This video continues at ua-cam.com/video/Yh1QUYn2f3I/v-deo.html and delves into so-called Untouchable Numbers. More Ben Sparks on Numberphile: bit.ly/Sparks_Playlist
Even though I know about number theory, and know about perfect, abundant, deficient, amicable, sociable, I had never heard of aspiring numbers before. Now because we have names for all of these categories, we seem to need one more. Doing the aliquot process once, divides all numbers into three categories: deficient, abundant, and perfect. But doing an aliquot sequence, we get (potentially) seven categories, but three of them don't seem to have names: Perfect - stay the same forever. Aspiring - eventually get to a perfect number. Amicable - bounce back and forth between two values. Sociable - cycle through a loop of more than two numbers. ?1? - the ones that never get to a loop or perfect number - there might not be any in this category. ?2? - numbers that eventually get to a loop. You might say they "aspire to be amicable or sociable, rather than aspiring to be perfect". ?3? - the numbers that get to 1 eventually. Note that both abundant and deficient numbers can fall into this category. I guess those ?1? numbers, if they are found to exist, can be named after whoever finally proves their existence. The ?2? numbers could be called "shy" numbers - they're trying to get into the amicable/sociable group. I suppose this category could be split into two. And the ?3? category in which the majority of numbers fall, should have some name, too. At first, I was thinking to propose calling them "mortal" numbers, because through the aliquot sequence, they eventually "die". But that seems too dark of a name.
The next puzzle for you to solve: The 300 Coins Problem. 300 coins are placed randomly on a table. A 300 letters long message (Signal) is written, one letter per coin, that would lead to a hidden treasure. Then the coins are flipped over and a randomly generated Noise 300 letters long is written on the other side of coins. The coins then get put in a bag and scrambled. Finally, the coins are put back on the table. Your task is to flip and move the coins around until the original message is recreated. Can you do it?
I checked wikipedia on sociable numbers for my own curiosity, and if it is accurate then: The only known loop lengths are 1 (perfect), 2 (amicable), 4, 5, 6, 8, 9 and 28. (and 5, 9 and 28 only have 1 known sequence each) "It is conjectured that if n is congruent to 3 modulo 4 then there is no such sequence with length n." So loops with length n=4k+3: 3,7,11,15,... is probably/maybe not possible.
To me, this is my favorite kind of numberphile video because it’s still very compelling while keeping the math below, like, a trigonometry level (which was where I stopped fully understanding math)
@@ToneHelix97 Sorry. I meant that the things we see and do, even the seemingly simple natural numbers, still hides a lot of complex reasoning. When things may seems obvious and intuitive, In reality it doesn't work like that.
For those interested, aliquot sequnce for 276 is currently at step 2146, not 2090. The last advance was made in January 2024, when a C209 was split into a P98 and P112. That means the number of digits, C for composite, P for prime. C209 is the supercomputer (or rather a distributed computing project) territory with months/years of GNFS sieving required to factor it. The previous hurdle was step 2140, passed in August 2022 after factoring a C213 which turned out to be P97 * P116.
I like to imagine that 276 goes all the way up straight to the first and only odd perfect number, and that number also happens to be the first number to start a loop that disproves the collatz conjecture.
Just want to point out that the first number that does a really wild ride was 138, and the next number he showed was 276, which is exactly double. And then the next Lehmer five is 552, again exactly double.
@AndyWitmyer 276 already has an index (sequence length) over 2,100 and 564 has an index near 3,500 currently. 138 ONLY took an index of 177 to resolve, thus both sequences are already at least one order of magnitude larger and show no signs of ending anytime soon.
So lets see how much magic 276 has in it. 3 times 276 is 828. The sequence is open. It joins the 660 sequence immediately. Also one of the Lehmer Five.
@@gordontaylor2815 if the numbers in the aliquot sequence are random, since they grow exponentially, the chance of finding a prime at any step n is proportional to 1/n, so the chance of finding a prime up to step n would be (somewhat) proportional to log(n), meaning you would expect orders of magnitude longer chains that are still finite if they just grow faster. aliquot sequences aren't random nor does it seem they just hit primes and stop, but if any similar situation is happening (which seems somewhat likely to me), then only 20 times the chain length would not be that strong of evidence for the chain suddenly being infinite
The first time I programmed this was in the 80s on a C64. I hit brick walls several times; first my algorithm to compute the proper divisor sum was too simple and thus too slow for the gigantic numbers I ran into for the 138. When I fixed that, they still kept growing beyond the numbers the programming language could handle. I had to restart the whole programming several times until I found what I really was looking for: These things which I now just learned are called sociable loops. I called them circles. Later I found them again in the OEISⓇ. Very nice to see all my steps again in this video now. ☺
For the rest of his days, Ben is going to wake in a cold sweat remembering the time he got 296 wrong. If his friend group is anything like mine, they'd never miss an opportunity to bring it up.
