Programa para la Conjetura de Goldbach (Dos primos con suma dada)

La conjetura de Goldbach es uno de los problemas sin resolver más antiguos y conocidos de la teoría de números de las matemáticas. Todo entero par mayor que 2 se puede expresar como la suma de dos números primos.

Ejemplos:  

Input :  n = 44
Output :   3 + 41 (both are primes)

Input :  n = 56
Output :  3 + 53  (both are primes) 

1. Encuentra los números primos usando Sieve of Sundaram
2. Compruebe si el número introducido es un número par mayor que 2 o no, si no hay retorno.
3. En caso afirmativo, reste uno por uno un número primo de N y luego verifique si la diferencia también es un número primo. En caso afirmativo, expréselo como una suma.

// C++ program to implement Goldbach's conjecture
#include<bits/stdc++.h>
using namespace std;
const int MAX = 10000;
 
// Array to store all prime less than and equal to 10^6
vector <int> primes;
 
// Utility function for Sieve of Sundaram
void sieveSundaram()
{
    // In general Sieve of Sundaram, produces primes smaller
    // than (2*x + 2) for a number given number x. Since
    // we want primes smaller than MAX, we reduce MAX to half
    // This array is used to separate numbers of the form
    // i + j + 2*i*j from others where 1 <= i <= j
    bool marked[MAX/2 + 100] = {0};
 
    // Main logic of Sundaram. Mark all numbers which
    // do not generate prime number by doing 2*i+1
    for (int i=1; i<=(sqrt(MAX)-1)/2; i++)
        for (int j=(i*(i+1))<<1; j<=MAX/2; j=j+2*i+1)
            marked[j] = true;
 
    // Since 2 is a prime number
    primes.push_back(2);
 
    // Print other primes. Remaining primes are of the
    // form 2*i + 1 such that marked[i] is false.
    for (int i=1; i<=MAX/2; i++)
        if (marked[i] == false)
            primes.push_back(2*i + 1);
}
 
// Function to perform Goldbach's conjecture
void findPrimes(int n)
{
    // Return if number is not even or less than 3
    if (n<=2 || n%2 != 0)
    {
        cout << "Invalid Input \n";
        return;
    }
 
    // Check only upto half of number
    for (int i=0 ; primes[i] <= n/2; i++)
    {
        // find difference by subtracting current prime from n
        int diff = n - primes[i];
 
        // Search if the difference is also a prime number
        if (binary_search(primes.begin(), primes.end(), diff))
        {
            // Express as a sum of primes
            cout << primes[i] << " + " << diff << " = "
                 << n << endl;
            return;
        }
    }
}
 
// Driver code
int main()
{
    // Finding all prime numbers before limit
    sieveSundaram();
 
    // Express number as a sum of two primes
    findPrimes(4);
    findPrimes(38);
    findPrimes(100);
 
    return 0;
}

// Java program to implement Goldbach's conjecture
import java.util.*;
 
class GFG
{
     
static int MAX = 10000;
 
// Array to store all prime less
// than and equal to 10^6
static ArrayList<Integer> primes = new ArrayList<Integer>();
 
// Utility function for Sieve of Sundaram
static void sieveSundaram()
{
    // In general Sieve of Sundaram, produces
    // primes smaller than (2*x + 2) for
    // a number given number x. Since
    // we want primes smaller than MAX,
    // we reduce MAX to half This array is
    // used to separate numbers of the form
    // i + j + 2*i*j from others where 1 <= i <= j
    boolean[] marked = new boolean[MAX / 2 + 100];
 
    // Main logic of Sundaram. Mark all numbers which
    // do not generate prime number by doing 2*i+1
    for (int i = 1; i <= (Math.sqrt(MAX) - 1) / 2; i++)
        for (int j = (i * (i + 1)) << 1; j <= MAX / 2; j = j + 2 * i + 1)
            marked[j] = true;
 
    // Since 2 is a prime number
    primes.add(2);
 
    // Print other primes. Remaining primes are of the
    // form 2*i + 1 such that marked[i] is false.
    for (int i = 1; i <= MAX / 2; i++)
        if (marked[i] == false)
            primes.add(2 * i + 1);
}
 
// Function to perform Goldbach's conjecture
static void findPrimes(int n)
{
    // Return if number is not even or less than 3
    if (n <= 2 || n % 2 != 0)
    {
        System.out.println("Invalid Input ");
        return;
    }
 
    // Check only upto half of number
    for (int i = 0 ; primes.get(i) <= n / 2; i++)
    {
        // find difference by subtracting
        // current prime from n
        int diff = n - primes.get(i);
 
        // Search if the difference is
        // also a prime number
        if (primes.contains(diff))
        {
            // Express as a sum of primes
            System.out.println(primes.get(i) +
                        " + " + diff + " = " + n);
            return;
        }
    }
}
 
