Counting Primes
February 9, 2016
We have studied functions for counting the prime numbers less than a given number in two previous exercises. All of them were based on Legendre’s phi function that counts the numbers less than x not stricken by sieving with the first a primes.
Today we look at a rather different method of counting the primes that is due to G. H. Hardy and Edward M. Wright. Their method is based on factorials, and their formula is
for n > 3, where ⌊n⌋ is the greatest integer less than n. The expression inside the big parentheses is 1 when n is prime and 0 when n is composite, by Wilson’s Theorem, which states that an integer n > 1 is prime if and only if (n − 1)! ≡ −1 (mod n); the theorem was first stated by Ibn al-Haytham (c. 1000AD), but is named for John Wilson, who first published it in 1770, and it was first proved by Lagrange in 1771.
Your task is to write a program that counts primes by the Hardy-Wright method. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.
Programming Praxis 
Very interesting algo, albeit somewhat impractical. It was a good refresher though for the BigInt type (Julia language) that makes it usable for inputs over 23.
(use-package :iterate) (defun prime-count (n) (+ -1 (iter (for j from 3 to n) (for f first 1 then (* f (- j 2))) (summing (- f (* j (floor f j)))))))My previous comment has a solution in Common Lisp.
from itertools import accumulate, count, islice from operator import mul def pi(n): factorials = accumulate(count(1), mul) summands = (f - (f//j)*j for j,f in enumerate(factorials, 3)) return sum(islice(summands, n-2)) - 1import functools def fact(n): return functools.reduce(lambda x, y: x * y, list(range(1, n + 1))) def primeCounting(n): return (-1 + sum([(fact(j - 2) - j * (fact(j - 2) // j)) for j in range(3, n + 1)]))Two Scheme implementations, or one in two ways. I felt quite stupid for quite a while before I understood that I want to update the factorial with j-1 instead of j-2 when I also update j. (Now I don’t understand why the model implementation also works :)
(define (pi n) ;for n > 3 (do ((S 0 (+ S (- f (* j (quotient f j))))) (f 1 (* (- j 1) f)) ; sic! (j 3 (+ j 1))) ((< n j) (- S 1)))) (define (pi1 n) ;for n > 3 (let loop ((S 0) (f 1) (j 3)) (if (< n j) (- S 1) (loop (+ S (- f (* j (quotient f j)))) (* (- j 1) f) ; sic! (+ j 1)))))[…] this weekend I was going through the Programming Praxis website, and this problem caught my […]