This was my thought. If you want to really chase a rabbit hole google Muratz Conjecture in relation to Collatz and you start to see real similarities. Wonder if there's something there.
You mean : the Collatz conjecture (or Hailstone or 3n+1) and variants that divide a number by 2 if it is even and else multiply it by 3 and add 1. Yes, a very similar situation that also came to my mind ( and many others I'm sure ). Great video Ben !
I LOVE THIS ONE! it's so exciting! this might be the first numberphile video that made me laugh out loud with joy and excitement. Also kudos to Ben; he's been responsible for several of my favorite numberphile videos.
You'd think to find things like this you need to invent something complicated. But here we have very easy algorithm that suddenly blows out and away so we don't even have enough computational power to check the end result. Loved the video!
There’s a lot of thing like that that amaze me. It’s trivial to prove that if you gather 6 people together, either you have 3 mutual acquaintances or 3 mutual strangers. 18 ensures 4 mutual acquaintances or strangers. But the minimum number to ensure 5 mutual acquaintances or 5 mutual strangers is still unknown (except that it’s between 43 and 48).
The 138 moment is how i feel about every sequence. Get kind of familiar with the general characteristics of the sequence, and then get blown away by a result.
Always a pleasure to see Ben. I was struggling with GCSE maths when he became my teacher and I went on to get a Masters Degree in Physics - one of the best teachers I have ever had
When I was 17 I saw James Grime’s video on amicable numbers and he showed us the keychains with 220 and 284. Being the nerdy 17-year-old I was, I bought them. I held onto those for about 6 years, until I finally had a long-term boyfriend to give one of them to. He’s an engineer so not quite as into pure math as I am, but he’s quite a good sport about his 220 wooden heart.
Parker: doing something that's almost right, but just wrong enough so it doesn't work Sparks: doing something wrong confidently, but knowing the right answer
I was sure this was going to be one of those situations like Collatz, where we're sure that everything goes to zero and it's just annoyingly difficult to prove... so it came as a big surprise, even knowing the title of the video, that we have a specific low number that we think might actually be a counterexample!
I think this is like Collatz, technically we don't know but (having spent a lot of time with these sequences) my suspicion is that infinity is an awfully long time for it to *not* end at some point. I think they will all end, it just takes enormous amounts of computations to check
@@TimSorbera the difference with collatz is that we know the answer for all starting numbers up to something like 2^60. It’s wild to me that we don’t know the answer for a starting number as low as 276.
I expected the answer to “are there any sequences that don’t collapse?” to be “we don’t know”. Especially since they’d already said it was a conjecture. But I’d never had guessed the first candidate would be such a low number unlike with the Collatz conjecture.
True, although the number 27 in the Collatz conjecture is a low number, yet blows all the way up to 9232 in a similarly shocking manner, but not quite like this! This is a more fundamental number theory, of which the Collatz conjecture is a more complex flavour.
Might be my favourite Numberphile video yet. Simple, pure maths that an 8-year-old can understand, but with a deep complexity that leaves the greatest mathematicians clueless. The content of this video is more universal than the Universe. It existed before the Big Bang, and will still exist after the Big Crunch. Perfect.
00:00 🔢 The discussion begins with a focus on number theory and exploring the properties of specific numbers. 00:08 🧮 The aliquot process involves finding the proper factors of a number, summing them, and repeating this process with the resulting number. 01:26 🚶♂️ Numbers like 8 and 24 are tested using the aliquot process to demonstrate how they can end up as deficient or abundant numbers. 03:35 🏛️ Perfect numbers, such as 6 and 28, are defined as numbers where the sum of their proper factors equals the number itself, and they never reduce to 1. 05:31 🔄 The Catalan-Dixon conjecture is introduced, questioning whether all aliquot sequences eventually end in 1 or enter a loop with perfect or amicable numbers. 07:32 🖥️ Computational methods are used to explore large numbers, like 138 and 276, demonstrating the limits of current technology and the potential for unknown behaviors in aliquot sequences. 10:23 💻 The use of logarithmic scales in visualizing large number sequences helps to manage and understand massive values that are difficult to represent linearly. 13:17 ❓ The sequence for 276 is highlighted as an unsolved problem in mathematics, suggesting it might not fit into known patterns and could challenge the Catalan-Dixon conjecture. 14:00 🔬 There are five specific numbers (276, 552, 564, 660, and 966) less than 1000 for which the behavior of their aliquot sequences remains unknown, indicating an area for future research.
