OK, but why just go forwards? Why not see if you can go backwards to find what leads to your starting number, and if the process goes on in that direction as well?
I'm curious: Why is there no analytical method, no proof? Why are we stuck generating sequences to see what they do? Is this impossible to reason about? Maybe if I were a regular viewer, I'd know that, but I've just been dipping into the channel now and then. It feels like something important has been left unsaid.
The proof that 5 is untouchable is easy, so easy I am surprised they didn't include it. You can only make 5 by summing 1 and 4. But if 4 divides a number so does 2, so it is impossible to have an aliquot sum in the first place.
by the same logic, though, any odd number is 2n + 1, and if 2n divides a number so does 2. so all odd numbers larger than 3 are untouchable? edit: nope, this doesn't work. you can also do 2n+1 with different factors. ie it's not necessary that you get 2n+1 by having the factors 1 and 2n.
@@renyhp It's that 5 is the only number where 2n+1 would be the ONLY way to make it. Other odd numbers can be formed by 1+2n, but they can also be formed in other ways. For example, 7 can be made by 2n+1 with n=3, but it can also be made by 1+2+4.
As a programmer, I am fascinated by videos showcasing something that we cannot compute. When watching the first video, as he explained "we do not know" my immediate reaction was "I'm gonna write something and find it", then I saw the scope of how how far it has been checked and I immediately switched to "how the heck did someone write something that could check that high".
This is unfortunately almost always the case for the "trivial" problems. There are multiple conjectures that are easy enough to understand in terms of simple Maths that are also fun to program and try for yourself. But for all of them, when you fancy the idea of looking into it, turns out somebody else with access to a super computer has already checked all the numbers up to a thousand digits. 😔
And my second thought about it was: and for several centuries all the greatest mathematicians, like Euler or Newton, had to calculate all their things manually. It is so much more convenient and error resistant now.
According to Martin Gardner's article on the topic (reprinted in his book, "Mathematical Magic Show"), the 28-cycle was announced by P. Poulet in 1918. (Or at least, Poulet announced a 28-cycle beginning with 14316; I assume it's the same one.)
The odd untouchable numbers are related to Goldbach's conjecture. If every even number greater than 4 can be written as a sum of two distinct primes, then every odd number greater than 5 is not untouchable. Say 2n + 1 > 5 and 2n = p + q, with p and q distinct primes, then 2n + 1 is the aliquot of pq.
Feels perilously close to 3x+1! I'd love to have seen some of the ways the analysis for this has been done mathematically rather than just computationally.
6:00 I also like, how the 2 zig-zaggy patterns perfectly intertwine, because 1 graph hit the same amicable number 1 turn later. It’s like an amicable pair of amicable pairs, with that nice DNA-pattern 🧬💞. 😊
What a brilliant conjecture, just the kind of thing that really interests me, trivial to understand and if a solution is ever found then it'll be incredibly complex in comparison :D
So many tjmes a physicist discovers something profound about reality, and then realizes a mathematician has already been there 10 years ago just for fun. Im all for having fun with math - for the joy of it, and also for the chance of a true insight into reality
@var67 hyperbolic/non-euclidean geometry came first as a lark... then Einstein found it useful to describe reality. Early group theory; turns out extremely applicable to conservation laws, re Emmy Noether. -1/12ths turns out to give correct answers in some calculations. Complex numbers came first when mathematicians were playing around with quadratics, etc... ended up very useful for quantum physics. There's 4.. could probably come up with more
@var67 (p.s. i never said "a physicist thinking..." ... I was just pointing out that mathematicians have often come across something while just having a lark that ends up being important for physicists later on.)
@@publiconions6313 I misunderstood, as you may have figured out. I did think you meant: physicist "discovers" something but no the mathematician discovered it earlier. So yes you're right, physicists get their grubby paws on ANY old maths. I should know, as a (failed) physicist. Btw, I loved the Journal of Recreational Mathematics back in the day.
@@var67 word! ; ) I figured we were just swinging past each other a bit there. I wonder, do you listen to Daniel Whiteson's podcast?.. it may be a little layman, I was never even close to a physicist (failed or otherwise.. lol, I sell insurance - snore) but he recently had an episode entirely based on the idea that "hey, octonions are cool, wonder if they apply somewhere" .. really peaked my imagination, especially considering how quarternions ended up making a lot of sense for QCD?.. it's over my head ,so I might have the specifics wrong. But my dream scenario is that some Numberphile in some corner of the world makes a connection like that
5 is the only odd untouchable number if a slight strengthening of Goldbach's conjecture holds. Goldbach's conjecture states that ever even number greater than 2 is the sum of two prime numbers. A stronger statement that also seems true is that every even number greater than 6 is the sum of two _distinct_ prime numbers. If this is true, then given any odd number n > 7, we can write n= p + q + 1 with p and q distinct primes. But the only proper factors of pq are 1, p, and q, so its aliquot sum is s(pq) = 1 + p + q = n. That leaves the special cases of 1, 3, 5, and 7. For any prime p, s(p) = 1, s(4) = 3, and s(8) = 7. So only 5 is untouchable.
