Crazy Math Patterns that stop working......eventually

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Just because a mathematical claim is true for a huge number of values, doesn't mean it is ALWAYS true, right? How many values do you think you would check before you said ok, ok, it's pretty much certainty true? In this video we see four pretty insane examples where a pattern appears to work, sometimes for outrageously long time, but eventually fail.

First we look at the pattern 1,2,4,8,16,31 which comes from dividing a circle into the maximal number of areas by connecting n points along the boundary. Then we see Borwein Integrals that appears to be pi/2 forever, but then eventually stop. Check out https://johncarlosbaez.wordpress.com/2018/09/20/patterns-that-eventually-fail/ for more explanations of this one. Next up is prime generators and how we can get the next 39 primes in a row by starting at the number 41 using the formula n^2+n+41. Finally we see the function li(x) which approximates the prime counting function pi(x) which converges via the prime number theorem.

0:00 Thanks for 100k!
0:34 Circle Division Formula
2:36 Browein Integrals
6:21 Prime Generators
9:19 Prime Approximating Functions

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Видео Crazy Math Patterns that stop working......eventually канала Dr. Trefor Bazett