I didn't know about it either. Looking at his channel he doesn't *really* have one either (There are only 3 videos): Of course, 3 videos is a lot better than the 0 I was aware of ^^
"... concept is so pure and simple. Yet, the result looks completely random." And the same is true of Pell's Equation solutions (one of my own favorite, pet math topics). Fred
@@tomc.5704 How do you halve infinity? I would define "Small" number to mean calculable. The number of calculable numbers are finite while "Large" numbers are incalculable and infinite. Conjecture: most of the numbers are large.
@@tomc.5704 You can't talk about a fraction of infinite because it isn't a number. Infinite is the size of a set with no end. The cardinality (cardinal numbers describe an amount of things) of an infinite set is a special number which is not in the set of the integers called aleph null. You can only think about density with infinite sets, how often you expect something to occur, not how many times it occurs.
@@04LightningFan What numbers? Integers? Rationals? reals? Wish i could remember what i read somewhere about throwing an ideal dart at the number line, and seeing what the probability would say about the number you hit. i.e. that the numbers that you would hit, that are transcendental, incomputable, etc and would occur with probability, one. Probably got these details completely wrong, but i'm sure someone could correct me on this. Something to do with measure theory, i guess???
when i was in high school i used to think maths is for genius people only but this lockdown changed everything, because of people like you and other mathematicians who contribute to numberphile channel now i see maths as something which is logical and simple abstract ideas which can describe the world and sometimes it just blows my mind by these simple ideas and this video blew my mind....
Me too! Watching Numberphile made me realize I wanted to study math in university this year. There is so much more out there than high school shows us. Thank you Ben and everyone in the only mathematics community!
Because maths is generally taught very badly in schools imo. I nearly failed maths at school, yet went to university at 30 and got a masters in physics. My analogy is this: It's like being taught French, you learn the grammer and the rules of how to construct sentences except they don't tell you what any of the words mean. I remember being taught calculus. I asked the teacher what the point of it was and got sent out of class, because the teacher didn't understand well enough to explain it.
@Calen Crawford 1/n approaches zero as n goes to infinity. Your second quotient I dont understand as 1 divided by any positive integer apart from 1 does not equal 1. Where do surreal numbers come in?
@@fossil98 1/infinity%, meaning almost 0%, so the confusing part of the comment roughly translates to: "... then 1 equals [almost 0]% of all whole integers." With the surreal numbers coming in to explain exactly what "almost 0" means, I think the surreals are basically what you get if you take the reals but add infinity and 1/infinity an numbers.
@Calen Crawford Firstly, infinity is not a number in the surreal number system. You're probably thinking of the extended real line, but in that system 1/infinity = 0. Secondly, there are many ways of making sense of infinity as a number and it's misleading to act like any one way is the "correct" one.
Hi Ben! I met you outside a pub inside Bath and got a pic with you! Very much enjoyed this project (I think this is the one you told me about) and all these mega numbers 😁
I think this was the best fav number of the series. Yes it's amazing that this simple conjecture first fails at such a large number, but it's also amazing that it was so much *smaller* than the estimate from the proof that such a number existed.
Well, it was really a proof that there was *a* region where the summatory Liouville function was positive, around 10^361. The proof never said it was the first region where this was true. (And yes, you might think you'd have to calculate the SLF one value at a time, so if you reached 10^361, you'd have to go past 906,150,257 first. But... analytic number theory is more complicated than that, in ways I don't understand.) It's similar to Graham's number. You know that's only an upper bound on the solution to a problem, not the actual answer, right? And the *lower* bound to that problem, as of 2020, is... 13.
@@alexpotts6520 Yes, like a first stab at the twin prime conjecture had a limit of 70,000,000 and I think it's now down to 6. I was also puzzled when i read about this, that *a* counter-example was found already in 1960. If that was found by just counting up, you would hit the *first* counter-example but that wasn't found until 1980.
@@bjornmu Like I said, analytic number theory is weird, there are a bunch of similar results of the "small numbers obey this pattern, but at some really big number it breaks" kind. A lot of these number-theoretical functions can be written as series with error terms like log(log(log(x))), etc, that only become relevant and make their effects visible when you're looking at truly gargantuan numbers.
5:19 Thanks for giving us this hint! I find the oscillation even more fascinating than the general bias towards numbers that are the product of an odd number of prime factors. (That bias can be explained by the density of primes).
Because of a famous argument due to Littleeood, if Polya's conjecture was true ( at least for all sufficiently large numbers), then Riemann hypothesis would follow. Littlewood used his argument to disprove the conjecture that pi(x) is less than li(x).