"Because it's fun" needs to be more accepted as a reason to do anything. Why art, why music, why dance? Because it's fun. Why science? Why mathematics. Grrr, you're not allowed to have fun!
If the sequence hits a prime, it collapses right away, right? And I appreciate that there's other ways to start the collapse as well. But just as a first path into understanding this: isn't there some kind of competition going on between the speed at which the sequence increases and the spacing of the primes at ever larger orders of magnitude? I other words: do we know anything about how big the abundancy of high numbers will be in relative terms, just like we know that the prime density behaves like n/log(n)?
The sequences don't just increase, though. They go through periods of decreasing as well. Determining a heuristic for how quickly they grow sounds difficult since, even averaging it out, there doesn't seem to be a steady growth rate.
There are no other ways to collapse. To hit 0, it first has to hit 1 (unless it started at 0), and to hit 1, it has to first hit a prime (unless it started at 1). So every sequence that terminates at 0 goes a -> b -> ... -> p -> 1 -> 0 with p prime. More specifically, p will always be an odd prime other than 5 (since 2 and 5 are untouchable), unless you start at p. And to answer your question, the asymptotic density of abundant numbers is known to be between 0.2474 and 0.2480. That means that as n increases without bound, the proportion of numbers less than n that are abundant approaches some limit that is about 24.77%.
Loved this video, and how it connected primes, perfects, amicables and sociable sequences together! Great you also included the very long sequence of 28 I think. There have been a lot of aliquot sequences of length 4 discovered as well. A few years back I was very involved in searching for all 14 and 15 digit amicable pairs, now all are known to 20 digits or more! Was also co-discover of a couple of largest known amicable pairs, but these were later beaten quite well.
2856 blasting into my top 10 favorite numbers. 28 doubled is 56 so it's easy to remember and it creates that awesome "megaloop" which has a 28 step cycle!
When I am looking at these graphs I am convinced that we are looking at some strange unexplained feature of the universe and how it works, why do specific numbers have specific properties and why are there patterns and shapes it feels like a sublime mystery hidden in there
In the graph at 5:00, one number grows considerably faster then any other number. Like, even within the possibly unbounded ones, that one grows muc faster. What number is that?
One thought looking at the graph of all the sequences, is what does it look like if you line up all the end points, so if two sequences merge, rather than parallel lines, it is just a single line that merged. Social loops would need to do something like aligning the first repeat of the lowest point of the cycle.
It's crazy to think that if it can be shown that just 1 of these sequences is unbounded, then we immediately know that infinitely many numbers will never hit 1, a perfect number or a loop, blowing the whole thing wide open
Must have missed it, can you explain why? If a sequence beginning with N is unbounded, then obviously any number whose aliquot is N will also be unbounded (and equally for any point on the unbounded path). But how do you show that there is a number with an aliquot of N.
@@silver6054 If the aliquot sequence for N goes to infinity, then the aliquot sequence for every number in that same sequence also goes to infinity, so you get infinite counterexamples for free from just N
@@silver6054OP may he been thinking of something a bit less trivial, but there is an easy reasoning: If the aliquote of x is unbounded, it has an infinite amount of numbers in its aliquote sequence. Every one of the numbers in its aliquote sequence also has an unbounded aliquote sequence.
When someone asks, "what's the point " i think the simple answer is that understanding comes from the analysis of factsand it's impossible to predict which facts are going to be a part of that process. Most people out there will have figured this out already
Just started learning Python this year and an aliquot sequence generator was one of the first things I wrote. I don't have a graph though. That's cool.. how do you do that? Also.. very good seeing you again Ben. How is it that you're always interested in what I'm interested in and making videos about it?
5:09 He doesn't call attention to these, but notice 5 parallel lines terminating from a height of 6 digits just before reaching 40 terms. That's the 318 sequence. And the 7 that terminate after about 50 terms: that's 180. Then there's that blue line that takes off like a rocket and hits the ceiling in 50 terms. Not even one of the Lehmer Five. That's 840.
When adding the divisors you get 1 + something. 5 is 1+ 4, but if a number has 4 as a divisor it also has 2 as a divisor, sothat doesn't work. 5 = 1 + 1 + 3 doesn't work either, as you can't have two ones. 1 + 2 + 2 also doesn't work, as you instead have two twos. Quite easy to prove.
Prove what exactly? That summing the divisors always yields a number which is 1+something? Thats trivial, and im not sure what your examples are getting at.
The end of the video. It talks about an untouchable number. A number which no other number can reach with the aliquot sequence. In order to get 5 you need 1+2+2 which has two of the same number. A number cannot have the same number twice as a factor since it's only counted once. The nearest numbers are 4 and 6 from 1+3 and 1+2+3. Therefore there is no way for the factors of another number to sum to 5 Therefore untouchable.
