The Troublemaker Number - Numberphile
Vložit
- čas přidán 26. 06. 2024
- Dr Harini Desiraju discusses Somos Sequences and a number which breaks a streak.
More links & stuff in full description below ↓↓↓
Dr Harini Desiraju is a postdoctoral fellow at The University of Sydney. This video was recorded at MSRI.
Like sequences - see these videos with Neil Sloane: bit.ly/Sloane_Numberphile
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outr...
And support from The Akamai Foundation - dedicated to encouraging the next generation of technology innovators and equitable access to STEM education - www.akamai.com/company/corpor...
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Videos by Brady Haran
Patreon: / numberphile
Numberphile T-Shirts and Merch: teespring.com/stores/numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9 - Věda a technologie
I think it would have been interesting to describe what was happening to get from somos 3 to somos 4. I mean it wasn't shocking that all the equations so far could only give 1s since you were just multiplying and dividing 1s. And it was also really obvious why that broke at 4 since that's when they added in that pesky +. It just seemed almost arbitrary I'm pretty sure that if somos-3 had been a_n = (a_n -1)*13 / (a_n-2)*sqrroot(2), then non-integers would have started a lot earlier. I mean, I know that isn't correct and I can extrapolate what each next one is going to be after seeing 5 of them, but at some point it would have been nice if they had mentioned explicitly what the pattern for forming the equations was and why anyone was interested in looking at those particular sequence-generating equations.
The numerator is the sum of products pairs such that the offsets of each pair sum to k. So in domos 5 you get 4 witg 1, 2 with 3. At somos 2 and 3, tgeres only one such pairing each (1 with 1 and 1 with 2 respectively). Tgat vganges at somo4 (1 with 3 and 2 with itself)
@@Yxiomel Yes, I said I figured it out, but you explaining it does not explain why they didn't explain it in the video. And I'm going to suggest that you make sure your fingers are on the home keys before you start typing.
@@ApesAmongUs When they wrote down somos 8, the pattern becomes so obvious that even I got it, so i don't think there's harm done with not mentioning it explicitly. And when it's written on the page with the finger "pinching" to get the factors, it seems like a fairly "natural" thing to do, which might be enough to answer why anyone would be interested in it.
@@muskyoxes I agree with OP. It does make you hang unnecessarily. IMO it would be better to give somos-k general formula, and then say for 1st 3 terms it's all ones. It makes you feel like video time and effort was spent on trivial things instead of context and combined with "we don't know why it breaks" makes a subpar impression. It's not the kind of dopamine cycle people enjoy =D
@@muskyoxes I started skipping a lot back and ahead because I felt I had missed it. That was harm done. Never irritate your customer.
The video misses the mark.
At first, the algorithm seems boring, then it grows without an explanation why. What happened to the part of the video where we learn why something works the way it does
Did I miss something? I don't think it is ever explained in the first place what a Somos-K sequence is. What determines the Kth algorithm? Without giving a rule for how the Somos-K algorithms are defined, and why we would define them that way, this just looks arbitrary. I'm sure there is some definition, but it needs to be part of the presentation.
So basically the algorithm for k-somos is
a_n = (a(n-1) a(n-k+1) + a(n-2)a(n-k+2) + ...) /a(n-k)
Where the the sum of the indices of every multiplication pair is 2n-k, for example: in the case of 4-somos we have
a_n = (a(n-3) a(n-1) + a(n-2) a(n-2)) /a(n-4)
And indeed n-3 + n-1 = 2n-4 and n-2 + n-2 = 2n-4. Similarly for 5-somos a_n = (a(n-4) a(n-1) + a(n-3) a(n-2)) /a(n-5)
Again the indices add up to 2n-k:
n-4 + n-1 = 2n-5 and n-3 + n-2 = 2n-5. I hope that helps a bit!
@@tipeg8841 Yeah, I looked it up. Pretty simple, but in video we only see up to k=5 and only a brief glance -- no chance to really see the pattern, only a some polynomial of a_(n-i)'s/a_(n-k). The pattern is simple and this presentation would have been much stronger if it had been shown from the beginning.
I agree it looks arbitrary, more so than usual for numberphile
is it used for something, an example please
They show the pattern after 7:46. It would have been nice if it were sooner.