3:20 I just thought of an interesting proof of that: For each factor k of a number n exists a duel factor, n/k such that their product is n. As long as k ≠ n/k the number of factors must be even since we can pair up all the factors like so: (k,n/k). Therefore, if the number of factors is odd there must be a factor that doesn't pair up and doesn't satisfy k ≠ n/k. Which therefore means k = n/k and *k² = n*
You made a typo in your description, you wrote "Most numbers have an odd number of prime numbers!" instead of "Most numbers have an odd number of prime factors!"
How could there possibly be a proof for this! I can't even imagine where I would start for something that seems as random as this. I would be incredibly interested in a video on the proof (no matter how complicated it is)
You can find order in these kinds of sets with observations. Let p(x) equal -1 when x has an odd number of prime factors and 1 when x has an even number of prime factors. You can observe that p(xy) = p(x)p(y). And we're here considering the sums of the p(x)s. Already you might feel like some factoring techniques might work, but they don't, but then you start to see that it isn't completely unapproachable.
Thanks Ben. Similar to the numberphile video with Holly Krieger on Merten's conjecture. Now that I know you have a channel, I've subscribed and turned on notifications.
Might there still be more numbers with an odd number of factors, just in a weaker sense? For example, the limit of the proportion might still slightly favor odd numbers of factors, even if it moves around a bit along the way.
I know it's a discrete graph, but you can look at the graph @9:16 and consider the area above and under the graph. Saying that that area approaches -infinity as the number approaches infinity would be a weaker version of this statement. Still not easy to prove, but I would bet that that would be a true statement.
Just subscribed! I've watched you for years on Numberphile, but found your other content to be a bit too long for me to be able to watch in one sitting. This lenght was perfect, more like this would be fabulous! :)
Every prime number has one prime factor, and early on there are a lot of primes. This is probably the driving force behind the function’s initial dive below zero. However, as the primes begin to thin out, so does their impact on the sum.
I knew it was gonna be a mind bender. I knew it would involve some really unique property. I knew it would be fascinating. I did not know Matt Parker would stick the DJ horn sound effect in.
At 3:12 : "Conjecture: n is a square if and only if it has an odd number of factors" . It is true , not a conjecture ( 2 minutes on the back of an enveloppe). In fact we may say more : Theorem: n is a k'th power if and only if it has a number of factors congruent to one modulo(k). I am wrong ?
So the next challenge would be to find the first number under which positives and negatives cancel each other, if it ever happen. This would mark a domain of well distributed odd and even numbers if picked up randomly.
I postulate that there is a prime number around 30'102 that for some odd reason should count as 2 factors on it own. And it's because we count it as only 1 that this conjecture fail.
New fave. What is the area of that region/running area under the curve as the upper bound increases? It seems doubtful, but does the running area ever become positive?
So you're basically saying the running sum of the sign of the shown function is always negative. It certainly feels like it but I wouldn't be surprised if it also turns out wrong
so what i'm curious about is, what would it look like if you were to add up the area underneath the curve of that function up to any given point? that should give some kind of idea of... how much the value of that function tends to be negative or positive for all the previous numbers. and as far as I can tell for everything we've seen in this video up to the ~906 million mark, that area would be very decidedly negative even after you add in that little chunk sticking into the positives. if we graph out the area underneath that curve as its own line, will *that* line, that overall count of how even or odd the prime factors tend to be up to any given point, keep trending into the negatives forever, or will it eventually cross into the positives too? i have no idea if this question is at all meaningful or relevant to the conjecture or anything else, but it's just the first place my brain goes to and kinda fun to think about!
There is an interesting corollary in nature and cosmology to the strong law of small numbers, and this is in the isotropy of the universe which only becomes apparent when viewed across the largest of distance scales. Scales we have only been able to measure in recent decades. It is not necessarily obvious at all that the universe is extremely even in distribution. On scales even up to superclusters the universe looks very lumpy. But pan out far enough and we now know that the distribution of mater is astoundingly uniform.
He nonchalantly says at the beginning that only square numbers have odd number of factors.... wait, what?!.. my mind blown just by that. Heh.. had to pause the video and take some time to digest just that little tid bit.
Ha. It is a nice number fact, and I was a little blown away the first time I realised it. :) For what it's worth - this vid makes use of the fact: ruclips.net/video/-UBDRX6bk-A/видео.html
Is this just because you can't really have 0 prime factors, so you have to start at 1 prime factor (i.e. the primes), and 1 is odd, so this biases the number of prime factors being odd?