30 also does an Arithmetic progression of an impressive length: 5 rounds (6, if you count the starting number, itself), right, at the start; growing by exactly 12, at each step 😮.
13:57 Well; if it keeps on growing infinitely; then, we’ll never know, because there’s infinitely many steps. Unless we work out some kind of formula for Aliquot-sequences and abundant numbers.
I'd stopped like 5 minutes into the video and programmed up some code in python to see the aliquotity (what im calling it) of numbers, and it got to 138 and i was thinking "I must've coded this wrong" but i kept following it in the debugger and it kept adding and more importantly, making sense, and so i was like "why is it doing this" and i thought id continue watching the video to try and see it, thought around the like 8 minute mark you guys would've mentioned it but then i kept watching and now at 10:07 it all makes sense. Let's see how many iterations it takes
I spent a few years factoring aliquot sequences with my computer in its spare time. It can be a lot of fun to see the sequences progress and learn the math of the ups and downs as well as the factoring algorithms and tools.
Aliquot comes from the old french aliquote, which means "contained an exact number of times in a whole". For example, a person is an aliquot of a crowd. For numbers, the aliquots are the proper divisors.
Love this... particularly the animations. Is @SparksMaths going to do a live build video for the GeoGebra applet that he used? I hope so! Ben and Brady, thanks for another great video!
4 years ago, Holy Krieger on this channel about the Mertens Conjecture "yeah it zig zags around to zero like crazy", one commentator said "yeah like my bank account". Conclusion was "if we knew it we could never write it down, because we would need all of the atoms in the uiverse to write it down"
Part of the problem is that if odd perfect numbers DO exist (many people think they don't) they're going to be very large numbers to work with - the current best estimate of the smallest one is at LEAST 2300 digits with 48 factors!
The Lehmar 5 also have some interesting interactions, from a couple that I noticed: 966 - 660 = 306 which you mentioned falls into the 276 series. 966 - 552 = 414, then minus 276 = 138, the half of 276. 966/138 = 7
Watching (very intelligent) people dissect numbers like this and find patterns is such a raw showing of human brain power and ingenuity. I've always been in awe of people with mathematical brains like this.
I wonder if one way to go about solving this problem is to count the number of members in a group/cycle and to see if there’s a limit to that number If for any loop you could find a different loop with one additional member/if there is no limit to the number of members in a group, then you know there have to be numbers that go forever
I'm surprised this doesn't attract more attention, if only because it would imply there are trajectories that can flawlessly avoid primes without being a trivial sequence of multiples. If there are numbers that trend to infinity, then the patterns they follow would be another insight into the patterns that primes follow
Those doing research on these sequences have noticed a few patterns (the technical term is "guides") generally based on two principles: * How many powers of two the number you're looking at has (fewer means smaller numbers and more means larger numbers) * Is there any power of three in the number (if yes -> bigger numbers, if no -> smaller numbers) You generally want the terms in the sequence getting smaller because that increases your odds of it terminating by hitting a prime (or some kind of cycle of numbers).
This is similar to the 3n+1 problem. The problem goes like this: For even numbers, divide by 2; For odd numbers, multiply by 3 and add 1. With enough repetition, do all positive integers converge to 1? Veritasium did a video about 3n+1, highly recommend to see that video to see the similarities between the AliQuot sequence and 3n+1.
You might also recognise 276 as the horsepower rating for a lot of Japanese performance cars. Honda NSX; Toyota Supra, Chaser, Crown, Soarer, etc.; Nissan Skyline, Stagea, Laurel, 300ZX, etc.; Mazda RX-7, Cosmo, etc.; Mitsubishi Lancer, Galant, Legnum, 3000GT, etc. - they've were all offered with their top-spec model officially having 276 bhp. It's the result of a gentlemen's agreement between Japanese manufacturers. They were worried that a looming "power war" would cause their government to restrict the speed of cars sold in Japan, so the manufacturers decided to stop it by self-imposing a limit on the advertised horsepower to 276 hp (280 ps / 206 kW), which matched the most powerful engine they were producing at that time.