Yeah. A randomly defined sequence outputs random numbers. Oh well.
This video is missing the part that actually explains the algorithms. As it stands they seem arbitrary at first (of course they are not).
I think they're arbitrary like the Fibonacci sequence algorithm. To me it seems like it is, in fact, arbitrary but it produces interesting results.
This video is completely pointless, because the algorithms seem arbitrary, and it‘s not explained why they might not be arbitrary.
@@CompanionCube I agree
@@CompanionCube I disagree.
"seem arbitrary at first"
Gotta say it: You are delusional.
"Somos Sequences" are just regular sequences introducing themselves in Spanish. ** FLIES AWAY **
Not to be confused with "Samos sequences", which are lines of cream cheese.
Or in Portuguese,
@@therealnotanerd_account2 Fair point xD
And they’re all… Juan, Juan, Juan, Juan….
Nor samosa sequences. Which is the topology of tessellation in indian pastry based triangles fitting onto round plates that include irregular bhaji and circular pakora
For the first time in a Numberphile video, I was lost in the first half minute. It felt like this video was part 2 or part 3 of a series and I hadn’t seen the previous videos.
same
They didn't explain:
The concept
How you generated the sequences
The application
and finally what it mattered or was interesting
11:27 "That's a hard question." The sense of mystery is palpable.
ÓSingle jodi ke liye call ya WhatsApp I'll get ok ok
I also loved that quote.
Numberphile is the best ASMR channel out there :)
11:47 "That's the buzz-word" *lol*
??..
I don’t have the mathematical education to fully grasp this (or most other videos) but I find it pleasing and soothing to watch and I love knowing that there are people out there beavering away at seemingly obscure and strange mathematics, with or without any practical applications. And I love the shirt.
I feel the same way.
I met a geologist once who has synesthesia. He said that his brain interpreted numbers as having personalities, and that 7 was an asshole. Confirmed.
17 is way worse.
I agree. He eight nine
7 may be an a-hole but it almost always gets picked before the other single digits!
That's so strange. I also have synesthesia and 7 is a very prickly number for me as well.
There are three truths: one is the loneliest number, three is a magic number and seven is an asshole. I don't have synesthesia and I can agree with that.
These "Somos sequences" (which are a form of elliptic sequences) were first described in the paper "Step into the Elliptic Realm" by Michael Somos (29 Jan 2000).
Wow, it's really recent then, I thought it was a lot older.
I used to work with Michael's brother Leslie, who sadly passed away many years ago. I heard of the Somos sequence back around the time the paper was published and really only understood it as a mathematical oddity, but then Michael is a genius and I am not. Now that I've been reminded of the Somos sequence I'll have to take a closer look at why mathematicians find it interesting, possibly I can grasp some of it!
I have been watching and loving Numberphile videos for many years. But this was the first one in all that time that completely lost me at some point. It is missing ANY kind of grounding. Why are those sequences important? What are the reasons the don't break in the first 7 incarnations? Why would they break with different starting numbers?
The way the video is now is just basically: "Here we have a system of creating sequences. They stay integers for 7 of them and then start to break apart into fractions. Enjoy!" - Which is just kind of pointless, isn't it?
Exactly! This video described and overly emphasized all the most obvious, trivial and completely non-exciting details, and did not supply ANY explanation of the important claims or even the definitions.
I agree that this video kind of throws us into the deep end of the pool quickly. It doesn't have, for example, Neil Sloane talking about Avatar. But some numberphile videos are like that. Some give a backstory, some just jump in. And it is still 12 minutes---setting it up might have added more time.
We'll put - after several years of enjoying Numberphile videos, this was the first that seems arbitrary and uninteresting to me, and I think you've explained why.
You're mostly right, but it's far from the first
@@haibrenner Yes
For Somos 1-7 the correct seed for integers is all ones, I wonder if there is a seed for Samos 8 which gives all integers?
Underrated comment.
8 zeros works ;)
@@42ArthurDent42 you can not divide by 0
@@42ArthurDent42 You cant divide by 0.
@@MrBiggles168 you can’t... everything is clearly positive here, so you get +infinity. And you get zero back when you divide by infinity.
It's pretty clear that Somos-k would only have the first fraction at or after the 2*k'th term.