I don't know if I'm right but my idea at the onset was to say : assuming the number of prime factors is perfectly random for all numbers, then it still has to be odd for all primes, and this creates a bias. The numbers that are even powers would pull the other way but they are less dense than the primes.
1 is the product of 0 primes. What you say explains the initial downward trend, but doesn't explain later intervals where upward steps predominate over downward steps.
0:53 You have misstated it. But you're in good company, as Wikipedia has misstated it, too. It's not "have an odd number of prime factors", it's "are the product of an odd number of primes". But thanks for going on to explain the correct idea.
Well, yes. The statements are equivalent if you count the repeated prime factors. But your statement is perhaps neater at capturing what we need to count. I liked the reference to the prime *factors* though as an intrinsic property of the number.
What if we put the intuitive idea into a weaker conjecture? The Liouville sum doesn't _always_ stay nonpositive, but does it stay nonpositive _most of the time?_ What if we conjecture that most numbers have an odd number of prime factors... most of the time?
5:19 " it's kind of got a self-similar fractally look to it there's some interesting consequences from that um go look up the riemann zeta function if i see a connection with the riemann hypothesis in this " 6:20 " and as i mentioned before there is to do with the riemann zeta function and look up the connections on wikipedia or wolfram "
Another examples of stong law of small numbers where we see issue very very late is 111111111111111... 19 1s is a prime number even though after 11 we didn't have any. Are there more such examples??
How to properly state conjectures in number theory: "Most numbers have between log(N) & log(N)+1 prime factors" is bullshit "For each positive integer, N, let A(N) = the number of integers, k, 1 1/2, then that is what we mean by "Most numbers, N, have between log(N) & log(N)+1 prime factors" ".
This conjecture reminded me of some talk on Numberphile, and it was Mertens conjecture. ruclips.net/video/uvMGZb0Suyc/видео.html They are so much alike!
That one ignored numbers with repeated prime factors didn't it? I wondered at the time what would happen if you included them. Well now I know. Kind of. I guess there's a third sequence that considers the number of distinct prime factors.
But this summatory function looked unlike a random walk. There are long intervals where upward steps are much more common than downward ones, to be followed by long intervals where the reverse happens.
@@rosiefay7283 Wrong, this is a classic example of human misconception of randomness. There is even a statistical test exploiting this ('run' test for randomness). Humans tend to think randomness is somehow connected to perfectly shuffled sequences but in reality randomness also leads to longer sequences in mostly just one direction. Just plot some brownian motions, sometimes they will explode in one direction, sometimes not.
Voila! A problem based on this came in the IMEO 2020 paper as Q3 if I remember correctly. There I wondered if this function L(n) ever was positive, and was absolutely convinced that it could never happen. Opened my eyes! I wish this video was made before I gave that test 🙃. Have a look at the AoPS threads for the same to check out that problem. It's pretty interesting.
Question, if it's proven that the Liouville Sum crosses the x-axis an even amount of times, would that then prove that Pólya's conjecture is true after all? Because if it crosses the x-axis an even amount of times it will be negative as x goes to infinity (we start by going negative after all).
Is it proven that there are an infinite number of possible n for which this function is larger than 0? Or could there possibly be a largest number for which it is larger than 0? If so you could still argue that "most numbers" is fulfilled as a criteria. That would also be closer to how I would interpret the initial formulation of the conjecture.
It's actually proven (despite what I say around 10:00) that it crosses between negative and positive infinitely many times. Haselgrove proved this in 1958 in fact. So, unfortunately, even a modified version of the conjecture is false. :(
Even though the polya conjecture is wrong how it was originally stated, isn't it still weird that the liouville function looks that way? I mean if the number of prime factors was completely random, i.e. equally likely even or odd, the liouville function would most probably cross 0 much earlier.
so far have read about: * 4 comments on marten's conjecture/constant * 4 comments asking for the python script (not counting my own comment regarding this) * 3 comments wondering about when the area will be equal to zero * 2 comments confusing "random" with evenly distributed ... wow, there are many recurring themes in this comment section
If the conjecture is about 'most', then if the sum of odd and even is zero, it is already disproven, isn't it? Doesn't have to cross the x-axis into the positive arena for that.
exp(10^120) billion years is clearly the best number - that is the Poincaré recurrence time for our universe - it is somewhat sad how few people have an idea how smart Poincaré was.