This video continues at ua-cam.com/video/Yh1QUYn2f3I/v-deo.html and delves into so-called Untouchable Numbers.
More Ben Sparks on Numberphile: bit.ly/Sparks_Playlist
Do 5
Even though I know about number theory, and know about perfect, abundant, deficient, amicable, sociable, I had never heard of aspiring numbers before. Now because we have names for all of these categories, we seem to need one more. Doing the aliquot process once, divides all numbers into three categories: deficient, abundant, and perfect.
But doing an aliquot sequence, we get (potentially) seven categories, but three of them don't seem to have names:
Perfect - stay the same forever.
Aspiring - eventually get to a perfect number.
Amicable - bounce back and forth between two values.
Sociable - cycle through a loop of more than two numbers.
?1? - the ones that never get to a loop or perfect number - there might not be any in this category.
?2? - numbers that eventually get to a loop. You might say they "aspire to be amicable or sociable, rather than aspiring to be perfect".
?3? - the numbers that get to 1 eventually. Note that both abundant and deficient numbers can fall into this category.
I guess those ?1? numbers, if they are found to exist, can be named after whoever finally proves their existence.
The ?2? numbers could be called "shy" numbers - they're trying to get into the amicable/sociable group. I suppose this category could be split into two.
And the ?3? category in which the majority of numbers fall, should have some name, too. At first, I was thinking to propose calling them "mortal" numbers, because through the aliquot sequence, they eventually "die". But that seems too dark of a name.
The next puzzle for you to solve: The 300 Coins Problem.
300 coins are placed randomly on a table.
A 300 letters long message (Signal) is written, one letter per coin, that would lead to a hidden treasure.
Then the coins are flipped over and a randomly generated Noise 300 letters long is written on the other side of coins.
The coins then get put in a bag and scrambled.
Finally, the coins are put back on the table.
Your task is to flip and move the coins around until the original message is recreated.
Can you do it?
I checked wikipedia on sociable numbers for my own curiosity, and if it is accurate then:
The only known loop lengths are 1 (perfect), 2 (amicable), 4, 5, 6, 8, 9 and 28. (and 5, 9 and 28 only have 1 known sequence each)
"It is conjectured that if n is congruent to 3 modulo 4 then there is no such sequence with length n."
So loops with length n=4k+3: 3,7,11,15,... is probably/maybe not possible.
U still exist?
The Numberphile Conjecture: If you give numberphile enough time, every integer will have a video about it.
That's the plan
Absolutely beautiful and simple conjecture! And i love that that's the plan!
the numberphile playlist of all videos will then become an OEIS sequence since it will have a unique sequence of integers by age of video.
Fun fact: 9538 is the smallest number that can't be defined in 30 English words or less.
@@thewhitefalcon8539”Nine thousand five hundred thirty-eight”
296 🤦♀
(my wife is now not speaking to me for 284 days apparently)
Was it just a brainfart, or did you think about 296 for different reasons and got it mixed up?
I think I had 496 in my head (for perfect reasons) and it contaminated my thoughts. Mea culpa. 🫤
Epic fail )
🫂
@@NorlanderGT the answer is at 4:02
The fact that he doesn't know the number that's on his wife's half of the heart is concerningly humorous
something something keychain parties
I think he's ending up in the dog house for a while ;)
Time stamp?
4:58
284
8:58 "It's so over!"
9:01 "We're so back!"
9:04 "It's so over!"
9:12 "We're so back!"
in the midst of "its so over", I found there was within me, an invincible "we're so back!"
I looked specifically for this comment
Brought to you by... Jelle's Marble Runs!
Reminds me of Tetris gameplay shooting for some crazy world record breakthrough.
WHEEEEEEE
brady commentating the 138 graph has me hysterical oh my lord
Here before this comment is popular
masterpiece
It was the "go son!!" That sent me
They need him in as a guest commentator on @jellesmarbleruns
Made my day and it's not even 8am!
That amicable number heart keychain is one of the nerdiest romantic thing I've ever heard of - it's very cute
Didn't James Grime mention this as a thing to do when he taught us amicable numbers like 10 years ago?
Here before this comment is popular
@@HasekuraIsuna I wonder if that's where Ben got the idea from.
So romantic to forget your wife's number
You can buy the keyrings at Maths Gear.
