.youtube.com/watch?v=j5s0h42GfvM

I also wrote up a quick summary in text form, here: https://shreevatsa.net/post/willans-prime-formula/

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