Just to be annoying, wouldn't the first case be n=1? Since 1 has zero prime factors? I know it is a corner case, so it doesn't matter all that much. Unless you somehow argue 0 has and odd number of prime factors? :p
Mathematically, we know that if Polya's conjecture is true then the Riemann hypothesis is also true. Outside of math? I dunno. But I'll tell you now that if you go into math expecting every to turn single idea into laser guns and warp drives, then you're in for a rough time.
Восьмияче́йник sure, no explicit counterexample has been found, but we know for a fact it exists between 10^16 and 10^(7.91*10^39). Seeing how huge these are the exact number will probably not be found anytime soon.
You could very much reduce the amount of computation required by choosing say every 5th or 10th number, and selectively doing all computation for areas when it gets close to zero. I can think of other ways to reduce the amount too. You see, the trick here is to Compute Smart 😀
Didn't even know you had your own channel. You're one of my favorite numberphile contributors.
Same here!
Yes. I think the ‘Tree Gaps and the Orchard Problem’ by Sparks is the best video in the numberphile collection.
@@bemusedindian8571
No way, I still haven't pieced together my brain since it was mindblown by Chaos Game!
I didn't know about it either. Looking at his channel he doesn't *really* have one either (There are only 3 videos):
Of course, 3 videos is a lot better than the 0 I was aware of ^^
He's tied for first with at least half of them
My mind is completely blown. I mean, the concept is so pure and simple. Yet, the result looks completely random. Wow.
Smarten up then
This of one of the big reasons we mathematicians love what we do
It's like the mandelbrot set.. These patterns come from nowhere. They are literally built-in to logic and reality.
Markets behave in the same way
"... concept is so pure and simple. Yet, the result looks completely random."
And the same is true of Pell's Equation solutions (one of my own favorite, pet math topics).
Fred
conjecture: most of the numbers are small.
I could write a Python script to disprove it!
Counter conjecture: "Small" is a relative term best defined as "below average"; exactly half of the numbers are small.
@@tomc.5704 How do you halve infinity? I would define "Small" number to mean calculable. The number of calculable numbers are finite while "Large" numbers are incalculable and infinite. Conjecture: most of the numbers are large.
@@tomc.5704 You can't talk about a fraction of infinite because it isn't a number. Infinite is the size of a set with no end. The cardinality (cardinal numbers describe an amount of things) of an infinite set is a special number which is not in the set of the integers called aleph null. You can only think about density with infinite sets, how often you expect something to occur, not how many times it occurs.
@@04LightningFan What numbers? Integers? Rationals? reals?
Wish i could remember what i read somewhere about throwing an ideal dart at the number line, and seeing what the probability would say about the number you hit. i.e. that the numbers that you would hit, that are transcendental, incomputable, etc and would occur with probability, one.
Probably got these details completely wrong, but i'm sure someone could correct me on this. Something to do with measure theory, i guess???
when i was in high school i used to think maths is for genius people only but this lockdown changed everything, because of people like you and other mathematicians who contribute to numberphile channel now i see maths as something which is logical and simple abstract ideas which can describe the world and sometimes it just blows my mind by these simple ideas and this video blew my mind....
I love hearing this sort of thing. Welcome to the fold!
Me too! Watching Numberphile made me realize I wanted to study math in university this year. There is so much more out there than high school shows us. Thank you Ben and everyone in the only mathematics community!
Because maths is generally taught very badly in schools imo. I nearly failed maths at school, yet went to university at 30 and got a masters in physics.
My analogy is this: It's like being taught French, you learn the grammer and the rules of how to construct sentences except they don't tell you what any of the words mean.
I remember being taught calculus. I asked the teacher what the point of it was and got sent out of class, because the teacher didn't understand well enough to explain it.
@@benbooth2783 that's so touching. Thank you for your inspiration.
@@benbooth2783 You were unfortunate then, not to get a truly enthusiastic teacher. That can make all the difference.
Fred
It takes 0% of the numbers to get people’s hopes up, but also 0% to shoot them down.
@Calen Crawford 1/n approaches zero as n goes to infinity. Your second quotient I dont understand as 1 divided by any positive integer apart from 1 does not equal 1. Where do surreal numbers come in?
@@fossil98 1/infinity%, meaning almost 0%, so the confusing part of the comment roughly translates to:
"... then 1 equals [almost 0]% of all whole integers."
With the surreal numbers coming in to explain exactly what "almost 0" means, I think the surreals are basically what you get if you take the reals but add infinity and 1/infinity an numbers.