2:55 "This is like properties of numbers that we didn't put there." I can see why Ben would say this, but I'm not entirely convinced. I would argue that these patterns and properties are a product of a human decision, because we decided the rules for determining an aliquot sequence. This is like the Collatz conjecture, in that it explores the properties of a sequence, but that sequence is the product of a specific algorithm humans came up with. It's interesting to study, but it's only useful if that specific algorithm is useful for anything. I want to believe that there is something useful to be learned from studying these patterns, but the more I think about it the less I think there can be. The rules for these sequences don't strike me as having any direct connection to some inherent property of numbers, like the primes do. Is there anything inherently useful about the sum of an integer's factors? Where does that ever come up in mathematics outside of these novel sequences? It seems to me that these sequences are little more than a toy for mathematicians to play around with. They are useful insofar as they are an exercise in searching for patterns. Maybe the pursuit of these questions leads mathematicians to develop new tools and techniques that have applications elsewhere in mathematics, and that could be a good thing. But I think it would be a mistake to think that any of the patterns themselves (like whether 276 diverges) actually tells us anything useful about numbers in general, because that pattern is entirely dependent upon the arbitrary algorithm we used to generate it.
Beautiful! 😊 Give me five! “What’s the Point!?” Well.. a single point is kind of boring..that’s why you pick another, and another, and start connecting em, and you discover one by mistake, another by coincidence ..and, there Is the Point :)
New at loving number theory so possibly a very noob question. If there is one Aliquot number that does not converge, wouldn't that imply there are an infinite number of non-convergent numbers? If the prime factors of x sum to y, and x never converges wouldn't that mean that y never converges, and then the same logic apply to y's decedents, as well?
@@smicksatusadotnet That kind is stuff is fascinating to me. We’ve found none, but if we find one, we’ve found an infinite number of them. Math is like magic and alchemy sometimes.
What's the distribution of sequence lengths before reaching a resolution? Are the Lehmer Five just the tail of a distribution or are they outliers with lengths much longer than any other numbers? If it is the latter, that suggests something interesting is going on with those numbers. If it is the former, it could just be random and some numbers had to be the longest and it just so happens to be them.
5 then has one of the most strange meanings of all numbers, maybe the relation with the halving of things due to the nature of the base we're using, maybe there are some other intuitions to grasp if we search for this function in other number bases
7:32 - Recall: 5 is the only odd prime greater than 3 that is less than all the primes greater than 5. So there's that to ponder... It's a tautology folks, just like _ALL_ the rest of math(s). Don't overthink it!
I imagine it like the (nonnegativ) integers become vertices of a (infinite) directed graph, some are unconnected like 5. The graph has cyclic subsets, and leafs (vertices with only one connection). Its just a different language but it helps me think about it.
Unless a particular sequence can model a "prime-avoiding" algorithm, which may not even exist, and would require the sum of factors to NEVER be a prime, then I'd say the answer is yes (that is, if it doesn't loop!). The main issue is that the computation time required grows as the numbers to factor grow. If we weren't bottlenecked by computation power, we would probably have found the end points for all but the most insanely long sequences, because prime gaps in very large numbers can also affect how long a sequence can dodge primes.
This untouchable number business is very interesting. 5 seems obvious since you can't add up 1+different primes to get 5. Wonder what the business is with the other untouchable numbers, could enjoy a whole video on that.
3:52 I laughed sooo hard "all of them" 3:56 sooo funny. Like literally bricks an HPC cluster of non-quantum supercomputers and if unsupervised causes like 300 grand worth of network traffic lag damage and cooling system issues. boss comes in: _"what the_ *_Fibonacci_* _happened while I was out?"_ answer: 😊 "Well we lost a battle but won a war!"
Funny, it reminds me of the behavior of the "3 x + 1" problem (or Syracuse problem) : some numbers (not so big !) have a very long flight compared to others... 🙂
5 being the only odd untouchable number is related to the goldbach conjecture, that every even number bigger than 4 is the sum of two primes. If a number is 1 more than the sum of two distinct primes p and q then it will follow p times q in an aliquot sequence. This is more restrictive than goldbach as it requires the primes to be distinct.
1:39 if your EKG looks like that you should probably be in the ER or the ICU, but that's a cute name :P ...yeah, I might have studied bioengineering in college
The question why to do it is valid but to me there is one exceptionally good answer to that. Basic science is done for curiosity, but it has immense proof that it's worthwhile. All the technology that makes it possible to pose that question has roots in those curiosities. We have not figured out how to make progress better than let some freeloading free thinker do their stuff and build engineering on their results.
If the sequence is chaotic (or random) it should eventually hit a prime, given infinite time and infinite primes. The larger numbers would end up avoiding primes due to prime gaps, so the sequences could end up quite long until they inevitably hit a mine, so to speak. It's not too dissimilar from Voyager leaving our star system, with so much empty space, the chance of collision was significantly reduced, but it will eventually hit something given infinite time and constant speed. But our lack of computing power prevents constant speed, and so it would be like time slowing down exponentially for Voyager, which would drastically increase the time to collide from a constant-time observer.
Well, you need to prove that sequence is completely chaotic. On the other hand, there are approximately 24.74% to 24.8% abundant numbers on the number line, while density of prime numbers is 1/ln(x), which is reducing the further you are on the number line. For 276 the highest number is bigger than 10^130, so if you will hit, it will be really unlikely. And there are already lik 10k numbers like that, which didn't hit anything so far and still growing, so chances are higher that abundant numbers will be main presence in the chain.