This is really what numberphile is all about
This is the video I'm going to cite for the foreseeable future when someone asks what number theory is. And I'm going to foist it on my kids tonight
Just need Tadashi Tokeida to incorporate some weird toy into it
+
To me, this is my favorite kind of numberphile video because it’s still very compelling while keeping the math below, like, a trigonometry level (which was where I stopped fully understanding math)
8:46
What an absolute roller coaster ride of emotions!
I'm still recovering
@@numberphile I think we should call them "rollercoaster numbers"
This is why I love mathmatics: a relatively simple question leads to a whole mini world of calculations and mysteries.
The universe doesn't care about intuition 😂
@@daniel_77.your comment makes no sense
@@ToneHelix97 Sorry. I meant that the things we see and do, even the seemingly simple natural numbers, still hides a lot of complex reasoning. When things may seems obvious and intuitive, In reality it doesn't work like that.
Brady's commentary of the highs and lows of 138 was awesome
For those interested, aliquot sequnce for 276 is currently at step 2146, not 2090. The last advance was made in January 2024, when a C209 was split into a P98 and P112. That means the number of digits, C for composite, P for prime. C209 is the supercomputer (or rather a distributed computing project) territory with months/years of GNFS sieving required to factor it. The previous hurdle was step 2140, passed in August 2022 after factoring a C213 which turned out to be P97 * P116.
He's gonna have to sleep on the couch tonight because he forgot his wife's amicable number... AGAIN!
It should be easy enough to recalculate if you forgot shouldn't it
😂
11:16
"The answer is... We don't know"
Brady, utterly disappointed: "Of course not..."
you mathematicians don't know shi...
I like to imagine that 276 goes all the way up straight to the first and only odd perfect number, and that number also happens to be the first number to start a loop that disproves the collatz conjecture.
Lol that would be funny
@@Lucashallaltrying to square each digit of 276, and add them together, you will then arrive into the melancoil, which is problematic.
Just want to point out that the first number that does a really wild ride was 138, and the next number he showed was 276, which is exactly double. And then the next Lehmer five is 552, again exactly double.
If you look at it in terms of using 138 as a base number n, 3 of the 5 numbers are multiples of n. 2n, 4n, 7n.
@AndyWitmyer 276 already has an index (sequence length) over 2,100 and 564 has an index near 3,500 currently. 138 ONLY took an index of 177 to resolve, thus both sequences are already at least one order of magnitude larger and show no signs of ending anytime soon.
So lets see how much magic 276 has in it. 3 times 276 is 828. The sequence is open. It joins the 660 sequence immediately. Also one of the Lehmer Five.
@@patrickmckinley8739you ought to try something like, 4+49+36, which becomes the number 89, and this point, we land in the melancoil.
@@gordontaylor2815 if the numbers in the aliquot sequence are random, since they grow exponentially, the chance of finding a prime at any step n is proportional to 1/n, so the chance of finding a prime up to step n would be (somewhat) proportional to log(n), meaning you would expect orders of magnitude longer chains that are still finite if they just grow faster. aliquot sequences aren't random nor does it seem they just hit primes and stop, but if any similar situation is happening (which seems somewhat likely to me), then only 20 times the chain length would not be that strong of evidence for the chain suddenly being infinite
This was an awesome sort of "back to the roots of Numberphile" video, and the general excitement overall from both Ben and Brady were just great.
The first time I programmed this was in the 80s on a C64. I hit brick walls several times; first my algorithm to compute the proper divisor sum was too simple and thus too slow for the gigantic numbers I ran into for the 138. When I fixed that, they still kept growing beyond the numbers the programming language could handle. I had to restart the whole programming several times until I found what I really was looking for: These things which I now just learned are called sociable loops. I called them circles. Later I found them again in the OEISⓇ.
Very nice to see all my steps again in this video now. ☺
If you ever doubted yourself after all these years Brady - you still got it. Absolute banger of a Numberphile video!
The best part of the video is where he watches the Price of Bitcoin
I was about to say!… the path for 138 looks like a stock price.
I am a handsome man
For the rest of his days, Ben is going to wake in a cold sweat remembering the time he got 296 wrong. If his friend group is anything like mine, they'd never miss an opportunity to bring it up.
It'll be his version of the Parker square
This feels like the 3n+1 conjecture, but finding an actual number that blows to infinity!
You noticed that, too!
It practically is, in more ways than one.
This was my thought. If you want to really chase a rabbit hole google Muratz Conjecture in relation to Collatz and you start to see real similarities. Wonder if there's something there.
