The Mysterious Mersennes: Mersenne Primes and Perfect Numbers
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- Опубликовано: 9 авг 2022
- Almost all the biggest primes known are Mersenne primes, and every Mersenne prime is an even perfect number. But are there are an infinite number of Mersenne primes, and are there any odd perfect numbers?
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My high school teacher defined prime numbers:
"these are numbers that have no factors, they are their own factors"
Great watch after a long day, thanks
Thanks
I had never ever thought of finding such an incredible video. Thank you Mr discoverthaths!.
Muchas gracias José Antonio!!
if a mersenne prime is defined by mersenne = 2ᵖ-1, can we make an ever bigger prime much faster by putting the new result as the power?
bigger mersenne = 2^(mersenne)-1? after checking for the other conditions
It will be a Mersenne number but not necessarily - and probably not - prime. It is the "checking for other conditions" that consumes so much computing time.
Your speculation that if n is a Mersenne prime, then 2ⁿ - 1 must be prime turns out to be true for the first four Mersenne primes, those being 3, 7, 31, and 127. In fact, Lucas manually tested 2¹²⁷ - 1 usng his primality test (the Lucas-Lehmer test) to confirn that this is indeed a Mersenne prime. That's a record that should stand forever as the largest number even proven to be prime without the use of a computer. However, when computers made it possible to test much larger numbers, it was found that this pattern ends with the next Mersenne prime: 2¹³ - 1 = 8191 is a Mersenne prime, but 2⁸¹⁹¹ - 1 is not.
I once heard that if 1 is not counted in the sum, we don't know of any number whose nontrivial divisors add to itself, nor do we know of a proof that there isn't one. Is that still true?
There probably aren't any of these "quasiperfect numbers," but to date this hasn't been proven. What is known is that if such a number exists, then it must be an odd perfect square greater than 10³⁵ with at least 7 distinct prime factors.