Regarding the question "what's the point?", I think it's worth pointing out that there is such a thing as recreational mathematics. So the real question is : to which degree is number theory part of it?
problem with negative numbers - they will cancel each other. Like factors of -4 are -1,1,-2,2, so sum is 0, if you do same logic for 4, you can also have negative numbers like -1,1,-2,2 and also have 0. You can't use negatives, because they won't give any progression anyway. And you can't add random negative nubmers, because they won't have any logic.
This is a continuation of our video about 276 and Aliquot Sequences with Ben Sparks. See the first part at ruclips.net/video/OtYKDzXwDEE/видео.html
What if it ends up at the odd perfect number? 😂
Love your love for numbers.❤
Can you post the Python script?
OK, but why just go forwards? Why not see if you can go backwards to find what leads to your starting number, and if the process goes on in that direction as well?
I'm curious: Why is there no analytical method, no proof? Why are we stuck generating sequences to see what they do? Is this impossible to reason about? Maybe if I were a regular viewer, I'd know that, but I've just been dipping into the channel now and then. It feels like something important has been left unsaid.
296, the Parker amicable number
Haha well played
I think we need to let Matt Parker off the hook on this one. 220 & 296 should be forever known as a Sparks pair.
Lol
You've heard of the Parker Square, now it's time for the Sparker Pair
@@jackeea_ brilliant
"I WANT TO KNOW BECAUSE IT'S THERE AND NONE OF US PUT IT THERE."
-Ben Sparks, 2024 - absolute legend
Channeling his inner Edmund Hillary.
Ah, but our hands are not at all clean because we asked the question. Why this question and not the infinity of others we did not ask?
"Let's explore the jungle a bit... and the ones we have come across, some of them are nice" 😊
The proof that 5 is untouchable is easy, so easy I am surprised they didn't include it. You can only make 5 by summing 1 and 4. But if 4 divides a number so does 2, so it is impossible to have an aliquot sum in the first place.
by the same logic, though, any odd number is 2n + 1, and if 2n divides a number so does 2. so all odd numbers larger than 3 are untouchable?
edit: nope, this doesn't work. you can also do 2n+1 with different factors. ie it's not necessary that you get 2n+1 by having the factors 1 and 2n.
This whole video mentions untouchable numbers disappointingly shortly considering it's the video title.
@@tim..indeed agreed
@@renyhp It's that 5 is the only number where 2n+1 would be the ONLY way to make it. Other odd numbers can be formed by 1+2n, but they can also be formed in other ways. For example, 7 can be made by 2n+1 with n=3, but it can also be made by 1+2+4.
@@renyhp ah so goldbach conjecture is sufficient so that 5 is the only one. If 2n=p+q, then p*q has factors p, q and 1.
As a programmer, I am fascinated by videos showcasing something that we cannot compute. When watching the first video, as he explained "we do not know" my immediate reaction was "I'm gonna write something and find it", then I saw the scope of how how far it has been checked and I immediately switched to "how the heck did someone write something that could check that high".
How powerful is your computer
This is unfortunately almost always the case for the "trivial" problems. There are multiple conjectures that are easy enough to understand in terms of simple Maths that are also fun to program and try for yourself. But for all of them, when you fancy the idea of looking into it, turns out somebody else with access to a super computer has already checked all the numbers up to a thousand digits. 😔
You can’t calculate your way to proving something is endless.
@@samlevi4744 Exactly 🎯!
And my second thought about it was: and for several centuries all the greatest mathematicians, like Euler or Newton, had to calculate all their things manually. It is so much more convenient and error resistant now.
The happiness in Brady's voice when he got to name something is amazing
I adore Ben Sparks
I've grown to love that little corner and table that all his videos have. 😂
@@pennnel agreed!
He's the Russell Crowe of maths
@@stevemattero1471 Funnily enough, Russell Crowe has played a Mathematician (John Nash, in ”Beautiful Mind”) 😅.
As far as I know, that 28-cycle that 2856 hits is the longest discovered one.
28 is also a perfect number, cue X-files music.
Wow
According to Martin Gardner's article on the topic (reprinted in his book, "Mathematical Magic Show"), the 28-cycle was announced by P. Poulet in 1918. (Or at least, Poulet announced a 28-cycle beginning with 14316; I assume it's the same one.)
@@JohnDoe-ti2np Same loop- 14316 is the smallest number in the cycle.
@@ianstopher9111 and additionally the number 2856 consists of two numbers 28 and 56 (2x 28). Cue the x-files music in loop
The number 2856, (where 56 is 28*2) discovers a cycle of 28 numbers (which is also a loop of 56 numbers)! Impressive!
...what?
how can cycle differ from loop?