You mean : the Collatz conjecture (or Hailstone or 3n+1) and variants that divide a number by 2 if it is even and else multiply it by 3 and add 1. Yes, a very similar situation that also came to my mind ( and many others I'm sure ). Great video Ben !
5:30 Whoops, that's worth at least an extra flower in the next bouquet.
I can't believe y'all is still coming up with videos like this are all these years. You're legends
I LOVE THIS ONE! it's so exciting! this might be the first numberphile video that made me laugh out loud with joy and excitement.
Also kudos to Ben; he's been responsible for several of my favorite numberphile videos.
You'd think to find things like this you need to invent something complicated. But here we have very easy algorithm that suddenly blows out and away so we don't even have enough computational power to check the end result. Loved the video!
There’s a lot of thing like that that amaze me. It’s trivial to prove that if you gather 6 people together, either you have 3 mutual acquaintances or 3 mutual strangers.
18 ensures 4 mutual acquaintances or strangers.
But the minimum number to ensure 5 mutual acquaintances or 5 mutual strangers is still unknown (except that it’s between 43 and 48).
The 138 moment is how i feel about every sequence. Get kind of familiar with the general characteristics of the sequence, and then get blown away by a result.
This is the best Numberphile that I've seen in years
some of these numberphile videos genuinely shock me to my core
well done
Always a pleasure to see Ben. I was struggling with GCSE maths when he became my teacher and I went on to get a Masters Degree in Physics - one of the best teachers I have ever had
Bumping so he may see this
@SparksMaths see the parent comment :)
5:27 it was almost physical the amount of relief I felt seeing the correct number on the other half of the heart. ❤️
Never this channel fail to amaze me.
This is one of those that are so simple to understand that is mind blowing
When I was 17 I saw James Grime’s video on amicable numbers and he showed us the keychains with 220 and 284. Being the nerdy 17-year-old I was, I bought them. I held onto those for about 6 years, until I finally had a long-term boyfriend to give one of them to. He’s an engineer so not quite as into pure math as I am, but he’s quite a good sport about his 220 wooden heart.
You don’t often read a story so wholesome and heartwarming on the internet.
Brady cheering on 138 is so funny.
GO ON SON!!!
220 and 296. The Parker Heart.
HAHA
The Sparks Amicable
Parker: doing something that's almost right, but just wrong enough so it doesn't work
Sparks: doing something wrong confidently, but knowing the right answer
I was sure this was going to be one of those situations like Collatz, where we're sure that everything goes to zero and it's just annoyingly difficult to prove... so it came as a big surprise, even knowing the title of the video, that we have a specific low number that we think might actually be a counterexample!
Yeah I was shocked how low the first number is where we haven’t figured out the answer.
I think this is like Collatz, technically we don't know but (having spent a lot of time with these sequences) my suspicion is that infinity is an awfully long time for it to *not* end at some point. I think they will all end, it just takes enormous amounts of computations to check
@@TimSorbera the difference with collatz is that we know the answer for all starting numbers up to something like 2^60.
It’s wild to me that we don’t know the answer for a starting number as low as 276.
@@TimSorbera Richard K Guy presented some evidence for a counter-conjecture that there are unbounded aliquot sequences.
One of the most exciting and touching video i've seen on UA-cam. thanks again Numberphile
I expected the answer to “are there any sequences that don’t collapse?” to be “we don’t know”.
Especially since they’d already said it was a conjecture.
But I’d never had guessed the first candidate would be such a low number unlike with the Collatz conjecture.
True, although the number 27 in the Collatz conjecture is a low number, yet blows all the way up to 9232 in a similarly shocking manner, but not quite like this! This is a more fundamental number theory, of which the Collatz conjecture is a more complex flavour.
Love the old school style videos, love Ben's enthusiasm, great video for my sunday morning, thanks lads
8:49 I guess I'm a Nerd, I was genuinely excited & cheering the number on as it went. lol
An instant classic! Great job guys
Cheers - glad you enjoyed it
The Australian accent is perfect for providing passionate commentary on an evolving graph!
"Are there any that don't come back." My immediate thought was, "It's a conjecture--we don't know."
These are my favorite numberphile videos. Great stuff
You ought to test the number 276 out for happiness and perfection.
In chemistry, an aliquot is taking off part of your solution and then only doing something with that part rather than the whole solution.
"Of course." - I will never regret subbing to your channel.
