/sci/ - Science & Math

>>9950380
The method that Euler deduced for generating Euler's totient function and Euler's function (partition function) using generalised pentagonal numbers is quite stunningly beautiful.

Norman Wildberger has an excellent, and probably the best, video on it:
https://www.youtube.com/watch?v=1mSk3J3GlA8

I made a gif of an Ulam spiral spiralling towards Euler's polynomial: [math]x^2 + x + 41[/math], which generates 40 consecutive primes when [math]0 \leqslant x \leqslant 39[/math]. These primes are (41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601), and they can be seen on the gif forming the long diagonal passing through the centre of the final frame.

I am not sure if he is "THE" smartest man in the history of humanity, but he is likely in a shortlist of candidates who qualify for such a statement.