Euclid Euler Theorem

Difficulty Level : Hard
Last Updated : 05 Apr, 2021

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 = $\text{[math]}$ 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

Python3

# Python3 code to verify Euclid Euler Theorem
#define show(x) cout << #x << " = " << x << "\n";
def isprime(n):
    i = 2
 
    # check whether a number is prime or not
    while(i * i <= n):
        if (n % i == 0):
            return False;
        i += 1
    return False;
 
def isperfect(n): # perfect numbers
 
    # check is n is perfect sum of divisors
    # except the number itself = number
    s = -n;
    i =1
    while(i * i <= n):
 
        # 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;   
        i += 1
    return (n == s);
 
# Driver code
if __name__=='__main__':
 
    # storing powers of 2 to access in O(1) time
    power2 = [1<<i for i in range(61)]
 
    # generation of first few numbers
    # satisfying Euclid Euler's theorem
    print("Generating first few numbers satisfying Euclid Euler's theorem");
    for i in range(2, 26):  
        no = (power2[i] - 1) * (power2[i - 1]);
        if (isperfect(no) and (no % 2 == 0)):
            print("(2^{} - 1) * (2^({} - 1)) = {}".format(i, i, no))
     
    # This code is contributed by rutvik_56.

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
?>

Javascript

<script>
 
// JavaScript program to verify Euclid Euler Theorem
 
    function isprime(n)
    {
        // check whether a number is prime or not
        for (let 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
        let s = -n;
        for (let i = 1; i * i <= n; i++)
        {
  
            // is i is a divisor of n
            if (n % i == 0)
            {
                let 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
        let power2 = [];
        for (let i = 0; i <= 60; i++)
        {
            power2[i] = 1 << i;
        }
  
        // generation of first few numbers
        // satisfying Euclid Euler's theorem
        document.write("Generating first few numbers " +
                         "satisfying Euclid Euler's theorem" + "<br/>");
        for (let i = 2; i <= 25; i++)
        {
            let no = (power2[i] - 1) * (power2[i - 1]);
            if (isperfect(no) && (no % 2 == 0))
            {
                document.write("(2^" + i + " - 1) * (2^(" +
                                 i + " - 1)) = " + no + "<br/>");
            }
        }
     
    // This code is contributed by code_hunt.
</script>

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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