An Exact Formula for the Primes: Willans' Formula

Thanks to my supporters on Patreon! Get early access to videos and more: https://www.patreon.com/EricRowland Formulas for the nth prime number actually exist! One was cleverly engineered in 1964 by C. P. Willans. But is it useful? ---------------- References: Herbert Wilf, What is an answer?, The American Mathematical Monthly 89 (1982) 289–292. https://doi.org/10.1080/00029890.1982.11995435 C. P. Willans, On formulae for the nth prime number, The Mathematical Gazette 48 (1964) 413–415. https://doi.org/10.2307/3611701 Further reading: Jeffrey Shallit, No formula for the prime numbers?. http://recursed.blogspot.com/2013/01/no-formula-for-prime-numbers.html ---------------- # Python code import math def prime(n): return 1 + sum([ math.floor(pow(n/sum([ math.floor(pow(math.cos(math.pi * (math.factorial(j - 1) + 1)/j), 2)) for j in range(1, i+1) ]), 1/n)) for i in range(1, pow(2, n)+1) ]) ---------------- (* Mathematica code *) prime[n_] := 1 + Sum[Floor[(n/Sum[Floor[Cos[Pi ((j - 1)! + 1)/j]^2], {j, 1, i}])^(1/n)], {i, 1, 2^n}] ---------------- 0:00 A formula for primes? 1:24 Engineering a prime detector 4:00 Improving the prime detector 5:46 Counting primes 6:29 Determining the nth prime 9:42 The final step 11:36 What counts as a formula? 12:56 What's the point? 13:51 Who was Willans? ---------------- Animated with Manim. https://www.manim.community Thanks to Ken Emmer for supplying the microphone. Web site: https://ericrowland.github.io