Sieve of Eratosthenes (Forth)

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The Sieve of Eratosthenes is an algorithm for rapidly locating all the prime numbers in a certain range of integers. It operates by marking as composite all nontrivial multiples of each prime in sequence until only primes remain unmarked. It is most well-suited to implementation using arrays, which are compact and feature random access.

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This program counts the number of primes up to 7919 (the 1000th prime) using the Sieve of Eratosthenes. The algorithm is optimized by assuming 2 is prime, and thus only examining odd numbers.

( dup .) can be uncommented to list the primes found.

7919 2/ CONSTANT maxp      \ 2/ because we only count odd primes
: PRIMES ( -- n )
  HERE maxp 1 FILL
  1 ( count, including 2 )
  maxp 0 DO
    I HERE + C@ IF
      I 2* 3 + ( dup .) DUP  I + ( prime current )
      BEGIN  DUP maxp U<
      WHILE  0 OVER HERE + C!
             OVER +
      2DROP 1+
  LOOP ;
PRIMES .    \ 1000
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