Untouchable Numbers - Numberphile 

This is a continuation of our video about 276 and Aliquot Sequences with Ben Sparks. See the first part at ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-OtYKDzXwDEE.html

296, the Parker amicable number

"I WANT TO KNOW BECAUSE IT'S THERE AND NONE OF US PUT IT THERE." -Ben Sparks, 2024 - absolute legend

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.

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".

The happiness in Brady's voice when he got to name something is amazing

As far as I know, that 28-cycle that 2856 hits is the longest discovered one.

The number 2856, (where 56 is 28*2) discovers a cycle of 28 numbers (which is also a loop of 56 numbers)! Impressive!

@1:40 if your ECG looks like this, please stop this video and phone an ambulance immediately! 😆

I adore Ben Sparks

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.

"because it's there to explore" wonderful

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.

4:47 this should have been in the main video! what an amazing graph

3:29 "prime numbers, factorizing them is hard." -ben sparks

numbers like 980460, which converge to an amicable pair, could be called "voyeuristic numbers"

Combine this with the Collatz Conjecture, soda, orange juice, triple-sec, lime, muddled ginger, and whiskey for a refreshing Smashed Gödel.

5:53 Awww, they're dancing together ^ _ ^

I would love to see a version of this animation that goes on for longer and bigger. Like those Mandelbrot deep dives you get.

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

His wife: "296. Who's that and why is she texting you?"

Cupid number - eventually hits upon an amicable pair.

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.

I like "Go go Gadget Aliquot Sequence!"

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 🧬💞. 😊

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

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.

Many: What's the point? Tolkien: well, shut up.

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)?

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

ben sparks always delivering some fascinating mathematics!

Thank you Numberphile for such great content!

what a great double-feature!!!

Better names for the mega-loops : Cabal Numbers Sewing Circles Parlements (they talk in circles) Labyrinth Numbers (like in Chartres) Charybdis Numbers (whirlpool)

Awesome video here. I am completely blown away that such a low number shows this unbounded behavior.

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.

That animation is a piece of art!

I love how excited Brady sounds about this.

Very cool! Might be my new favorite mathematical conjecture

Amazing structures.

the graph of "all" the numbers would be cool to see with the last step at the same x axis point.

"What's the point?" Probably the most asked question on Numberphile

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

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.

2:59 That's the most honest answer I've heard from a mathematician to the question "why do that?" until now.

The fact that 5 is the only one, mindblowing.

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.

I liked that the social loop had 28 numbers, and 28 itself is a perfect number.

fascinating, absoloutely fascinating, why...

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

We need to start the OEISS: The Online Encyclopedia of Integer Sequence Sequences.

Why explore it? For the same thing that sets us apart from most (but not all) of the animal kingdom: pure and simple curiosity.

GO GO GADGET ALIQUOT SEQUENCE!!!1!1!!!!

I hope Ben has 284 comfy pillows for his impending couch exile! 😁

"what's the point? why explore this stuff?" "it's there, to explore... i wanna know"

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.

I wish ( and i am sure and positive )that every real number will have a numderphile video on it( or will atleast be mentioned among others ).

I love the pure maths fun

I had almost forgotten about Hair Matt.

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

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.

It wouldn't surprise me if somehow there was a way of plotting it that showed some close relationship with the Mandelbrot set haha.

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.

This feels much like the 3n+1 problem, also known as the hailstone numbers. Is there any relationship between the two problems?

1:28 BATMANs

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

Interesting.

Uploaded 5 hours ago! As far as I'm concerned I'm just in time.

Yet another reason to love 5.

Unlike most everything else discussed here, 5 being untouchable is pretty easy to show. These are the ways to write 5 as a sum of unordered positive integers: 5 × 1 + 0 × 2 + 0 × 3 + 0 × 4 + 0 × 5 = 1 + 1 + 1 + 1 + 1 3 × 1 + 1 × 2 + 0 × 3 + 0 × 4 + 0 × 5 = 1 + 1 + 1 + 2 2 × 1 + 0 × 2 + 1 × 3 + 0 × 4 + 0 × 5 = 1 + 1 + 3 1 × 1 + 0 × 2 + 0 × 3 + 1 × 4 + 0 × 5 = 1 + 4 1 × 1 + 2 × 2 + 0 × 3 + 0 × 4 + 0 × 5 = 1 + 2 + 2 0 × 1 + 0 × 2 + 0 × 3 + 0 × 4 + 1 × 5 = 5 0 × 1 + 1 × 2 + 1 × 3 + 0 × 4 + 0 × 5 = 2 + 3 Going back to "proper factors", `1` is always a factor, and factors are counted only once. So, throwing out sums with duplicate terms, or those missing `1`, we're left with just `1 + 4`. The problem is that any number that has `4` as a factor must also have `2` as a factor, which would then be included in the sum, so the sum `1 + 4` is not achievable in an aliquot sequence. On the other hand, if we included square roots twice, then we would have `4 -> 1 + 2 + 2 = 5`.

It has to be noted that factoring big numbers into their prime factors is not *the* thing that secures the internet today. The current mainstream big thing is elliptic curve cryptography

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 :)

Easy to prove 5 is untouchable every aliquot sum contains 1 if you add 2 or 3 to that you'll be left with remainders of 2 and 1 respectively which will be already represented in the sum, ergo neither are possible so the only possible aliquote sum that gives 5 is 1+4 but anything that has 4 has a factor will also have 2 as a factor ergo no number as aliquot number 5 QED :D

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.

@0:55 They should be called matchmaker numbers! (The ones that wander around til they find an amicable pair)

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.

I always end these videos wanting more...

Could we get a circular pattern loop arc or even an arc?

Reminds me of the collatz conjecture. Does every starting number eventually reach 1?

Reminds me of hyperbolicity a la the klein quatic.

It might sound odd, but I find the Implications this might have in quantum mechanics very intriguing.

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.

What does the graph look like?

An untouchable number would have to have certain properties. Since the aliquot process is always adding proper factors, an untouchable number must never be able to be the sum of proper factors. 5 is untouchable, because the only possible way to get to 5 with a sum of distinct, positive integers (where you must include 1) is 1 + 4, but no numbers have *only* 1 and 4 as proper factors (because any number with 4 as a proper factor also has 2 as a proper factor). It's kind of amazing that there are infinitely many untouchable numbers (proven by Erdős), since an untouchable number has to have every single possible way of summing a list of distinct positive integers (including 1) be an impossible list of proper factors. It seems quite improbable that such would happen at all, never mind an infinite number of times, and yet not only does it happen, it happens relatively frequently (wikipedia says the natural density is > 0.06).

If I had learned about aliquot sequences in 7th grade I might have talked about nothing else in high school.

Romantic numbers is certainly a good name

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

Is the Aliquot number of 2520 easy to find. You could even try numbers like 360,360 or 720,720.

Aliquot Sequences is adding the primes, right?

Good thing we're soon getting efficient factorisation of large numbers with quantum computers. Never mind it breaks the internet. We need answers!

How do we know 5 is untouchable? We need a Numberphile3 video

I think we need a video about 296 ;-)

Computational Irreducibility in action?

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.

does any fractal pattern emerge from this algo?

"Go go Gadget, Aliquot sequence!"

296…I wonder who has that number on their keychain?

I think I disproved the conjecture that 5 is the only odd untouchable number, because I'm odd and untouchable and definitely a one!

"...because it is there to explore."

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?