// Driver code
public static void main (String[] args)
{
    // Finding all prime numbers before limit
    sieveSundaram();
 
    // Express number as a sum of two primes
    findPrimes(4);
    findPrimes(38);
    findPrimes(100);
}
}
 
// This code is contributed by mits

# Python3 program to implement Goldbach's
# conjecture
import math
MAX = 10000;
 
# Array to store all prime less
# than and equal to 10^6
primes = [];
 
# Utility function for Sieve of Sundaram
def sieveSundaram():
     
    # In general Sieve of Sundaram, produces
    # primes smaller than (2*x + 2) for a
    # number given number x. Since we want
    # primes smaller than MAX, we reduce
    # MAX to half. This array is used to
    # separate numbers of the form i + j + 2*i*j
    # from others where 1 <= i <= j
    marked = [False] * (int(MAX / 2) + 100);
 
    # Main logic of Sundaram. Mark all
    # numbers which do not generate prime
    # number by doing 2*i+1
    for i in range(1, int((math.sqrt(MAX) - 1) / 2) + 1):
        for j in range((i * (i + 1)) << 1,
                        int(MAX / 2) + 1, 2 * i + 1):
            marked[j] = True;
 
    # Since 2 is a prime number
    primes.append(2);
 
    # Print other primes. Remaining primes
    # are of the form 2*i + 1 such that
    # marked[i] is false.
    for i in range(1, int(MAX / 2) + 1):
        if (marked[i] == False):
            primes.append(2 * i + 1);
 
# Function to perform Goldbach's conjecture
def findPrimes(n):
     
    # Return if number is not even
    # or less than 3
    if (n <= 2 or n % 2 != 0):
        print("Invalid Input");
        return;
 
    # Check only upto half of number
    i = 0;
    while (primes[i] <= n // 2):
         
        # find difference by subtracting
        # current prime from n
        diff = n - primes[i];
 
        # Search if the difference is also
        # a prime number
        if diff in primes:
             
            # Express as a sum of primes
            print(primes[i], "+", diff, "=", n);
            return;
        i += 1;
 
# Driver code
 
# Finding all prime numbers before limit
sieveSundaram();
 
# Express number as a sum of two primes
findPrimes(4);
findPrimes(38);
findPrimes(100);
 
# This code is contributed
# by chandan_jnu

// C# program to implement Goldbach's conjecture
using System;
using System.Collections.Generic;
 
class GFG
{
     
static int MAX = 10000;
 
// Array to store all prime less
// than and equal to 10^6
static List<int> primes = new List<int>();
 
// Utility function for Sieve of Sundaram
static void sieveSundaram()
{
    // In general Sieve of Sundaram, produces
    // primes smaller than (2*x + 2) for
    // a number given number x. Since
    // we want primes smaller than MAX,
    // we reduce MAX to half This array is
    // used to separate numbers of the form
    // i + j + 2*i*j from others where 1 <= i <= j
    Boolean[] marked = new Boolean[MAX / 2 + 100];
 
    // Main logic of Sundaram. Mark all numbers which
    // do not generate prime number by doing 2*i+1
    for (int i = 1; i <= (Math.Sqrt(MAX) - 1) / 2; i++)
        for (int j = (i * (i + 1)) << 1; j <= MAX / 2; j = j + 2 * i + 1)
            marked[j] = true;
 
    // Since 2 is a prime number
    primes.Add(2);
 
    // Print other primes. Remaining primes are of the
    // form 2*i + 1 such that marked[i] is false.
    for (int i = 1; i <= MAX / 2; i++)
        if (marked[i] == false)
            primes.Add(2 * i + 1);
}
 
// Function to perform Goldbach's conjecture
static void findPrimes(int n)
{
    // Return if number is not even or less than 3
    if (n <= 2 || n % 2 != 0)
    {
        Console.WriteLine("Invalid Input ");
        return;
    }
 
    // Check only upto half of number
    for (int i = 0 ; primes[i] <= n / 2; i++)
    {
        // find difference by subtracting
        // current prime from n
        int diff = n - primes[i];
 
        // Search if the difference is
        // also a prime number
        if (primes.Contains(diff))
        {
            // Express as a sum of primes
            Console.WriteLine(primes[i] +
                        " + " + diff + " = " + n);
            return;
        }
    }
}
 
// Driver code
public static void Main (String[] args)
{
    // Finding all prime numbers before limit
    sieveSundaram();
 
    // Express number as a sum of two primes
    findPrimes(4);
    findPrimes(38);
    findPrimes(100);
}
}
 
/* This code contributed by PrinciRaj1992 */

<?php
// PHP program to implement Goldbach's
// conjecture
$MAX = 10000;
 
