Goldbach's conjecture is one of the oldest and best-known unsolved problems in the number theory of mathematics. Every even integer greater than 2 can be expressed as the sum of two primes.
Examples:
Input : n = 44 Output : 3 + 41 (both are primes) Input : n = 56 Output : 3 + 53 (both are primes)
Approach: 1
- Find the prime numbers using Sieve of Sundaram
- Check if the entered number is an even number greater than 2 or not, if no return.
- If yes, then one by one subtract a prime from N and then check if the difference is also a prime. If yes, then express it as a sum.
Below is the implementation of the above approach:
// 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>
Output
2 + 2 = 4 7 + 31 = 38 3 + 97 = 100
Time Complexity: O(n log n)
Auxiliary Space: O(MAX)
A Goldbach number is a positive integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.