Abundant, Deficient, and Perfect Numbers ← Number Theory ← Socratica

Рет қаралды 3,512

9 ай бұрын

𝙎𝙞𝙜𝙣 𝙪𝙥 for Number Theory course (coming soon) on our website:
www.socratica.com/courses/num...
In ancient times, integers were simply called "numbers", hence "number theory". Scholars pulled apart numbers - slicing and dicing them - looking for interesting properties which might explain things like why a month is ~28 days long. One approach involved looking at the divisors of an integer. Integers with a LOT of divisors were called "abundant". Those were very few were called "deficient". And some integers seemed to have a near-magical number of divisors and were called "perfect".
In this lesson we'll explore abundant, deficient, and perfect numbers.
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Written & Produced by Michael Harrison
Edited by Megi Shuke
About our Instructor:
Michael earned his BS in Math from Caltech, and did his graduate work in Math at UC Berkeley and University of Washington, specializing in Number Theory. A self-taught programmer, Michael taught both Math and Computer Programming at the college level. He applied this knowledge as a financial analyst (quant) and as a programmer at Google.
#numbertheory
#mathematics
#socratica

Пікірлер: 9

@VidSummaryAI 9 ай бұрын

Hope you have a nice day! We've summarized the main points discussed for convenience, but encourage watching the full video to get maximal value from the creator's knowledge. Check my profile if you need summaries. 0:00 - 1:00 - Introduction to divisibility and the fundamental theorem of arithmetic. Every integer greater than 1 can be uniquely factored into prime numbers. 1:00 - 2:00 - Definitions of key terms like divisor, composite numbers, prime numbers. Introduces divisor functions to measure divisibility. 2:00 - 3:00 - Explanations of divisor functions - sigma, tau, and the sum of proper divisors s(n). Examples provided. 3:00 - 4:00 - History of perfect, abundant, and deficient numbers based on s(n). 6 and 28 called perfect numbers by ancient Greeks. 4:00 - 5:00 - Proof that there are infinite deficient and abundant numbers based on properties of s(n). Questions raised about perfect numbers. 5:00 - 6:00 - Patterns noticed in the first few perfect numbers. Later examples broke assumed patterns. Unanswered questions remain about perfect numbers. 6:00 - 7:00 - Facts about even perfect numbers. Must be of the form 2p-1(2n) where 2p-1 is a Mersenne prime. Connections to unsolved problems around Mersenne primes. 7:00 - 8:00 - Question of whether odd perfect numbers exist also remains unresolved, with some partial results. 8:00 - End - Summary that many open problems remain around perfect numbers and divisibility.

@andyl.5998 9 ай бұрын

This seems to be the same as an earlier video on the channel named "Perfect Numbers, Mersenne Primes, Abundant Numbers & Deficient Numbers".

@jagadishgospat2548 9 ай бұрын

I like the new format, makes it faster to produce more videos right?

@mistercorzi 9 ай бұрын

Largest known Mersenne prime (as of now) is 2^82,589,933 − 1 and was discovered in 2018. There are 51 Mersenne primes known hence 51 perfect numbers known. Loved the video - well explained. Thanks

@kirbymarchbarcena 9 ай бұрын

Well done!!!

@B._Smith 9 ай бұрын

I used to get upset that 1 is not a normal prime but it makes life a lot more convenient. Unique is the key word. Imagine if 12 = 1 * 2 * 2 *3 but also 12 = 1 *1 * 2 *2 * 3 ...it could go on forever and not be unique anymore. 😅