Euclid Euler Theorem - GeeksforGeeks

According to Euclid Euler Theorem, a perfect number which is even, can be represented in the form $(2^n - 1)*(2^n / 2) ))$ where n is a prime number and $2^n - 1$ is a Mersenne prime number. It is a product of a power of 2 with a Mersenne prime number. This theorem establishes a connection between a Mersenne prime and an even perfect number.

Some Examples (Perfect Numbers) which satisfy Euclid Euler Theorem are:

6, 28, 496, 8128, 33550336, 8589869056, 137438691328

Explanations:
1) 6 is an even perfect number.
So, is can be written in the form 
(2² - 1) * (2^{(2 - 1)}) = 6
where n = 2 is a prime number and 2^n - 1 = 3 is a Mersenne prime number.

2) 28 is an even perfect number.
So, is can be written in the form 
(2³ - 1) * (2^{(3 - 1)}) = 28
where n = 3 is a prime number and 2^n - 1 = 7 is a Mersenne prime number.

3) 496 is an even perfect number.
So, is can be written in the form 
(2⁵ - 1) * (2^{(5 - 1)}) = 496
where n = 5 is a prime number and 2^n - 1 = 31 is a Mersenne prime number.

Approach(Brute Force):

Take each prime number and form a Mersenne prime with it. Mersenne prime = $2^n - 1$ where n is prime. Now form the number (2^n – 1)*(2^(n – 1)) and check if it is even and perfect. If the condition satisfies then it follows Euclid Euler Theorem.

C++

// CPP code to verify Euclid Euler Theorem 
#include <bits/stdc++.h> 
using namespace std; 
  
#define show(x) cout << #x << " = " << x << "\n"; 
  
bool isprime(long long n) 
{ 
    // check whether a number is prime or not 
    for (int i = 2; i * i <= n; i++) 
        if (n % i == 0) 
            return false; 
    return false; 
} 
  
bool isperfect(long long n) // perfect numbers 
{ 
    // check is n is perfect sum of divisors 
    // except the number itself = number 
    long long s = -n; 
    for (long long i = 1; i * i <= n; i++) { 
  
        // is i is a divisor of n 
        if (n % i == 0) { 
            long long factor1 = i, factor2 = n / i; 
            s += factor1 + factor2; 
  
            // here i*i == n 
            if (factor1 == factor2) 
                s -= i; 
        } 
    } 
    return (n == s); 
} 
  
int main() 
{ 
    // storing powers of 2 to access in O(1) time 
    vector<long long> power2(61); 
    for (int i = 0; i <= 60; i++) 
        power2[i] = 1LL << i; 
  
    // generation of first few numbers 
    // satisfying Euclid Euler's theorem 
  
    cout << "Generating first few numbers "
            "satisfying Euclid Euler's theorem\n"; 
    for (long long i = 2; i <= 25; i++) { 
        long long no = (power2[i] - 1) * (power2[i - 1]); 
        if (isperfect(no) and (no % 2 == 0)) 
            cout << "(2^" << i << " - 1) * (2^(" 
                << i << " - 1)) = " << no << "\n"; 
    } 
    return 0; 
} 

Java

// Java code to verify Euclid Euler Theorem 
class GFG  
{ 
    static boolean isprime(long n) 
    { 
        // check whether a number is prime or not 
        for (int i = 2; i * i <= n; i++)  
        { 
            if (n % i == 0)  
            { 
                return false; 
            } 
        } 
        return false; 
    } 
  
    static boolean isperfect(long n) // perfect numbers 
    { 
        // check is n is perfect sum of divisors 
        // except the number itself = number 
        long s = -n; 
        for (long i = 1; i * i <= n; i++)  
        { 
  
            // is i is a divisor of n 
            if (n % i == 0)  
            { 
                long factor1 = i, factor2 = n / i; 
                s += factor1 + factor2; 
  
                // here i*i == n 
                if (factor1 == factor2)  
                { 
                    s -= i; 
                } 
            } 
        } 
        return (n == s); 
    } 
  
    // Driver Code 
    public static void main(String[] args)  
    { 
        // storing powers of 2 to access in O(1) time 
        long power2[] = new long[61]; 
        for (int i = 0; i <= 60; i++) 
        { 
            power2[i] = 1L << i; 
        } 
  
