Step by Step solution about How to resolve the algorithm Sieve of Eratosthenes step by step in the Mathematica/Wolfram Language programming language

Eratosthenes Sieve Algorithm in Wolfram Language

The given Wolfram code implements the Eratosthenes sieve algorithm to find all prime numbers up to a given integer n. The sieve algorithm works by iteratively marking off multiples of each prime number starting at 2.

First Implementation:

Eratosthenes[n_] := Module[{numbers = Range[n]},
 Do[If[numbers[[i]] != 0, Do[numbers[[i j]] = 0, {j, 2, n/i}]], {i, 2, Sqrt[n]}];
 Select[numbers, # > 1 &]]

Explanation:

1. Initialize numbers as a list of integers from 1 to n.
2. Iterate from 2 to the square root of n. For each i in this range:
   • If numbers[[i]] is not 0 (meaning it hasn't been marked off), it is a prime number.
   • Mark off all multiples of i in the numbers list.
3. Use Select to filter out the non-prime numbers (# > 1 & condition) and return a list of prime numbers.

Second Implementation:

Eratosthenes[n_] := Module[{numbers = Range[n]},
 Do[If[numbers[[i]] != 0, Do[numbers[[j]] = 0, {j,i i,n,i}]], {i,2,Sqrt[n]}];
 Select[numbers, # > 1 &]]

Explanation:

This implementation is similar to the first one but uses a different way to generate multiples of i. It iterates through the remaining non-marked numbers starting from i i with a step size of i.

Third Implementation:

Eratosthenes2[n_] := Module[{numbers = Range[3, n, 2], limit = (n - 1)/2}, 
 Do[c = numbers[[i]]; If[c != 0,
   Do[numbers[[j]] = 0, {j,(c c - 1)/2,limit,c}]], {i,1,(Sqrt[n] - 1)/2}];
 Prepend[Select[numbers, # > 1 &], 2]]

Explanation:

This implementation optimizes the algorithm by considering only odd numbers (since even numbers greater than 2 are not prime). It initializes numbers as a list of odd numbers from 3 to n. The limit variable is calculated to avoid unnecessary iterations.

1. Iterate from 1 to (Sqrt[n] - 1)/2. For each i:
   • Let c be the corresponding odd number in the numbers list.
   • If c is not 0 (meaning it hasn't been marked off), mark off all multiples of c in the numbers list.
2. Prepend the number 2 to the list of selected prime numbers because it is a special case and is not included in the odd number range.

Eratosthenes[n_] := Module[{numbers = Range[n]},
  Do[If[numbers[[i]] != 0, Do[numbers[[i j]] = 0, {j, 2, n/i}]], {i, 
    2, Sqrt[n]}];
  Select[numbers, # > 1 &]]

Eratosthenes[100]

Eratosthenes[n_] := Module[{numbers = Range[n]},
  Do[If[numbers[[i]] != 0, Do[numbers[[j]] = 0, {j,i i,n,i}]],{i,2,Sqrt[n]}];
  Select[numbers, # > 1 &]]

Eratosthenes[100]

Eratosthenes2[n_] := Module[{numbers = Range[3, n, 2], limit = (n - 1)/2}, 
  Do[c = numbers[[i]]; If[c != 0,
    Do[numbers[[j]] = 0, {j,(c c - 1)/2,limit,c}]], {i,1,(Sqrt[n] - 1)/2}];
  Prepend[Select[numbers, # > 1 &], 2]]

Eratosthenes2[100]