// Array to store all prime less than
// and equal to 10^6
$primes = array();
 
// Utility function for Sieve of Sundaram
function sieveSundaram()
{
    global $MAX, $primes;
     
    // In general Sieve of Sundaram, produces
    // primes smaller than (2*x + 2) for a
    // number given number x. Since we want
    // primes smaller than MAX, we reduce
    // MAX to half. This array is used to
    // separate numbers of the form i + j + 2*i*j
    // from others where 1 <= i <= j
    $marked = array_fill(0, (int)($MAX / 2) +
                                100, false);
 
    // Main logic of Sundaram. Mark all
    // numbers which do not generate prime
    // number by doing 2*i+1
    for ($i = 1; $i <= (sqrt($MAX) - 1) / 2; $i++)
        for ($j = ($i * ($i + 1)) << 1;
             $j <= $MAX / 2; $j = $j + 2 * $i + 1)
            $marked[$j] = true;
 
    // Since 2 is a prime number
    array_push($primes, 2);
 
    // Print other primes. Remaining primes
    // are of the form 2*i + 1 such that
    // marked[i] is false.
    for ($i = 1; $i <= $MAX / 2; $i++)
        if ($marked[$i] == false)
            array_push($primes, 2 * $i + 1);
}
 
// Function to perform Goldbach's conjecture
function findPrimes($n)
{
    global $MAX, $primes;
     
    // Return if number is not even
    // or less than 3
    if ($n <= 2 || $n % 2 != 0)
    {
        print("Invalid Input \n");
        return;
    }
 
    // Check only upto half of number
    for ($i = 0; $primes[$i] <= $n / 2; $i++)
    {
        // find difference by subtracting
        // current prime from n
        $diff = $n - $primes[$i];
 
        // Search if the difference is also a
        // prime number
        if (in_array($diff, $primes))
        {
            // Express as a sum of primes
            print($primes[$i] . " + " .
                  $diff . " = " . $n . "\n");
            return;
        }
    }
}
 
// Driver code
 
// Finding all prime numbers before limit
sieveSundaram();
 
// Express number as a sum of two primes
findPrimes(4);
findPrimes(38);
findPrimes(100);
 
// This code is contributed by chandan_jnu
?>

<script>
// Javascript program to implement Goldbach's
// conjecture
let MAX = 10000;
 
// Array to store all prime less than
// and equal to 10^6
let primes = new Array();
 
// Utility function for Sieve of Sundaram
function sieveSundaram()
{  
    // In general Sieve of Sundaram, produces
    // primes smaller than (2*x + 2) for a
    // number given number x. Since we want
    // primes smaller than MAX, we reduce
    // MAX to half. This array is used to
    // separate numbers of the form i + j + 2*i*j
    // from others where 1 <= i <= j
    let marked = new Array(parseInt(MAX / 2) + 100).fill(false);
 
    // Main logic of Sundaram. Mark all
    // numbers which do not generate prime
    // number by doing 2*i+1
    for (let i = 1; i <= (Math.sqrt(MAX) - 1) / 2; i++)
          for (let j = (i * (i + 1)) << 1;
            j <= MAX / 2; j = j + 2 * i + 1)
            marked[j] = true;
 
    // Since 2 is a prime number
    primes.push(2);
 
    // Print other primes. Remaining primes
    // are of the form 2*i + 1 such that
    // marked[i] is false.
    for (let i = 1; i <= MAX / 2; i++)
        if (marked[i] == false)
            primes.push(2 * i + 1);
}
 
// Function to perform Goldbach's conjecture
function findPrimes(n)
{
    // Return if number is not even
    // or less than 3
    if (n <= 2 || n % 2 != 0)
    {
        document.write("Invalid Input <br>");
        return;
    }
 
    // Check only upto half of number
    for (let i = 0; primes[i] <= n / 2; i++)
    {
        // find difference by subtracting
        // current prime from n
        let diff = n - primes[i];
 
        // Search if the difference is also a
        // prime number
        if (primes.includes(diff))
        {
            // Express as a sum of primes
            document.write(primes[i] + " + " + diff + " = " + n + "<br>");
            return;
        }
    }
}
 
// Driver code
 
// Finding all prime numbers before limit
sieveSundaram();
 
// Express number as a sum of two primes
findPrimes(4);
findPrimes(38);
findPrimes(100);
 
// This code is contributed by gfgking
</script>

Producción:  

2 + 2 = 4
7 + 31 = 38
3 + 97 = 100

Un número de Goldbach es un número entero positivo que se puede expresar como la suma de dos números primos impares. Dado que cuatro es el único número par mayor que dos que requiere el primo par 2 para poder escribirse como la suma de dos primos, otra forma del enunciado de la conjetura de Goldbach es que todos los enteros pares mayores que 4 son números de Goldbach.
 
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