Let's try more bonkers things that have this pattern of digits
142857 should be tried
@@NoNameAtAll2 loop of 28 is always also a loop of 56 cuz it repeats itself after 56 terms as well
142856 better yet
The odd untouchable numbers are related to Goldbach's conjecture. If every even number greater than 4 can be written as a sum of two distinct primes, then every odd number greater than 5 is not untouchable. Say 2n + 1 > 5 and 2n = p + q, with p and q distinct primes, then 2n + 1 is the aliquot of pq.
This doesn't work for 7 either, since 6 is not the sum of two distinct primes. But 7 is the aliquot of 8, so it's not a problem.
But like the 7 case, for any prime P, 2P+1 could be untouchable. Goldbach says nothing about the primes being distinct.
I just posted this and then saw your comment.
@@Phlosioneer That's true, but no counterexamples are known (greater than 6). It's just a stronger version of Goldbach's strong conjecture.
4:47 this should have been in the main video! what an amazing graph
Only real fans will see it 😎
Feels perilously close to 3x+1! I'd love to have seen some of the ways the analysis for this has been done mathematically rather than just computationally.
"because it's there to explore"
wonderful
3:29 "prime numbers, factorizing them is hard." -ben sparks
LOL
😂
"There are no prime numbers, only numbers that Bruce Schneier doesn't want us to factorize."
Lol
I would love to see a version of this animation that goes on for longer and bigger.
Like those Mandelbrot deep dives you get.
we're gonna need a bigger computer for that :)
@1:40 if your ECG looks like this, please stop this video and phone an ambulance immediately! 😆
5:53 Awww, they're dancing together ^ _ ^
numbers like 980460, which converge to an amicable pair, could be called "voyeuristic numbers"
That number found true love later in life
Aha! Knew I wasn't making up that I've seen you outside GD.
6:00 I also like, how the 2 zig-zaggy patterns perfectly intertwine, because 1 graph hit the same amicable number 1 turn later. It’s like an amicable pair of amicable pairs, with that nice DNA-pattern 🧬💞. 😊
What a brilliant conjecture, just the kind of thing that really interests me, trivial to understand and if a solution is ever found then it'll be incredibly complex in comparison :D
This video sequence offers abundant proof maths is interesting.
ben sparks always delivering some fascinating mathematics!
So many tjmes a physicist discovers something profound about reality, and then realizes a mathematician has already been there 10 years ago just for fun. Im all for having fun with math - for the joy of it, and also for the chance of a true insight into reality
Name one example of a physicist thinking they discovered something when a mathematician already did.
@var67 hyperbolic/non-euclidean geometry came first as a lark... then Einstein found it useful to describe reality. Early group theory; turns out extremely applicable to conservation laws, re Emmy Noether. -1/12ths turns out to give correct answers in some calculations. Complex numbers came first when mathematicians were playing around with quadratics, etc... ended up very useful for quantum physics.
There's 4.. could probably come up with more
@var67 (p.s. i never said "a physicist thinking..." ... I was just pointing out that mathematicians have often come across something while just having a lark that ends up being important for physicists later on.)
@@publiconions6313 I misunderstood, as you may have figured out. I did think you meant: physicist "discovers" something but no the mathematician discovered it earlier. So yes you're right, physicists get their grubby paws on ANY old maths. I should know, as a (failed) physicist. Btw, I loved the Journal of Recreational Mathematics back in the day.
@@var67 word! ; ) I figured we were just swinging past each other a bit there. I wonder, do you listen to Daniel Whiteson's podcast?.. it may be a little layman, I was never even close to a physicist (failed or otherwise.. lol, I sell insurance - snore) but he recently had an episode entirely based on the idea that "hey, octonions are cool, wonder if they apply somewhere" .. really peaked my imagination, especially considering how quarternions ended up making a lot of sense for QCD?.. it's over my head ,so I might have the specifics wrong. But my dream scenario is that some Numberphile in some corner of the world makes a connection like that
0:32 It can't be 220 and 284, the log is just above 3. It is hard to see on the graph, but 1184 and 1210 are more probable. Maybe 2620 and 2924.
I just ran it. 980460 hits 2620,2924.
5 is the only odd untouchable number if a slight strengthening of Goldbach's conjecture holds. Goldbach's conjecture states that ever even number greater than 2 is the sum of two prime numbers. A stronger statement that also seems true is that every even number greater than 6 is the sum of two _distinct_ prime numbers. If this is true, then given any odd number n > 7, we can write n= p + q + 1 with p and q distinct primes. But the only proper factors of pq are 1, p, and q, so its aliquot sum is s(pq) = 1 + p + q = n.
That leaves the special cases of 1, 3, 5, and 7. For any prime p, s(p) = 1, s(4) = 3, and s(8) = 7. So only 5 is untouchable.
2:59 That's the most honest answer I've heard from a mathematician to the question "why do that?" until now.
"Because it's fun" needs to be more accepted as a reason to do anything.
Why art, why music, why dance? Because it's fun.
Why science? Why mathematics. Grrr, you're not allowed to have fun!
Awesome video here. I am completely blown away that such a low number shows this unbounded behavior.
Combine this with the Collatz Conjecture, soda, orange juice, triple-sec, lime, muddled ginger, and whiskey for a refreshing Smashed Gödel.
