The Mysterious Mersennes: Mersenne Primes and Perfect Numbers

[@joseantoniobarreranunez9949]

I had never ever thought of finding such an incredible video. Thank you Mr discoverthaths!.

[@juanmemol]

Muchas gracias José Antonio!!

[@sphakamisozondi]

My high school teacher defined prime numbers: "these are numbers that have no factors, they are their own factors"

[@GeoffryGifari]

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

[@discovermaths]

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.

[@zanti4132]

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.

[@iwersonsch5131]

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?

[@zanti4132]

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.