Videos about a specific number like this are the best
I loved the commentary for 138 :D It gave me a really good laugh! And also the youtube channel idea hahaha, brilliant
I love all the Numberphile alumni but I always come back to Ben. Top 10 Numberphile videos are probably 40% Ben Sparks here.
I love Ben's videos. Also he looks different than the previous video whenever he has been gone for awhile.
Might be my favourite Numberphile video yet. Simple, pure maths that an 8-year-old can understand, but with a deep complexity that leaves the greatest mathematicians clueless. The content of this video is more universal than the Universe. It existed before the Big Bang, and will still exist after the Big Crunch. Perfect.
This is in my top 3 Numberphile episodes. Awesome stuff!
This is amazing. Humble little 276 is beyond our means.
That #: 276 described in Acts 27!
2️⃣▶️7️⃣▶️6️⃣
Acts 27:37 - And we were in all in the ship two hundred threescore and sixteen souls.
I love how absolutely inspired Brady was by 138
"The Himilayas..."
:,D
00:00 🔢 The discussion begins with a focus on number theory and exploring the properties of specific numbers.
00:08 🧮 The aliquot process involves finding the proper factors of a number, summing them, and repeating this process with the resulting number.
01:26 🚶♂️ Numbers like 8 and 24 are tested using the aliquot process to demonstrate how they can end up as deficient or abundant numbers.
03:35 🏛️ Perfect numbers, such as 6 and 28, are defined as numbers where the sum of their proper factors equals the number itself, and they never reduce to 1.
05:31 🔄 The Catalan-Dixon conjecture is introduced, questioning whether all aliquot sequences eventually end in 1 or enter a loop with perfect or amicable numbers.
07:32 🖥️ Computational methods are used to explore large numbers, like 138 and 276, demonstrating the limits of current technology and the potential for unknown behaviors in aliquot sequences.
10:23 💻 The use of logarithmic scales in visualizing large number sequences helps to manage and understand massive values that are difficult to represent linearly.
13:17 ❓ The sequence for 276 is highlighted as an unsolved problem in mathematics, suggesting it might not fit into known patterns and could challenge the Catalan-Dixon conjecture.
14:00 🔬 There are five specific numbers (276, 552, 564, 660, and 966) less than 1000 for which the behavior of their aliquot sequences remains unknown, indicating an area for future research.
"LIKE MARBLE RACING" I LOVE THIS MAN
This is a very cool one, exactly the kind of stuff that interests me. Ben is so good at explaining this kind of stuff
The expresions and the sparkling enthusiastic eyes of Ben is invalueable.
Perhaps my favorite Numberphile to date!
Fabulous video! Always a mindblowing experience watching Numberphile videos! This one was particularly inspiring 🙏🙏🙏
I frickin' love Ben
Ben always has the best videos
30 also does an Arithmetic progression of an impressive length: 5 rounds (6, if you count the starting number, itself), right, at the start; growing by exactly 12, at each step 😮.
8:47 is one of the most satisfying rides in numberphile history ❤
13:57 Well; if it keeps on growing infinitely; then, we’ll never know, because there’s infinitely many steps. Unless we work out some kind of formula for Aliquot-sequences and abundant numbers.
I'd stopped like 5 minutes into the video and programmed up some code in python to see the aliquotity (what im calling it) of numbers, and it got to 138 and i was thinking "I must've coded this wrong" but i kept following it in the debugger and it kept adding and more importantly, making sense, and so i was like "why is it doing this" and i thought id continue watching the video to try and see it, thought around the like 8 minute mark you guys would've mentioned it but then i kept watching and now at 10:07 it all makes sense. Let's see how many iterations it takes
I spent a few years factoring aliquot sequences with my computer in its spare time. It can be a lot of fun to see the sequences progress and learn the math of the ups and downs as well as the factoring algorithms and tools.
For those new to the topic - you can check the known factorizations for any sequence on factordb
i love this channel so much
We love the people who watch it!
Classic Numberphile!!!!!! ❤❤❤
Write phyton code to check ❌️
Write code in geogebra ✅️
12:56 That was so ominous. "...and I've gone a bit further... and so have others." Then the atmospheric sounds. 😂
Ben's vids are always so interesting.
Aliquot comes from the old french aliquote, which means "contained an exact number of times in a whole". For example, a person is an aliquot of a crowd. For numbers, the aliquots are the proper divisors.
I still come back to this video from time to time to watch Brady's commentary on 138.
I squealed with glee when Ben's face popped up.