        // generation of first few numbers 
        // satisfying Euclid Euler's theorem 
        System.out.print("Generating first few numbers " +  
                         "satisfying Euclid Euler's theorem\n"); 
        for (int i = 2; i <= 25; i++)  
        { 
            long no = (power2[i] - 1) * (power2[i - 1]); 
            if (isperfect(no) && (no % 2 == 0))  
            { 
                System.out.print("(2^" + i + " - 1) * (2^(" +  
                                 i + " - 1)) = " + no + "\n"); 
            } 
        } 
    } 
}  
  
// This code is contributed by PrinciRaj1992 

C#

// C# code to verify Euclid Euler Theorem 
using System; 
using System.Collections.Generic; 
      
class GFG  
{ 
    static Boolean isprime(long n) 
    { 
        // check whether a number is prime or not 
        for (int i = 2; i * i <= n; i++)  
        { 
            if (n % i == 0)  
            { 
                return false; 
            } 
        } 
        return false; 
    } 
  
    static Boolean isperfect(long n) // perfect numbers 
    { 
        // check is n is perfect sum of divisors 
        // except the number itself = number 
        long s = -n; 
        for (long i = 1; i * i <= n; i++)  
        { 
  
            // is i is a divisor of n 
            if (n % i == 0)  
            { 
                long factor1 = i, factor2 = n / i; 
                s += factor1 + factor2; 
  
                // here i*i == n 
                if (factor1 == factor2)  
                { 
                    s -= i; 
                } 
            } 
        } 
        return (n == s); 
    } 
  
    // Driver Code 
    public static void Main(String[] args)  
    { 
        // storing powers of 2 to access in O(1) time 
        long []power2 = new long[61]; 
        for (int i = 0; i <= 60; i++) 
        { 
            power2[i] = 1L << i; 
        } 
  
        // generation of first few numbers 
        // satisfying Euclid Euler's theorem 
        Console.Write("Generating first few numbers " +  
                      "satisfying Euclid Euler's theorem\n"); 
        for (int i = 2; i <= 25; i++)  
        { 
            long no = (power2[i] - 1) * (power2[i - 1]); 
            if (isperfect(no) && (no % 2 == 0))  
            { 
                Console.Write("(2^" + i + " - 1) * (2^(" +  
                                i + " - 1)) = " + no + "\n"); 
            } 
        } 
    } 
} 
  
// This code is contributed by Rajput-Ji 

PHP

<?php 
// PHP code to verify  
// Euclid Euler Theorem 
  
// define show(x)  
// cout << #x << " = " << x << "\n"; 
  
function isprime($n) 
{ 
    // check whether a number 
    // is prime or not 
    for ($i = 2; $i * $i <= $n; $i++) 
        if ($n % $i == 0) 
            return false; 
    return false; 
} 
  
function isperfect($n) // perfect numbers 
{ 
    // check is n is perfect sum  
    // of divisors except the  
    // number itself = number 
    $s = -$n; 
    for ($i = 1;  
         $i * $i <= $n; $i++)  
    { 
  
        // is i is a divisor of n 
        if ($n % $i == 0)  
        { 
            $factor1 = $i; 
            $factor2 = $n / $i; 
            $s += $factor1 + $factor2; 
  
            // here i*i == n 
            if ($factor1 == $factor2) 
                $s -= $i; 
        } 
    } 
    return ($n == $s); 
} 
  
// Driver code 
  
// storing powers of 2 to  
// access in O(1) time 
$power2 = array(); 
for ($i = 0; $i <= 60; $i++) 
    $power2[$i] = 1<< $i; 
  
// generation of first few  
// numbers satisfying Euclid  
// Euler's theorem 
echo "Generating first few numbers " .  
     "satisfying Euclid Euler's theorem\n"; 
       
for ($i = 2; $i <= 25; $i++)  
{ 
    $no = ($power2[$i] - 1) *  
          ($power2[$i - 1]); 
    if (isperfect($no) &&  
                 ($no % 2 == 0)) 
        echo "(2^" . $i . " - 1) * (2^(" .  
                     $i . " - 1)) = " .  
                     $no . "\n"; 
} 
  
// This code is contributed by mits 
?> 

Output:

Generating first few numbers satisfying Euclid Euler's theorem
(2^2 - 1) * (2^(2 - 1)) = 6
(2^3 - 1) * (2^(3 - 1)) = 28
(2^5 - 1) * (2^(5 - 1)) = 496
(2^7 - 1) * (2^(7 - 1)) = 8128
(2^13 - 1) * (2^(13 - 1)) = 33550336
(2^17 - 1) * (2^(17 - 1)) = 8589869056
(2^19 - 1) * (2^(19 - 1)) = 137438691328

Explanation of the outputs are provided in the the explanations to the examples above.

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