The first k terms are always 1 and the next k terms are always only divided by one of those 1s. Those are all integers by definition. The first possible non-integer term is the one at 2*k (when starting at 0) because that's the first time the divisor is not going to be 1 since it's the k+1'th term.
It's possible that 2*k still works because the term at k+1 happens to divide the rest evenly, but there can never be a fraction earlier.
I was thinking the same thing. The first non-integer has a strict lower bound on where it can appear, and it seems like the trend is that once you pass that bound it breaks very quickly, with a few sequences getting an extra integer term and some breaking immediately.
Personally, I think it's more interesting that some of the sequences never break.
@@phiefer3 ikr ...
false.
The algorithm of how the equation grows should have been explained first, and the trivial infinite strings of 1 should have been skimmed over.
I couldn't agree more. Interesting video nonetheless but definitely could have benefited from that
yea obviously
"The sequence is not interesting, but the algorithm is getting juicier."
Where does the algorithm actually come from? There seems to be a random selection of a’s being multiplied or added or squared without explaining where the algorithm itself comes from?
Like… is it possible to generalise the algorithm?
ÓSingle jodi ke liye call ya WhatsApp I'll get
Yes. It's on the Wikipedia page for the Somos sequences, if you want to see it, but it's not easy to type out. Probably should've been mentioned in this video, but, regardless, it exists and is well-defined.
I was thinking this as well. Seemed arbitrary from the video. I'll check out the Wikipedia.
The algorithm is a(8)*a(0) = a(7)*a(1) + a(6)*a(2) + ... + a(4)*a(4), and you obtain a(8) by dividing both sides by a(0) (this one in particular is for somos-8)
It is clearly explained at 8:56
I'd love to see a part 2 for this video that goes deeper into the theory!
Thank you for keeping the blue "heading" consistently on each diagram. That is IMMENSELY beneficial for scanning through the video, whether with full images or with youtube thumbnails. I wish everyone considered that on video graphics of all sorts of disciplines.
Remember when Fox sports started maintaining the score and time remaining on football broadcasts? That was a world changer in terms of navigation for skimming and non-realtime playback.
> Remember when Fox sports started maintaining the score and time remaining on football broadcasts
... no, not at all. What an oddly specific thing to refer to.
What? You think it was Fox that originated that?
While it wasn't invented by Fox, it was invented by Fox Sports' sibling Sky Sports in 1992, coming to Fox as the FoxBox in 1994 after their acquisition of the rights to NFL content. Fox News also was the first channel to feature a permanent news ticker, 30 minutes before CNN did on 9/11
@@PeterNjeim Thank you for the intelligent response. That detail is nice to know.
(BTW I tape sporting events and use the constant reference point "progress indicator"(effectively) in the corner to navigate. Just like I do with THIS video. I'm including this in case any smart alecks without sufficient IQ to grasp that obvious concept need a little help. :) )
Many CZcamsrs also use chapters, which you can see on the time bar and skip to directly from the description
I saw the terms 314 and 1529 around the 5:40 mark and freaked out because I thought the next terms were gonna be consecutive strings of the decimal expansion of pi. Turns out I was just being silly, it's supposed to be 1592
Oh, boy, silly you. :´) Tnx for a tear. But the fear of wth is real.
Year ago I started to play with primes for the lulz. Eventually I freak out because I discoverd a method. But it turned out that Euler was there first. I had tottaly forgotten about Eulers Totient funtion. 25+ years had been passed. Oh boy. But that dopamine hit, ah yeah :)
11:15 The first n terms are ones by definition, and the next n terms have ones as the respective ones out of the first n terms as denominators, so the first 2n terms are integers
I noticed this too. They seem impressed by how it makes integers for so long but the formula is designed to give you at least 2n integers.
@@TuberTugger It's still impressive how instantly it fails after its guaranteed integers run out. It's as if it is an extremely chaotic sequence which is in direct oppositon to the previous Somos stages.
Yeah, I think it's no surprise at all that you get fractions pretty much instantly after the guaranteed integers run out. There is no a priori reason why it should still be integers. Imo the only interesting thing is that the previous sequences give only integers.
@@GynxShinx yea cause the sequence is so random the probability that it fails to give int after its not defined to do so is very low so obviously its not impressive its just more probable
@@rocketsandmore6505 The fact that it is random is what is interesting and counter to the consistency that we are made to expect from previous Somos stages.
