@thomasrinschler6783
13:25 "But Euler wasn't finished yet." I think this sentence appears in most histories of mathematical concepts.
@deanrinehart
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 surprise…and finding out he’s 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!
@janviersechan9670
Factorizing 2^67-1 without using calculator or any mechanical computing device is insane 😮
@jonahmishaga1995
As a physics undergrad. I’ve come to realize that Euler is a Titan alongside Einstein and Newton. Every single bit of modern physics has Euler to thank for providing the mathematical Tools to construct a vivid picture of the universe and its underlying principles. Absolute legend.
@nathanaelhahn
4:03 "Euclid was actually thinking along similar lines" Euclid: calculates perfect numbers with actual lines
@Oodle-ox2vf
I am an old, retired scientist/engineer/educator, who refuses to quit. I enjoy the interesting discussion that comes from many of your videos. They are also on my list of insomniac pastimes. Thank you. Keep them coming. 🙂
@lifthras11r
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.
@VintageBlacklist
I have a research project due tomorrow and I was really looking for something distracting. My procrastination thanks you.
@logician1234
There is something so bizarre about Euclid and Euler having a collaboration. If the history of mathematics was a book of fiction, I would call this a fan service 😂
@Color_Splsh
Wow, this topic throws me back. I remember in middle school, one of my friends was a big math nerd and he told me that his one goal in life was to find the odd perfect number. I had completely forgotten about that until i saw this video, thank you veritasium.
@haleyroe2647
I love consistently understanding the first 25% of veritasium maths videos.
@concrete401
I took a class from Dr. Nielsen in 2009. He was a very engaging, dynamic teacher, to the point that when he wrote an answer on the board, followed by an exclamation point, someone asked, "Is that factorial or excitement?" and he responded, "EXCITEMENT!"
@BronsonMWhite
WOAH! Dr. Pace Nielsen was my professor for intro to proofs. I was NOT expecting him to show up in the video. He's a fantastic guy, exceptional professor, and brilliant number theorist.
@johnseo5525
Well done. I admire your work. Thank you.
@AA-100
21:15 As of Oct 2024, largest known prime is now 2^136,279,841 - 1
@joshuazelinsky5213
Video is well done. I'm a mathematician some of whose work has been on this topic (some of the results you put on at 23:51 are mine, and one is due to a joint paper of me with Sean Bibby and Pieter Vyncke). My apologies also for the length of this comment. I do have some quibbles about some of the history details but they are minor. (And it is possible that I'm getting some of the details wrong myself.) Descartes's construction of a spoof perfect number, shows he had a pretty good understanding of how sigma behaves. Descartes's spoof shows he had a pretty good understanding of sigma(n). Also, Descartes likely did prove that an odd perfect number must be of the form he suggested. What Euler did was a bit stronger. Euler showed that if n is an odd perfect number n= p^e m^2 where p is a prime , p does not divide m, and p and e are both 1 (mod 4). Notice that this implies Descartes's result. Regarding the Lenstra–Pomerance–Wagstaff conjecture, while it gives a specific estimate for how large the nth Mersenne prime is, there is some degree of doubt of if it is correct. We're much more confident that the conjecture is correct up to a multiplicative constant near 1. And we are much much confident that there are infinitely many Mersenne primes, even if LPW turns out to be wrong even on the order of growth of Mersenne primes. Regarding Pace's comment to high school students, I want to expand on that slightly. No one should be working on this problem with any hope of solving it any time soon. The problem is genuinely very difficult. The spoofs are in many respects a major obstruction to proving that no odd perfect numbers exist. In particular, many of the things we can prove about odd perfect numbers, also apply to spoofs. So if they were enough to prove that no odd perfect numbers existed, we would have proven that no spoofs exist, which is obvious nonsense. To use an analogy that my spouse suggested a while ago: If we are trying to convince ourselves that Bigfoot doesn't exist, but all we've done is list properties that all mammals have, we can't hope to show Bigfoot isn't real. There are few other big obstructions, one of which has a very similar flavor. But, Pace correctly notes that not that many people are working on the problem, so there may be more low hanging fruit than one would otherwise expect for aspects of the problem. For most really famous open math problems, like say the Riemann Hypothesis, or P ?= NP, lots of people have spent a lot of time thinking about aspects of it. So most mathematicians have a general attitude of not trying to bash their head against problems that a lot of other people have thought about. But in the odd perfect number situation, to some extent, the community may have overcorrected, and thus spent less time on it than they might otherwise. However, this may also be due in part to the odd perfect number problem being famous, but not by itself being very enlightening in terms of what it implies. Hundreds of papers prove theorems of the form "If the Riemann Hypothesis is true then " . And those papers are themselves very broad and varied in what follows after the then. In contrast, I'm aware of only a handful of papers with results of the form "If there are no odd perfect numbers then" and what follows after the then is always something involving divisors of a number in a somewhat straightforward fashion.
@seanshepherd1071
28:00 "Useless" problems are never really useless, because in the pursuit of attempting to solve them, something useful almost invariably gets created along the way. Entire fields of mathematics have been formalized because the tools that existed before them were insufficient to solve some problem with no practical application.
@Art_Vandelay_Industries
As someone that was never good at math it blows my mind how people could and can think in ways that can actually make sense of math so abstract. And without having computers to do the crunch for them back in the days.
@martafixarcoolt5993
I love when people have made up their mind on something, like there is a heuristic argument for that there is no odd perfect numbers, and then faced with a reasonable counter argument, imidiately recognize that their original argument is flawed. Just listening to reason and take that logic in, it is beautiful
@cupostuff9929