# Euclid Euler Theorem

According to Euclid Euler Theorem, a perfect number which is even, can be represented in the form where n is a prime number and 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, it 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, it 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, it 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 = 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 ` `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` `true``;` `}`   `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``<

## 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

 ``

## Javascript

 ``

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```

Time Complexity: O(sqrt(n))
Auxiliary Space: O(1)

Feeling lost in the world of random DSA topics, wasting time without progress? It's time for a change! Join our DSA course, where we'll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 geeks!

Previous
Next