This video is missing the 'motivation' part of the explanation. What are these SOMOS sequences and why should we care about them.
It is nice to see new faces on the channel
ÓSingle jodi ke liye call ya WhatsApp I'll get ok ok
The first three digits got me really interested
Is this video missing a section? It feels like it starts in the middle without any intro
They are missing where the algorithm comes from.
How are algorithms chosen ?
That's pretty puzzling because it seems there is a pattern, and then you add additions, then squaring things and...
How is it progressing ?
ÓSingle jodi ke liye call ya WhatsApp I'll get ok ok
It's a fairly simple definition that's hard to type out. It's on the Wikipedia page if you want to see it written in nice formatting, but you more of less multiply pairs of the previous k-1 terms, sum them, and divide by a_{n-k}. There's squaring when k is even because the final pair is the same term twice. When k is odd, that doesn't happen.
I think it was very briefly explained.
For Somos-k you calculate a_(n-1)*a_(n-k+1) + a_(n-2)*a_(n-k+2) + ... + a_(n-k/2)*a_(n-k/2) / a_(n-k)
The last term is exclusive for even k so k/2 is an integer and in that case you get a_(n-k/2)^2.
I don't know if I made it any clearer but I tried...
Thanks folks, that makes more sense now !
Somos-K is pairing up the last (K-1) amount of numbers as first with last, second with second last etc. (if K-1 is odd, the middle number is paired with itself), and these pairs' product are summed up and divided by the K-th last number.
Feels like a large important part of the setup to this video, why this sequence is important and how to generate it, was left on the cutting room floor.
It’s rather unsurprising that the sequences break into fractions around the 2n-th term - the first n terms were given 1s and then the next n terms all have 1 as its denominator by definition. 😉 This simply means that they stop being integers pretty much as soon as they can.
👍Single jodi ke liye call ya 👍👍WhatsApp 👍 ok
Yeah, I don't understand what's so magical about this.
@@JETAlone12 it doesn't seem that hard using modular arithmetic, I need to dig deeper into it.
??..
This video needed to explain the logic of the algorithms and how they progresses. why some elements are added, multiplied or squared. How those sequences were discovered. Without such an explanation it's a miss.
ÓSingle jodi ke liye call ya WhatsApp I'll get ok ok
ÓSingle jodi ke liye call ya WhatsApp I'll get ok ok
There's a formal definition that isn't too complicated but it's hard to type out. You basically take the previous k-1 terms, multiply together the opposite pairs (e.g., the first and the last of those terms, the second and second to last...), sum them and divide them by a_{n-k}. You get squaring when k is even, because the final pair you multiply is the same number twice.
somos algorithms do have a real world application, but in the end as with all high level mathematics, the true reason why always comes down to: "because I was messing around with a neat pattern" :P
You just take opposite pairs of preceding numbers (like the most recent and farthest back) and multiply them, then the next inner opposite pairs get added, then the next... And then divide it by the number than is the nth number before (where n is the somos number).
I have struggled with mathematics my whole life, but I didn't want it to stop me becoming an Engineer. In my last year of University (studying mechanical engineering) I approached my Math tutor for help, extra lessons, more practice examples... anything to help me. He told me that If was struggling with content so easy (the module was advanced mathematics for engineers), maybe I should think about leaving. It shook my already damaged confidence and I failed that exam, passing on a resit. I put everything I had into studying mathematics and I left University with a first-class bachelors degree in mechanical engineering and have been very happily working in CAE for over ten years. I love this channel, even though I still struggle to understand all of it's content. Never give up, if I can do it, you defintely can
🥂
maybe something gets lost in translation here for me, but who defines those algorithms? what are their properties or with which rules are they build? because it obviously is easy to come up with ANY algorithm with k>=8 that would still fall in the integer rule...
what am I missing?
Nah, you didn't miss anything. I kept thinking the same thing from the very start. The topic was poorly presented.
Are you talking about Somos?
It is falling a very simple set of rules. You can keep adding terms in the numerator spot as long at it is following the simple rules. It started in Somos 1 with the basic rule set.
@@Giganfan2k1 and the ruleset is?
@@DrazkurHW exactly, thats actually my point: very unusual for Numberphile :(
It only clicked for me with the visualization at 8:04 They should have shown this for all the ones from sonos-4 and up to show how the pattern develops.
@8:15 there is an error in the Somos-8 algorithm, a8 should be a7 in the numerator.
ÓSingle jodi ke liye call ya WhatsApp I'll get
I ALSO SPOTTED THAT TOO
That example was the first time I thought I'd be able to make sense of the logic behind all these equations because the video wasn't telling me for some reason - and then there was a typo so I still couldn't!
Put that in a square, does it break, put it in a circle does it break and so on. I quite liked it thanks
I have never in my life seen or read anything I understood less than the content of this video lol.
The OG kind of Numberphile video - one number and one number only. It’s nice when we get these videos again once in a while.
Is this one not meant to be about me?
You’re definitely the troublemaker mathematician tho!
ÓSingle jodi ke liye call ya WhatsApp I'll get
No tick of verification, why?
@@AnmolTheMathSailor perhaps not everyone feels the need for validation
@@solgato5186 yeah but I was wondering is it manually done or youtube automatically does it
This is all great but how are the algorithms derived for somos1, somos2, etc.? They all seem rather arbitrary.
ÓSingle jodi ke liye call ya WhatsApp I'll gets
They are not random, but the explanation is one of the things missing in this video.
a_n*a_{n-k}= a_{n-1}*a_{n-k+1} + a_{n-2}*a_{n-k+2}...+a_{n-(k-1)/2}*a_{n-(k+1)/2}, for odd values of k. For even values, the final term is (a_{n-k/2})^2.
Not easy to type out, but well-defined. You can look it up if you want to see it formatted as math and not in plain text.
@@jamielondon6436 It is clearly explained at 8:56
I can google the expansion too or a specific K algorithm but that doesn't mean a layman like myself understands how the full recursive expression manifests.
Random question, but can u get Niel Sloane to do more integer sequences and amazing graphs?
👍Single jodi ke liye call ya 👍👍WhatsApp 👍 ok
Neil is fantastic
This ^^
He can probably explain that series. In the end, they are in OEIS.
This seems to be above my head as to what makes this 'important',
It's not. Did anyone say it was? I think you missed the point of the video.
@@danielyuan9862 If it's not, there's no point.
8:11 - here is a typo: the formula for a_8 should not include a_8 itself! This should be an a_7 for anyone who was confused like me :)
Yes finally!! 🤣
These Somos sequences are fascinating and I think we’ll need more videos on them soon!
ÓSingle jodi ke liye call ya WhatsApp I'll get
Yes pls! 💝💝
This will be helpful in embedded systems , where you want to increase for DAC values exponentially but dont want a floating point.
I'm sorry I don't follow. Are you talking about digital to analog converters? How can this be used for that?
@@sophiophile I'd like to know that as well.
But for that, you can use a linear recurrence a_n = s a_{n-1} + t a_{n-2}. For example s=t=1 gives you the same rate of growth as the Fibonacci sequence.
Hardly surprising that it doesn't 'break' until addition is introduced, but that's a great T-shirt!
Fantastic t-shirt.
👍Single jodi ke liye call ya 👍👍WhatsApp 👍 ok
Everyone's been having a problem with this video but in reality its 420514/7 doing its magic
It really isn't. You need a foundation for a number to be interesting, which wasn't given propertly nor at the right moment.
@@AureliusEnterprises I believe he was making a joke that since this is the "troublemaker" number, it's doing its job by stirring up trouble in the comments.
I lost track of what was going on at 0:31
That's an interesting sequence of sequences. I wonder where it comes from.
It's always great to see a Dr. Who fan.
sums up math. keeping doing random things till you cant explain whats happening, and then lose your mind over it.
These sequences are quite interesting. The ratio of two subsequent numbers is very irregular. On average it's increasing but goes up and down quite randomly with no sign of any stabilizing. You wouldn't expect this with sequences built so systematically.
That Dalek T-shirt is rad.
It is somewhat tubular, isn't it?
Scrolled through the comments looking for someone else who noticed.
Wake me up when Neil Sloane enters
In all Somos-K sequences except Somos-1, the equation for the first calculated term: a(K), has a(0) as the denominator and the numerator consists of the terms between a(0) and a(K) multiplied pairwise and then added up. In the cases where K is even you get a term with no partner, which is then paired with itself, or in other words, squared. This continues with a(K+n) having a(n) as the denominator an so on.
But the equations for the terms in Somos-1 has no denominator, so it does not follow the pattern, so you can argue that Somos-1 is not a true Somos-K sequence.
It would have a denominator if written differently, but the first two equations were written in a simplified form. You could define somos-1 as a_n = {empty product = 1} / a_(n-1) and since the seed is 1, the formula is just always ones. And maybe they thought that bringing up the "empty product = 1" rule would be off-topic for this video, and since the output sequence is so trivial maybe even the original mathematician(s) didn't really care about how it was written.
I have no idea why they wrote somos-2 without explicit division though, that's just confusing, and it's not even like writing recursive formulas like that is common anywhere.
I liked the presentation of this video- spend a little time establishing a rule in your head, then something weird comes in. Hope we get to see more of this host, this was well done
Interesting way of teaching. I might actually use one of your methods when I teach my learners. I like. 🌱
Interesting presentation. Thank you for introducing 420514/7 ❤️ I never looked at somos sequences in this light
"Today I will tell you about this very fascinating number. Four... two... zero..."
Me: "say no more."
What is the font are ye using for the Number slides? It is gorgeous Edit: Found it, it's called "American Typewriter"
One of many algorythms with a quirky output. But most of them do not have a practical application other than for making an interesting list.
Never the less it's fun to see (and especially creating) one of them.
Amazing!
ÓSingle jodi ke liye call ya WhatsApp I'll get
Would have been interesting to learn about all the interesting math behind this algo.
They forgot to put the math part into this video.
I'm sure there are interesting maths to discover behind these sequences.
Maybe even more interesting than (1*1)/1=1
Something I missed in the first 8 minutes - given k, how do I write down the algorithm? It wasn't until sonos-8 at around 8:16 that I saw that the pattern for the algorithm was
I don't understand where the (a1 x a1) / 2 formula comes from at 1:23. I think I must be missing something.
They skipped that part for some reason.
"Just bear with me and you'll figure it out eventually"
Why don't you just tell us from the start? Lol
Everything aside, the general formula is a_n*a_{n-k}=a_{n-1}*a_{n-k+1}+a_{n-2}*a{n-k+2}+... all the way until the product of the center two numbers, or the center number squared.
Is it _just a coincidence_ that the _numerological sum of digits_ in 420514 = 7?
4 + 2 + 0 + 5 + 1 + 4 = 16, then 1 + 6 = 7
The number: 420-
me: say no more
what
ÓSingle jodi ke liye call ya WhatsApp I'll get
goddammit
lul
You understood the rest of the video from just that, didn't you?
When ur smoke break is delayed by 54 minutes...
Dr Harini is super cool! Explained all of this very well.
Im always confused about sequences like these. We’re mathematicians just sitting around one day and very bored then decided to make a “cool number sequence”? Or did this originate as a solution to solve a real-world problem?
@𝟗𝟑𝟓𝟎𝟓𝟕𝟗𝟕𝟕𝟑 Guru Ji yes
Probably the first one
@@sicapanjesis3987 agreed. No doubt the sequence is cool so job well done.
@@sicapanjesis3987 yep, normally the physicists and engineers are lagging behind, so when someone comes across a problem they pick the previously discovered and described sequence that fits "close enough". Otherwise it's all charts of random numbers
i'm pretty sure they do this in order to develop proof techniques and such
Great video!
With Somos-4, it seems that you may multiply the a_{n-3}a_{n-1} term by any integer m, e.g. 2 instead of 1, and the sequence will still be all integers. But avoid m=-1, for then a_5=0.
*this video is awesome*
I was so relieved to see that first "2".
Who and WHY made that sequence : (
Looks like tedious
ÓSingle jodi ke liye call ya WhatsApp I'll get ok ok
Uh, no comment that I can see here
Michael Somos. The point of them is mostly that it's weird they only make integers for the first few sequences; that isn't a property you'd expect, and it's mathematically interesting.
You never explain why. Something like "because x^2 is always divisible by y."
So someone just made up an algorithm that eventually breaks. Why does it matter?
I, for one, was very enthused and impressed with Somos-2. :D
I can never be a mathematician because I hate both chalk on chalkboard sounds and permanent marker on brown paper sounds.
You start with item 1. Equally, there is a rule for going BACKWARDS ... it's just the same sequence extended backwards. Is it like the Fibonacci sequence, in that it grows monotonically in the forward direction, but grows WITH ALTERNATING SIGNS in the backward direction?
_this is awesome_
420.000 is divisible by 7, 490 as well and 24 isn't. Therefore we get a periodic fraction of 60.000 + 70 + 24/7, making 60.073,428571, the part after the comma being periodical.
This video was really hard to watch and follow. Why did you not give the algorithm at the beginning? It wasn't until 8:03 (2/3 of the way through the video), and the 8th iteration, that enough information was shown to piece together what was even happening. And it still wasn't told, you have to work it out yourself.
Without looking it up myself, what I think the algorithm is is that the denominator is the n-Somos# index, and the numerator is the sum of the products of the remaining closer/more recent previous terms, paired from the outside in, with any lone middle term squared, i.e. [(n-1)*(n-Somos#-1) + (n-2)*(n-S#-2) +...+ (n-(S#/2))^2] / (n-S#)
Why not just state that at the beginning, and then show the weirdness that no fraction shows up in any of the sequences until the 8th iteration.
Somos2 is exactly solvable for any starting values: ln both sides to get ln(an) = 2ln(an-1)-ln(an-2). That's a linear recurrence characterized by [[2,-1],[1,0]]. Curiously, its jordan block is [[1,1],[0,1]], so its ln exhibits linear growth. So it grows exponentially.
You had me at 4-2-0.
They've already updated Wikipedia for Somos Sequences to include this video as an external link..
Is there ~any~ set of numbers from a sequence Somos-n, where n>8, that consists only of integers?
I like this!
I’m 8:00 minutes in and I’m missing something important: what’s the point? What is this “somos” thing for?
I'm sucha nerd. After sorting out some typing errors in the video, I was able to put the algorithms into a spread sheet. Then I had to learn some rounding to intergers intricacies of my software and behold. I finally got my sequences to match Dr Desaraju's. A little mental floss for a Saturday afternoon - somos- 4, 5, 6, 7, 8, 9, and 10. BTW the very things that made me dislike the video motivated me to keep playing with it.
1:43 If there is one thing I'm looking for in an algorithm, it's juiciness
If we run a Collatz Conjecture calculation on 420,514, it takes 130 steps to reach 1, and we get a very interesting graph.
I am sorry but can someone explain why did we add the square of a(n-2) in somos-4... Till then it was just multiplication is it? Plus what's the relevance of these somos-k sequences?
Thanks!
What is the series if you apply a modulo?
For those wondering about the algorithm:
It seems like for Somos k that a_n=( a_(n-1) a_(n-k+1) + a_(n-2) a_(n-k+2) + a_(n-3) a_(n-k+3) + ... + a_(n- floor(k/2) ) a_(n- ceil(k/2)) ) / a_(n-k)
where ceil(n)=floor(n)=2n and ceil(n + 1/2)= n+1, floor(n + 1/2)= n
👍Single jodi ke liye call ya 👍👍WhatsApp 👍 ok
I am deeply upset by how *close* to an integer that is visually. 420000. Divisible by 7. Dope. 14. Divisible by 7. Also dope. 500. Uhhhh. 500? Damn.
Of course, mathematically, you're a maximum of 3 away from being perfectly divisible by 7, but something about that number looks like it just WANTS it so bad.
I think you may be in base 10...
No clue what this was about.
My theory is somos-8 breaks because going from say 2 to 4 terms in the numerator, it looses an aspect like octominials vs quadrinomial or binomials. (Talked about in their video about quaternions)
Numberphillia is a big issue that should be resolved.
At last, an Indian mathematician being featured on numberphile
12:08 "there is very nice mathematics". The point of the video that's never been made. I know you know addition, ma'am.
It would be interesting to see how this varies as a function of base.
Hii, yeah, chemistry is my favourite subject.
ÓSingle jodi ke liye call ya WhatsApp I'll get
It is, imo, probably related to the density of primes. As primes become less dense there is less likelihood of total cancellation.