I love how excited Brady sounds about this.
what a great double-feature!!!
Thank you Numberphile for such great content!
If the sequence hits a prime, it collapses right away, right? And I appreciate that there's other ways to start the collapse as well. But just as a first path into understanding this: isn't there some kind of competition going on between the speed at which the sequence increases and the spacing of the primes at ever larger orders of magnitude?
I other words: do we know anything about how big the abundancy of high numbers will be in relative terms, just like we know that the prime density behaves like n/log(n)?
The sequences don't just increase, though. They go through periods of decreasing as well. Determining a heuristic for how quickly they grow sounds difficult since, even averaging it out, there doesn't seem to be a steady growth rate.
There are no other ways to collapse. To hit 0, it first has to hit 1 (unless it started at 0), and to hit 1, it has to first hit a prime (unless it started at 1). So every sequence that terminates at 0 goes a -> b -> ... -> p -> 1 -> 0 with p prime. More specifically, p will always be an odd prime other than 5 (since 2 and 5 are untouchable), unless you start at p.
And to answer your question, the asymptotic density of abundant numbers is known to be between 0.2474 and 0.2480. That means that as n increases without bound, the proportion of numbers less than n that are abundant approaches some limit that is about 24.77%.
Very cool! Might be my new favorite mathematical conjecture
Reminiscent of the Collatz conjecture.
Yeah maybe there's some similar structure to both
Loved this video, and how it connected primes, perfects, amicables and sociable sequences together! Great you also included the very long sequence of 28 I think. There have been a lot of aliquot sequences of length 4 discovered as well. A few years back I was very involved in searching for all 14 and 15 digit amicable pairs, now all are known to 20 digits or more! Was also co-discover of a couple of largest known amicable pairs, but these were later beaten quite well.
2856 blasting into my top 10 favorite numbers. 28 doubled is 56 so it's easy to remember and it creates that awesome "megaloop" which has a 28 step cycle!
That animation is a piece of art!
His wife: "296. Who's that and why is she texting you?"
The fact that 5 is the only one, mindblowing.
Only *proven odd* one. There are lots of proven even ones.
5 is the only known odd one. There are plenty of even ones. 5 isn’t even the smallest one. 2 is also untouchable
When I am looking at these graphs I am convinced that we are looking at some strange unexplained feature of the universe and how it works, why do specific numbers have specific properties and why are there patterns and shapes it feels like a sublime mystery hidden in there
I like "Go go Gadget Aliquot Sequence!"
@0:37, the amicable pair that it collapses to is not 220 and 284, it is 2620 and 2924.
Amazing structures.
@0:55 They should be called matchmaker numbers! (The ones that wander around til they find an amicable pair)
In the graph at 5:00, one number grows considerably faster then any other number. Like, even within the possibly unbounded ones, that one grows muc faster. What number is that?
276
One thought looking at the graph of all the sequences, is what does it look like if you line up all the end points, so if two sequences merge, rather than parallel lines, it is just a single line that merged. Social loops would need to do something like aligning the first repeat of the lowest point of the cycle.
how much of math is a result of intrinsic qualities in all counting systems vs just things arising out of a base 10 system? This is all so fascinating
I think these things covered in most numberphile videos like this one are base agnostic.
@@smicksatusadotnet very cool, I also like that term “base agnostic”
Better names for the mega-loops :
Cabal Numbers
Sewing Circles
Parlements (they talk in circles)
Labyrinth Numbers (like in Chartres)
Charybdis Numbers (whirlpool)
It's crazy to think that if it can be shown that just 1 of these sequences is unbounded, then we immediately know that infinitely many numbers will never hit 1, a perfect number or a loop, blowing the whole thing wide open
but then there will be a question "is this the only sequence"
Like collatz conjecture
Must have missed it, can you explain why? If a sequence beginning with N is unbounded, then obviously any number whose aliquot is N will also be unbounded (and equally for any point on the unbounded path). But how do you show that there is a number with an aliquot of N.
@@silver6054 If the aliquot sequence for N goes to infinity, then the aliquot sequence for every number in that same sequence also goes to infinity, so you get infinite counterexamples for free from just N
@@silver6054OP may he been thinking of something a bit less trivial, but there is an easy reasoning:
If the aliquote of x is unbounded, it has an infinite amount of numbers in its aliquote sequence. Every one of the numbers in its aliquote sequence also has an unbounded aliquote sequence.
I hope Ben has 284 comfy pillows for his impending couch exile! 😁
I misread that as 'cooch' for a moment 😂
You mean "296?"
When someone asks, "what's the point " i think the simple answer is that understanding comes from the analysis of factsand it's impossible to predict which facts are going to be a part of that process.
Most people out there will have figured this out already
Just started learning Python this year and an aliquot sequence generator was one of the first things I wrote. I don't have a graph though. That's cool.. how do you do that? Also.. very good seeing you again Ben. How is it that you're always interested in what I'm interested in and making videos about it?
