Every Perfect Number Is Even, and No One Knows Why

Take the number 6 and add up its divisors below itself: 1 + 2 + 3 = 6. The number equals the sum of its own parts. The next one to do this is 28, then 496, then 8128, then a jump past 33 million. These are the perfect numbers, and every single one ever found is even. Whether an ODD perfect number exists is unknown — it has been open since antiquity, and it is often called the oldest unsolved problem in mathematics. This video walks the whole story: where the even perfect numbers come from, why the tools that pin them down go completely silent on odd numbers, and the strange situation we are left in — we can describe an odd perfect number in extraordinary detail and still cannot say whether it exists.

The even side is completely understood. Euclid showed that a Mersenne prime (a prime of the form 2^p − 1) times the power of two beneath it always gives a perfect number, and Euler proved the converse: every even perfect number has exactly that form. The odd side has none of that — and so people went looking.

CHAPTERS
0:00 The Question
1:29 The Knife-Edge
2:55 Euclid's Recipe
4:15 Euler Closes the Even Case
5:44 The Odd Hole
7:07 The Empty Search
8:43 The Wanted Poster
10:24 Squeezed, Never Cornered
12:06 Coda

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Music by Vincent Rubinetti
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

Видео Every Perfect Number Is Even, and No One Knows Why канала Euclidea