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, 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.
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.
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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- Euclid's lemma
- Euclid's Algorithm when % and / operations are costly
- Check whether the given number is Euclid Number or not
- Euclid–Mullin Sequence
- Euler's criterion (Check if square root under modulo p exists)
- Optimized Euler Totient Function for Multiple Evaluations
- Euler Method for solving differential equation
- Euler's Four Square Identity
- Total nodes traversed in Euler Tour Tree
- Predictor-Corrector or Modified-Euler method for solving Differential equation
- Count integers in a range which are divisible by their euler totient value
- Euler zigzag numbers ( Alternating Permutation )
- Check if a number is Euler Pseudoprime
- Euler's Factorization method
- Count of elements having Euler's Totient value one less than itself
- Probability of Euler's Totient Function in a range [L, R] to be divisible by M
- Check if Euler Totient Function is same for a given number and twice of that number
- Euler's Totient Function
- Euler's Totient function for all numbers smaller than or equal to n
- Chinese Remainder Theorem | Set 1 (Introduction)
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