An Exact Formula for the Primes: Willans’ Formula

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. C. P. Willans, On formulae for the nth prime number, The Mathematical Gazette 48 (1964) 413-415. Further reading: Jeffrey Shallit, No formula for the prime numbers?. ---------------- # Python code import math def prime(n): return 1 sum([ (pow(n/sum([ (pow(( * ((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. Thanks to Ken Emmer for supplying the microphone. Web site: Twitter: