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).
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!
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
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?
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.
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)
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.
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 ?
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...
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.
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?
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.
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.
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.
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.
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.
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
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
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
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.
"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.
Thank-you. Forget the maths for a moment; I want to read all of what it says on her tee shirt! 🙂 Daleks have been a part of my education since they first appeared on BBC Television in the early 1960's. The Somos sequence is certainly interesting and curious but, as ordinary human beings, some of us want to know answers to questions such as where it is found, why it occurs, and what are its practical applications? A mathematician might be more concerned about why the apparent sequence of integers suddenly 'breaks' at /7, why it might never produce integers again, and whether it is true that no more integers are produced. Do you have more about this sequence planned for a future video? Also, what more can you tell us about the work of the MSRI, which would be worthy of inclusion on this channel? Thanks again. 🙂👍
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.
@@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.
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.
Another Indian-Italian. I once met a Indian couple in Wichita who ran a dollar store and when they told me they were from Italy I thought it was a bit odd.
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)
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?
Sorry, but this is just showing us a bunch of numbers. I feel like this would have been much better if the reasoning behind the behaviours shown was discussed, when it is dismissed away it feels like you're dumbing it down too much
Sorry but this was really bad. A number of sequences just thrown at us, no reason given, no general algorithm given, no idea what any of this is for. Plus endless trivial calculations performed really slowly.
Mathematicians sometimes define sequences without any known practical applications to see what will happen them. This class is sort of like a sequence of sequences and to me that makes it interesting.
Division by 7 has its own oddity. I did a great deal of work investigating different bases and their uses. We commonly use just three bases. Base 2,, digital anything, computers. Base 10,, we have ten fingers and count by tens, ubiquitous. and Base 12 is used with a base ten counting system, but degrees in a circle, hours on a clock, we use Base 12 in areas that demand easy divisions of quarters, sixths, eights,, etc. (Base 5 for some shepherds. One of my favourite counting words, bumfit. ) HOWEVER,,,, Base 7 is just plain weird. Divide any number 1 through 10 and you will get the same repeating fraction in Base 10,,, 142857,,, to infinity. The number may start on a different digit of the 6 number stream,, but it will always be the same sequence. Above 10? You will have an initial answer,, but then will go right back to the ,,, 142857 sequence. ?????? BTW the results of my number base investigation? It is just a shame we were not born with 12 fingers vs 10,,, Base 12 is sooooo much easier.
ah yes, base 7 gives you a DNA like structure. say "yo" if you want to see it. it will take me just a few minutes. Smth I saw when I was playing with bases and primes in multidimensional space. I also "discoverd" (for myself) 90 degree snakes vortices, maybe even vortices. Solution to your 12 finger problem. When just counting - use middle finger as six :P
A graphic solution with the integer and fractional terms different colors along the x-axis representing the solutions for each different somos sequences on the y-axis, is something i'd like to see-- ie somos 1-7 would be a single colored line stretching to infiniti, while somos 8 would change colors at 420514/7
Is there a proof that the first 6 somos sequences will never output a fraction? Like how do we know that the 2^(10^13)-1 -th output of somos 5 isn’t a fraction?
The first video in this series which was not enjoyable to watch. Pity. As others have already pointed out earlier, the algorithm wasn't explained (why? how? etc), the importance of this entire "somos" thing, the history, who, when, why did it bother with it in the first place, etc. The conclusion is also inconclusive: some are integers, some are fractions, so what? How is this important, interesting, does it lead to something? Also, admittedly there is no guarantee that the fractions continue ad infinitum. Actually anyone can write a simple subroutine (a program, a piece of code) which would simulate these sequences to millions of digits, if necessary. Anyway, Numberphile videos are always interesting (so far at least), but this one is strangely odd, alas.
The way my stilted mind works, after about 0.7 seconds of seeing the fraction I'm going, hey, that's not evenly divisible, after peeling off the 42 and 14 and subtracting 490 from 500. I'm guessing that's what the other 4.08 million subscribers do as well. I pondered that for about 7.0 seconds, anyway.
Trouble making out a couple words / CC is blank for some reason. Can anybody assist? 11:45 Some kind of *???* system (the buzzword) 12:03 *???* algebras?
I have a reasonable amount of mathematical training and I'm sorry, but I don't find this particularly interesting. Unless these sequences can be related to a deeper problem, or to a practical use, I'm not seeing the intrinsic interest.
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.
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!
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
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*
??..
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!
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.
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
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
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.
??..
"The sequence is not interesting, but the algorithm is getting juicier."
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.
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! 💝💝
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.
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.
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.
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
🥂
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.
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
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
That's an interesting sequence of sequences. I wonder where it comes from.
It's always great to see a Dr. Who fan.
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.
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!! 🤣
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.
Thank-you. Forget the maths for a moment; I want to read all of what it says on her tee shirt! 🙂 Daleks have been a part of my education since they first appeared on BBC Television in the early 1960's.
The Somos sequence is certainly interesting and curious but, as ordinary human beings, some of us want to know answers to questions such as where it is found, why it occurs, and what are its practical applications? A mathematician might be more concerned about why the apparent sequence of integers suddenly 'breaks' at /7, why it might never produce integers again, and whether it is true that no more integers are produced.
Do you have more about this sequence planned for a future video? Also, what more can you tell us about the work of the MSRI, which would be worthy of inclusion on this channel? Thanks again. 🙂👍
I have never in my life seen or read anything I understood less than the content of this video lol.
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.
What is the font are ye using for the Number slides? It is gorgeous Edit: Found it, it's called "American Typewriter"
That Dalek T-shirt is rad.
It is somewhat tubular, isn't it?
Scrolled through the comments looking for someone else who noticed.
It is, imo, probably related to the density of primes. As primes become less dense there is less likelihood of total cancellation.
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.
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.
So someone just made up an algorithm that eventually breaks. Why does it matter?
Wake me up when Neil Sloane enters
"Today I will tell you about this very fascinating number. Four... two... zero..."
Me: "say no more."
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
Another Indian-Italian. I once met a Indian couple in Wichita who ran a dollar store and when they told me they were from Italy I thought it was a bit odd.
I really liked the speaker (very engaging) but I think it would be cool to have an extra video to explain the algorithm
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)
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?
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 just watched it for the 4th time and what was bothering me is that the somos 2 formula was not written correctly
I can never be a mathematician because I hate both chalk on chalkboard sounds and permanent marker on brown paper sounds.
When ur smoke break is delayed by 54 minutes...
Sorry, but this is just showing us a bunch of numbers. I feel like this would have been much better if the reasoning behind the behaviours shown was discussed, when it is dismissed away it feels like you're dumbing it down too much
The missing point is the obvious "why?". Not why there are only integers for k
What was the point of this video?
Sorry but this was really bad. A number of sequences just thrown at us, no reason given, no general algorithm given, no idea what any of this is for. Plus endless trivial calculations performed really slowly.
Mathematicians sometimes define sequences without any known practical applications to see what will happen them. This class is sort of like a sequence of sequences and to me that makes it interesting.
@@realdealsd His concern is that he doesn't know the pattern of this sequence of sequences.
Many people can’t google Somos. Poor people
very interesting subject. slight mistake at 8:17 formula should be a8 = a7xa1 ... , not a8 = a8xa1...
Division by 7 has its own oddity. I did a great deal of work investigating different bases and their uses. We commonly use just three bases. Base 2,, digital anything, computers. Base 10,, we have ten fingers and count by tens, ubiquitous. and Base 12 is used with a base ten counting system, but degrees in a circle, hours on a clock, we use Base 12 in areas that demand easy divisions of quarters, sixths, eights,, etc. (Base 5 for some shepherds. One of my favourite counting words, bumfit. ) HOWEVER,,,,
Base 7 is just plain weird. Divide any number 1 through 10 and you will get the same repeating fraction in Base 10,,, 142857,,, to infinity. The number may start on a different digit of the 6 number stream,, but it will always be the same sequence. Above 10? You will have an initial answer,, but then will go right back to the ,,, 142857 sequence. ??????
BTW the results of my number base investigation? It is just a shame we were not born with 12 fingers vs 10,,, Base 12 is sooooo much easier.
ah yes, base 7 gives you a DNA like structure. say "yo" if you want to see it. it will take me just a few minutes.
Smth I saw when I was playing with bases and primes in multidimensional space.
I also "discoverd" (for myself) 90 degree snakes vortices, maybe even vortices.
Solution to your 12 finger problem. When just counting - use middle finger as six :P
in the great schema of things you could consider the numbers that are not fractions as being the "special ones" as fractions become the norm...
There is a mistake in the expansion of the algorithm displayed on the slide @ 8:39. See if you can find it ;)
This is an excellent introduction to a second video with more details!
Please make a video explaining how Somos-N is formed.
No explanation of why term was suddenly squared at somos4
@8:16 I think there's an "a8" that should be an "a7"
Yes, it's a misprint.
the generating functions seemed so arbitrary. This was quite dull , sorry.
And this is imortant because...
Hii, yeah, chemistry is my favourite subject.
ÓSingle jodi ke liye call ya WhatsApp I'll get
did i miss something, or did all of this have no practical application?
is it simply a mathematical curiosity?
You had me at 4-2-0.
But what's so special about Somos sequences?
How do you know algorithm for somos-k ?
What is the application for these sequences?
Numberphillia is a big issue that should be resolved.
It would be interesting to see how this varies as a function of base.
8:02 so also never explains how the Somos-x algorithm is derived
What a shitty explanation or idea?
Why we predicted that it shall all be integers
Also all the digits used in 420514/7 are those from the decimal expansion of 1/7 except '8'.
ÓSingle jodi ke liye call ya WhatsApp I'll get ok ok
Isn't that true for all non integer divisions by 7 (ignoring the 8 of course). 2/7, 3/7, 4/7 are the same digits just starting on a different one.
@@sophiophile That's not what he said. He's talking about the digits in the fraction itself, not the decimal expansion.
@@danielyuan9862 I see what you mean, but it's a bit of a stretch. There's two 4's, a 0 instead of an 8, and the 7 is on the denominator.
A graphic solution with the integer and fractional terms different colors along the x-axis representing the solutions for each different somos sequences on the y-axis, is something i'd like to see-- ie somos 1-7 would be a single colored line stretching to infiniti, while somos 8 would change colors at 420514/7
And the point was......?
What designates what the next algorithm is?
Is there a proof that the first 6 somos sequences will never output a fraction? Like how do we know that the 2^(10^13)-1 -th output of somos 5 isn’t a fraction?
As a mathematical idiot, this seems to follow the same idioms (mentally speaking) of the sea shell diagram of growth. Am I wrong?
How do you know there are no fractions before somos-8? Have they been tested to ∞ ?
There are algebraic proofs I believe.
You don't "test" math, you calculate math.
The proper term is 'checked'.
The first video in this series which was not enjoyable to watch. Pity. As others have already pointed out earlier, the algorithm wasn't explained (why? how? etc), the importance of this entire "somos" thing, the history, who, when, why did it bother with it in the first place, etc. The conclusion is also inconclusive: some are integers, some are fractions, so what? How is this important, interesting, does it lead to something? Also, admittedly there is no guarantee that the fractions continue ad infinitum. Actually anyone can write a simple subroutine (a program, a piece of code) which would simulate these sequences to millions of digits, if necessary. Anyway, Numberphile videos are always interesting (so far at least), but this one is strangely odd, alas.
How about *“The Troublemaker-Trendsetter”* ?
The graphics cover the video, it's unwatchable. In the future I suggest you actually watch the videos you upload.
lololol I thought it was going to be 42069
*"Still all Integers..."* The suspense was killing me.
This was interesting, but I don't get how each sequence algorithm is defined. Can anyone enlight me ?
I think the true magic is all the steps. The results matter of course but each step is also important. 118 is just 59 twice, 59 being a Prime.
8:11 it should be a7
this ladys awesome
T-shirt is awesome.
It's a short enough chuck that it's probably pure coincidence, but I was startled to see the first 7 digits of pi as a9 concatenated with a10.
I'm just going to define 420514/7 as a new integer.
ÓSingle jodi ke liye call ya WhatsApp I'll get ok ok
Based and limit pilled
I had to check that the thumbnail didn't equal 80085
Mistake at 8:30, a_8 = a_7 a_1... , whereas in the video a_8 = a_8 a_1...
I wonder if later somas have even more complexity in their results
I like this!
You've Made a mistake at 8:33
There are two 8's in the equation!
I think the 8 is ment to be a 7
The way my stilted mind works, after about 0.7 seconds of seeing the fraction I'm going, hey, that's not evenly divisible, after peeling off the 42 and 14 and subtracting 490 from 500.
I'm guessing that's what the other 4.08 million subscribers do as well. I pondered that for about 7.0 seconds, anyway.
Broke off one step before you. When I got to the 500 by your method I then factorise into 5 × 100 and I already happen to know neither divides by 7.
@@trueriver1950 Shrewd!
-Hello-
"Fractions!!" ^^
Trouble making out a couple words / CC is blank for some reason. Can anybody assist?
11:45 Some kind of *???* system (the buzzword)
12:03 *???* algebras?
I have a reasonable amount of mathematical training and I'm sorry, but I don't find this particularly interesting. Unless these sequences can be related to a deeper problem, or to a practical use, I'm not seeing the intrinsic interest.