I love his communication skills and his topics.
Love this... particularly the animations. Is @SparksMaths going to do a live build video for the GeoGebra applet that he used? I hope so!
Ben and Brady, thanks for another great video!
Link to the file is in the description, in case you missed it
4 years ago, Holy Krieger on this channel about the Mertens Conjecture "yeah it zig zags around to zero like crazy", one commentator said "yeah like my bank account".
Conclusion was "if we knew it we could never write it down, because we would need all of the atoms in the uiverse to write it down"
Excellent video! Original Numberphile :D
O.G.
The main TITLE=276 is at 11:30, all lehmer numbers (running horrendously away but less than a 1000) are 276, 552,564,660,966.
Next video better be Ben explaining why we haven't found an odd perfect number
Part of the problem is that if odd perfect numbers DO exist (many people think they don't) they're going to be very large numbers to work with - the current best estimate of the smallest one is at LEAST 2300 digits with 48 factors!
@@gordontaylor2815we might also test several numbers for happiness, as to whether you get to the number 1, or approach the melancoil.
"Maybe it's a perfect number?" "It's an aspiring number" haha i love it
This is just as amazing as the 196 Palindrome problem!
I love a classic numberphile "number " video ! Hope to see a lot more of them !
The Lehmar 5 also have some interesting interactions, from a couple that I noticed: 966 - 660 = 306 which you mentioned falls into the 276 series. 966 - 552 = 414, then minus 276 = 138, the half of 276. 966/138 = 7
Brady cheering has to be the funniest thing ever: "Go son! Go! Never stop!" hahahahahahahah
cool! a new video from Numberphile
yay!!!!!
Absolutely do that, that was a fun sequence to watch with the commentary
Watching (very intelligent) people dissect numbers like this and find patterns is such a raw showing of human brain power and ingenuity. I've always been in awe of people with mathematical brains like this.
5:26 Numberphile is always answering the really important questions.
I'm not a mathematician, yet, I find these curiosities fascinating. Thanks for the view.
I wonder if one way to go about solving this problem is to count the number of members in a group/cycle and to see if there’s a limit to that number
If for any loop you could find a different loop with one additional member/if there is no limit to the number of members in a group, then you know there have to be numbers that go forever
I'm surprised this doesn't attract more attention, if only because it would imply there are trajectories that can flawlessly avoid primes without being a trivial sequence of multiples. If there are numbers that trend to infinity, then the patterns they follow would be another insight into the patterns that primes follow
Those doing research on these sequences have noticed a few patterns (the technical term is "guides") generally based on two principles:
* How many powers of two the number you're looking at has (fewer means smaller numbers and more means larger numbers)
* Is there any power of three in the number (if yes -> bigger numbers, if no -> smaller numbers)
You generally want the terms in the sequence getting smaller because that increases your odds of it terminating by hitting a prime (or some kind of cycle of numbers).
Time to build a planet-sized supercomputer like in Hitchhiker's Guide to keep pushing 276!
276 always makes me think of the Pennsylvania Turnpike near Philly.
The genuine pain in his voice when he says "it stopped. oh:(" at 9:43 hahah
Investment for quantum computer technology must be enhanced in all over the world just because of seeing the journey of 276. It worths every penny :)
I do wonder if there is any connection to the collatz conjecture...
This is similar to the 3n+1 problem.
The problem goes like this:
For even numbers, divide by 2;
For odd numbers, multiply by 3 and add 1.
With enough repetition, do all positive integers converge to 1?
Veritasium did a video about 3n+1, highly recommend to see that video to see the similarities between the AliQuot sequence and 3n+1.
Ben: It's the first of what they call The Lehmer Five.
Me: Well *i* thought it was cool.
You might also recognise 276 as the horsepower rating for a lot of Japanese performance cars. Honda NSX; Toyota Supra, Chaser, Crown, Soarer, etc.; Nissan Skyline, Stagea, Laurel, 300ZX, etc.; Mazda RX-7, Cosmo, etc.; Mitsubishi Lancer, Galant, Legnum, 3000GT, etc. - they've were all offered with their top-spec model officially having 276 bhp.
It's the result of a gentlemen's agreement between Japanese manufacturers. They were worried that a looming "power war" would cause their government to restrict the speed of cars sold in Japan, so the manufacturers decided to stop it by self-imposing a limit on the advertised horsepower to 276 hp (280 ps / 206 kW), which matched the most powerful engine they were producing at that time.