Views : 7,141,370
Genre: Education
Date of upload: Mar 7, 2024 ^^
Rating : 4.954 (2,411/205,238 LTDR)
RYD date created : 2024-04-29T01:38:22.80087Z
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Top Comments of this video!! :3
13:25 "But Euler wasn't finished yet." I think this sentence appears in most histories of mathematical concepts.
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Watching a math related video strictly out of curiosity and having your general math professor Bill Dunham from 25 years ago pop up is a surpriseâŚand finding out heâs now a well respected mathematics historian and not just some guy who endlessly suffered non-math students struggles with train problems is absolutely fantastic. Go Mules!
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4:03 "Euclid was actually thinking along similar lines"
Euclid: calculates perfect numbers with actual lines
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At 15:42, to prove that the exponent of p is of the form 4k+1, you just have to remark that the sum of the divisors of p^(4k+3) is always divisible by 4 (the powers of p modulo 4 are all 1 if p =4a+1 or alternating 1 and 3 if p=4k+3), which would make 2n divisible by 4 hence n even. The alternating 1 and 3 must be excluded because in this case the sum of the divisors of p^(4k+1) would be divisible by 4 as well. So p is congruent to 1 modulo p (Euler's proof as well).
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What's also really cool is that if you divide the perfect number (at least the first four) by the last number in the line of numbers that make it then divide the perfect number by it, the result keeps doubling. To explain: 6 is 1+2+3, 6/3 is 2 or 2^1. 28 is 1+2+3+4+5+6+7, 28/7 is 4 or 2^2. 496 is 1+2++3...30+31, 496/31 is 16 or 2^4 or 4^2. 8128 is 1+2+3+...127+127, 8128/127 is 64 or 2^6 or 8^2. I don't know if the other perfect numbers fit that, but the first four do and I think that's funky
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One big application of Mersenne primes, that came from studying perfect numbers, is a good random number generator. RNGs had been historically very bad, until the introduction of Mersenne Twister in 1997, which uses a property of Mersenne primes to prove a good randomness. The most popular version uses a Mersenne prime 2^19937 - 1 for example, hence the name MT19937. There exist much more performant RNGs than Mersenne Twister now, but Mersenne Twister is still widely used thanks to its initial impact.
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I had a fun watch, definitely amazing to think about! I've been fascinated with numbers and problems since grade school and has been thinking about problems with patterns like this ever since. Not that I am any good at it nor am I sure when trying to come up with formulas based on these patterns. And sometimes, I tend to simplify these kind of problems based on what they look at. With that, I also think there is no odd perfect number for the fact that these perfect numbers we currently have all have the factor "2" which obviously makes it divisible by 2.
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@cupostuff9929
1 month ago
>walks up to blackboard >multiplies 2 numbers >walks away >round of applause Frank Nelson Cole was unfathomably based
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