Ben Sparks discusses aliquot sequences and why 276 holds a surprise. This video continues at • Untouchable Numbers - ... and delves into so-called Untouchable Numbers. More links & stuff in full description below ↓↓↓
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Perfect Numbers on Numberphile: • Perfect Numbers on Num...
Amicable Numbers: • 220 and 284 (Amicable ...
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This video continues at kzhead.info/sun/jMxqgbmRpWScZoE/bejne.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
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!
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
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
+
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.
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
@@vikashchandra9917 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.
He's gonna have to sleep on the couch tonight because he forgot his wife's amicable number... AGAIN!
Brady's commentary of the highs and lows of 138 was awesome
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.
One wonders if 276 and 552's aliquot trajectory would ultimately be in some way analogous to 138's, except by several orders of magnitude longer in sequence
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.
11:16 "The answer is... We don't know" Brady, utterly disappointed: "Of course not..."
you mathematicians don't know shi...
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 !
220 and 296. The Parker Heart.
HAHA
The Sparks Amicable
8:46 What an absolute roller coaster ride of emotions!
I'm still recovering
@@numberphile I think we should call them "rollercoaster numbers"
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
Brady cheering on 138 is so funny.
GO ON SON!!!
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.
It'll be his version of the Parker square
In chemistry, an aliquot is taking off part of your solution and then only doing something with that part rather than the whole solution.
5:30 Whoops, that's worth at least an extra flower in the next bouquet.
If you ever doubted yourself after all these years Brady - you still got it. Absolute banger of a Numberphile video!
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.
An instant classic! Great job guys
Cheers - glad you enjoyed it
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.
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.
5:27 it was almost physical the amount of relief I felt seeing the correct number on the other half of the heart. ❤️
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.
"Of course." - I will never regret subbing to your channel.
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).
"Are there any that don't come back." My immediate thought was, "It's a conjecture--we don't know."
8:49 I guess I'm a Nerd, I was genuinely excited & cheering the number on as it went. lol
Never this channel fail to amaze me. This is one of those that are so simple to understand that is mind blowing
These are my favorite numberphile videos. Great stuff
One of the most exciting and touching video i've seen on KZhead. thanks again Numberphile
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.
some of these numberphile videos genuinely shock me to my core well done
Videos about a specific number like this are the best
I miss these old kind of videos! And Ben is always a treat!
Threat?
@@Pathakin. Gotta love autocorrect 🙃
"LIKE MARBLE RACING" I LOVE THIS MAN
The Australian accent is perfect for providing passionate commentary on an evolving graph!
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.
This is the best Numberphile that I've seen in years
Fabulous video! Always a mindblowing experience watching Numberphile videos! This one was particularly inspiring 🙏🙏🙏
I love all the Numberphile alumni but I always come back to Ben. Top 10 Numberphile videos are probably 40% Ben Sparks here.
I can't believe y'all is still coming up with videos like this are all these years. You're legends
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
Love the old school style videos, love Ben's enthusiasm, great video for my sunday morning, thanks lads
8:47 is one of the most satisfying rides in numberphile history ❤
I frickin' love Ben
Bro is one digit away from summoning a fandom
what fandom
@@emperortgp2424 Based on the account, I assume they're referring to the number 2763 being mentioned multiple times in Battle For Dream Island episodes. Hope that helps! :)
The prophecy is spoken, we must test it.
@@MathNerd1729 yes
For those new to the topic - you can check the known factorizations for any sequence on factordb
Excellent video! Original Numberphile :D
O.G.
It's extremely funny to hear Python talked about as the "fast" option.
It's not as slow as people say, it just isn't as blazing fast as something like C. This only really becomes apparent when you start massively scaling your computations, so you wouldn't want to run Python on a scientific supercomputer.
Fortran77 ftw
Everything is relative :)
@@hammerth1421 except it does matter a lot - when your calculations are 10x (if not 100x) slower, it means you can do 100x less on your own PC before you have to resort to clusters/supercomputers. which of course is terrible news for hobbyists, not that that _really_ matters
I loved the commentary for 138 :D It gave me a really good laugh! And also the youtube channel idea hahaha, brilliant
I love Ben's videos. Also he looks different than the previous video whenever he has been gone for awhile.
Perhaps my favorite Numberphile to date!
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.
Tired: Forgetting your spouse's birthday Wired: Forgetting the amicable number on your spouse's keychain
This feels like just the collatz conjecture with extra steps! :D
I love a classic numberphile "number " video ! Hope to see a lot more of them !
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.
cool! a new video from Numberphile yay!!!!!
I squealed with glee when Ben's face popped up. I love his communication skills and his topics.
i love this channel so much
We love the people who watch it!
Simply amazing.
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.
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).
Classic Numberphile!!!!!! ❤❤❤
Write phyton code to check ❌️ Write code in geogebra ✅️
I'm so happy that Brady watches marble racing!
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
Brady still amazing with his genuinely excitement ❤🎉
Plot on log scale!! Edit: oh, thanks. Whew!
And the following day, he went home to an empty house, and only found a post-it note on the kitchen table with a single thing on it: 276
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!
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"
Stunning!
Appreciate ya. Thanks for sharing.
Great video, thanks!
5:26 Numberphile is always answering the really important questions.
"Maybe it's a perfect number?" "It's an aspiring number" haha i love it
Amazing! Love the fact that we don't know. Seems like too simple a question to not know the answer!
Thoroughly engaging
I was in Brady's shoes as he commentated about 138 😅 But 276 is as great as it can be
I'm all aboard for Number Racing channel! Is that what we're calling it? Number Racing?
Brady is a gift to humanity
brady cheering the graph on was so wholesome ^_^
Always interesting!
It is no coincidence that the fine-structure constant is very nearly 1/137, approaching 1/138 over time per my TOP DOWN cosmology previously posted at this site just now. Though the rate of change for alpha is slow, new resonances expressed as prime numbers are being rapidly made available for quantum expression, meaning that when 1/138 is reached, there will be a collapse, the universe divides, and the fine-structure constant will jump to 1/276, as in perfection.
Love Ben!
The genuine pain in his voice when he says "it stopped. oh:(" at 9:43 hahah
The entire dialog of Brady watching 138 going on was the most entertaining thing I've ever seen.
Ben Sparks 🤝 Geogebra files
Yay!!! Old school numberphile!
I am a simple beast. I see a numberphile thumbnail that's just a number written across the screen, and I click it
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
So is so interesting to watch 😮
One word: amazing