5:09 He doesn't call attention to these, but notice 5 parallel lines terminating from a height of 6 digits just before reaching 40 terms. That's the 318 sequence. And the 7 that terminate after about 50 terms: that's 180. Then there's that blue line that takes off like a rocket and hits the ceiling in 50 terms. Not even one of the Lehmer Five. That's 840.
Cupid number - eventually hits upon an amicable pair.
the graph of "all" the numbers would be cool to see with the last step at the same x axis point.
Many: What's the point?
Tolkien: well, shut up.
When adding the divisors you get 1 + something. 5 is 1+ 4, but if a number has 4 as a divisor it also has 2 as a divisor, sothat doesn't work.
5 = 1 + 1 + 3 doesn't work either, as you can't have two ones. 1 + 2 + 2 also doesn't work, as you instead have two twos.
Quite easy to prove.
I love a good proof by exhaustion
Prove what exactly? That summing the divisors always yields a number which is 1+something? Thats trivial, and im not sure what your examples are getting at.
The end of the video. It talks about an untouchable number.
A number which no other number can reach with the aliquot sequence. In order to get 5 you need 1+2+2 which has two of the same number. A number cannot have the same number twice as a factor since it's only counted once. The nearest numbers are 4 and 6 from 1+3 and 1+2+3.
Therefore there is no way for the factors of another number to sum to 5
Therefore untouchable.
You can't only have 2+3 due to every number having itself and 1 as a factor, and we do not count itself as explained in the original comment.
@@RepChris someone didn't watch the full video lol.
In the end Ben says "5 has been proven to be untouchable.. I think."
"what's the point? why explore this stuff?"
"it's there, to explore... i wanna know"
fascinating, absoloutely fascinating, why...
2:55 "This is like properties of numbers that we didn't put there."
I can see why Ben would say this, but I'm not entirely convinced.
I would argue that these patterns and properties are a product of a human decision, because we decided the rules for determining an aliquot sequence.
This is like the Collatz conjecture, in that it explores the properties of a sequence, but that sequence is the product of a specific algorithm humans came up with. It's interesting to study, but it's only useful if that specific algorithm is useful for anything.
I want to believe that there is something useful to be learned from studying these patterns, but the more I think about it the less I think there can be. The rules for these sequences don't strike me as having any direct connection to some inherent property of numbers, like the primes do. Is there anything inherently useful about the sum of an integer's factors? Where does that ever come up in mathematics outside of these novel sequences?
It seems to me that these sequences are little more than a toy for mathematicians to play around with. They are useful insofar as they are an exercise in searching for patterns. Maybe the pursuit of these questions leads mathematicians to develop new tools and techniques that have applications elsewhere in mathematics, and that could be a good thing. But I think it would be a mistake to think that any of the patterns themselves (like whether 276 diverges) actually tells us anything useful about numbers in general, because that pattern is entirely dependent upon the arbitrary algorithm we used to generate it.
Beautiful! 😊 Give me five!
“What’s the Point!?”
Well.. a single point is kind of boring..that’s why you pick another, and another, and start connecting em, and you discover one by mistake, another by coincidence ..and, there Is the Point :)
In geometry, a collection of points is called a pencil
We need to start the OEISS: The Online Encyclopedia of Integer Sequence Sequences.
New at loving number theory so possibly a very noob question. If there is one Aliquot number that does not converge, wouldn't that imply there are an infinite number of non-convergent numbers? If the prime factors of x sum to y, and x never converges wouldn't that mean that y never converges, and then the same logic apply to y's decedents, as well?
Yes. Finding one means finding an infinite number of them in this case.
@@smicksatusadotnet That kind is stuff is fascinating to me. We’ve found none, but if we find one, we’ve found an infinite number of them. Math is like magic and alchemy sometimes.
@@kingdomadventures we've found nine?
@@smicksatusadotnet No, but we found more problems with the iPhone's swipe-to-type :D :D :D :D :D :D
Yes it does. If an Aliquot sequence does not converge, then it is infinite by definition, and all numbers in said sequence also do not converge
What's the distribution of sequence lengths before reaching a resolution? Are the Lehmer Five just the tail of a distribution or are they outliers with lengths much longer than any other numbers? If it is the latter, that suggests something interesting is going on with those numbers. If it is the former, it could just be random and some numbers had to be the longest and it just so happens to be them.
These sequences remind me of the Collatz conjecture but much more satisfying 😅
5 then has one of the most strange meanings of all numbers, maybe the relation with the halving of things due to the nature of the base we're using, maybe there are some other intuitions to grasp if we search for this function in other number bases
I liked that the social loop had 28 numbers, and 28 itself is a perfect number.
4:13 At this point, Ben's wife has every right to start feeling jealous 😊
"What's the point?" Probably the most asked question on Numberphile
And it is such a worthless question.
Who cares? Why do it? Why not? Because it's fun.
Why does anyone do anything? Who cares?
7:32 - Recall: 5 is the only odd prime greater than 3 that is less than all the primes greater than 5. So there's that to ponder...
It's a tautology folks, just like _ALL_ the rest of math(s). Don't overthink it!
watching this video makes me feel like I just came home
I imagine it like the (nonnegativ) integers become vertices of a (infinite) directed graph, some are unconnected like 5. The graph has cyclic subsets, and leafs (vertices with only one connection). Its just a different language but it helps me think about it.
Reminds me of the collatz conjecture. Does every starting number eventually reach 1?
Unless a particular sequence can model a "prime-avoiding" algorithm, which may not even exist, and would require the sum of factors to NEVER be a prime, then I'd say the answer is yes (that is, if it doesn't loop!). The main issue is that the computation time required grows as the numbers to factor grow. If we weren't bottlenecked by computation power, we would probably have found the end points for all but the most insanely long sequences, because prime gaps in very large numbers can also affect how long a sequence can dodge primes.
Very cool
I think I disproved the conjecture that 5 is the only odd untouchable number, because I'm odd and untouchable and definitely a one!
I love the pure maths fun
This feels much like the 3n+1 problem, also known as the hailstone numbers. Is there any relationship between the two problems?
This untouchable number business is very interesting. 5 seems obvious since you can't add up 1+different primes to get 5. Wonder what the business is with the other untouchable numbers, could enjoy a whole video on that.
Yeah, I was a little put out that this whole video was called Untouchable, but it just teased them at the end.
If you counted the square number's root twice then 4 would go to 5 (but then nothing would go to 3 I guess)
3:52 I laughed sooo hard "all of them"
3:56
sooo funny.
Like literally bricks an HPC cluster of non-quantum supercomputers and if unsupervised causes like 300 grand worth of network traffic lag damage and cooling system issues.
boss comes in: _"what the_ *_Fibonacci_* _happened while I was out?"_
answer: 😊 "Well we lost a battle but won a war!"
0:50 I propose calling it a Thirsty Number
Could we get a circular pattern loop arc or even an arc?
Funny, it reminds me of the behavior of the "3 x + 1" problem (or Syracuse problem) : some numbers (not so big !) have a very long flight compared to others... 🙂
GO GO GADGET ALIQUOT SEQUENCE!!!1!1!!!!
5 being the only odd untouchable number is related to the goldbach conjecture, that every even number bigger than 4 is the sum of two primes. If a number is 1 more than the sum of two distinct primes p and q then it will follow p times q in an aliquot sequence. This is more restrictive than goldbach as it requires the primes to be distinct.
I always end these videos wanting more...
It wouldn't surprise me if somehow there was a way of plotting it that showed some close relationship with the Mandelbrot set haha.
Yet another reason to love 5.
1:39 if your EKG looks like that you should probably be in the ER or the ICU, but that's a cute name :P
...yeah, I might have studied bioengineering in college
I had almost forgotten about Hair Matt.
It might sound odd, but I find the Implications this might have in quantum mechanics very intriguing.
Is the Aliquot number of 2520 easy to find. You could even try numbers like 360,360 or 720,720.
Uploaded 5 hours ago! As far as I'm concerned I'm just in time.
I'd like to see it backwards. After you generate a line, translate it so the line ends at 1 or the start of a loop.
What's the point of asking what's the point?
The question why to do it is valid but to me there is one exceptionally good answer to that. Basic science is done for curiosity, but it has immense proof that it's worthwhile. All the technology that makes it possible to pose that question has roots in those curiosities. We have not figured out how to make progress better than let some freeloading free thinker do their stuff and build engineering on their results.
Why can't "because it's fun" be a valid answer?
If the sequence is chaotic (or random) it should eventually hit a prime, given infinite time and infinite primes. The larger numbers would end up avoiding primes due to prime gaps, so the sequences could end up quite long until they inevitably hit a mine, so to speak. It's not too dissimilar from Voyager leaving our star system, with so much empty space, the chance of collision was significantly reduced, but it will eventually hit something given infinite time and constant speed. But our lack of computing power prevents constant speed, and so it would be like time slowing down exponentially for Voyager, which would drastically increase the time to collide from a constant-time observer.
Well, you need to prove that sequence is completely chaotic. On the other hand, there are approximately 24.74% to 24.8% abundant numbers on the number line, while density of prime numbers is 1/ln(x), which is reducing the further you are on the number line. For 276 the highest number is bigger than 10^130, so if you will hit, it will be really unlikely.
And there are already lik 10k numbers like that, which didn't hit anything so far and still growing, so chances are higher that abundant numbers will be main presence in the chain.
does any fractal pattern emerge from this algo?
Regarding the question "what's the point?", I think it's worth pointing out that there is such a thing as recreational mathematics. So the real question is : to which degree is number theory part of it?
There is no useless math, it will all be useful to someone eventually (if people exist for long enough)
@@iamdigory not necessarily. "useful" is limited by what can be found in real world, but pure math, being an imaginary thing, isn't limited by it
So... what about negative numbers? Could you add the negative pairs of factors to positive sequences?
problem with negative numbers - they will cancel each other. Like factors of -4 are -1,1,-2,2, so sum is 0, if you do same logic for 4, you can also have negative numbers like -1,1,-2,2 and also have 0. You can't use negatives, because they won't give any progression anyway. And you can't add random negative nubmers, because they won't have any logic.
github link?
Computational Irreducibility in action?