@@petemagnuson7357 Ah I see. Misread. Yeah the surreals include all reals, +ordinal infinitie & infinitesimals.
Any%
@Calen Crawford Firstly, infinity is not a number in the surreal number system. You're probably thinking of the extended real line, but in that system 1/infinity = 0. Secondly, there are many ways of making sense of infinity as a number and it's misleading to act like any one way is the "correct" one.
Hi Ben! I met you outside a pub inside Bath and got a pic with you! Very much enjoyed this project (I think this is the one you told me about) and all these mega numbers 😁
I think this was the best fav number of the series. Yes it's amazing that this simple conjecture first fails at such a large number, but it's also amazing that it was so much *smaller* than the estimate from the proof that such a number existed.
Well, it was really a proof that there was *a* region where the summatory Liouville function was positive, around 10^361. The proof never said it was the first region where this was true. (And yes, you might think you'd have to calculate the SLF one value at a time, so if you reached 10^361, you'd have to go past 906,150,257 first. But... analytic number theory is more complicated than that, in ways I don't understand.)
It's similar to Graham's number. You know that's only an upper bound on the solution to a problem, not the actual answer, right? And the *lower* bound to that problem, as of 2020, is... 13.
@@alexpotts6520 Yes, like a first stab at the twin prime conjecture had a limit of 70,000,000 and I think it's now down to 6. I was also puzzled when i read about this, that *a* counter-example was found already in 1960. If that was found by just counting up, you would hit the *first* counter-example but that wasn't found until 1980.
@@bjornmu Like I said, analytic number theory is weird, there are a bunch of similar results of the "small numbers obey this pattern, but at some really big number it breaks" kind. A lot of these number-theoretical functions can be written as series with error terms like log(log(log(x))), etc, that only become relevant and make their effects visible when you're looking at truly gargantuan numbers.
Enjoyed the video greatly. Thanks much. Subscribed. Cheers
5:19 Thanks for giving us this hint! I find the oscillation even more fascinating than the general bias towards numbers that are the product of an odd number of prime factors. (That bias can be explained by the density of primes).
Because of a famous argument due to Littleeood, if Polya's conjecture was true ( at least for all sufficiently large numbers), then Riemann hypothesis would follow. Littlewood used his argument to disprove the conjecture that pi(x) is less than li(x).
3:20
I just thought of an interesting proof of that:
For each factor k of a number n exists a duel factor, n/k such that their product is n.
As long as k ≠ n/k the number of factors must be even since we can pair up all the factors like so:
(k,n/k).
Therefore, if the number of factors is odd there must be a factor that doesn't pair up and doesn't satisfy
k ≠ n/k.
Which therefore means k = n/k and
*k² = n*
Fantastic!
... and so k is the only unpaired divisor, and thus makes the total divisor count odd. Nice!
Fred
Out of the MegaFavNumber videos I watched this one is the best number!
Hey! :D
Hey Flammy
Hey!
u r going through the playlist? becoz I find u in almost all 13 videos before.
You are literally the reason I understand the Mandelbrot set
You should upload more regularly. Your content is always great
Really nice video. Favourite bit was about the Strong Law of Small Numbers (and Matt Parker's air horns, of course!).
You made a typo in your description, you wrote "Most numbers have an odd number of prime numbers!" instead of "Most numbers have an odd number of prime factors!"
Good spot! Thanks, fixed.
Lol
How could there possibly be a proof for this! I can't even imagine where I would start for something that seems as random as this. I would be incredibly interested in a video on the proof (no matter how complicated it is)
Proof by contradiction maybe?
You can find order in these kinds of sets with observations. Let p(x) equal -1 when x has an odd number of prime factors and 1 when x has an even number of prime factors. You can observe that p(xy) = p(x)p(y). And we're here considering the sums of the p(x)s. Already you might feel like some factoring techniques might work, but they don't, but then you start to see that it isn't completely unapproachable.
Thanks Ben. Similar to the numberphile video with Holly Krieger on Merten's conjecture. Now that I know you have a channel, I've subscribed and turned on notifications.
This is one of the best meganumbers so far.
I'd love to learn more about large first counterexamples. What other trends seem to go on forever, only to break an obscenely long time later?
Numberphile has a video on Merten’s conjecture, which has something similar
Great explanation. I really enjoyed your presentation. Thank you!
Might there still be more numbers with an odd number of factors, just in a weaker sense? For example, the limit of the proportion might still slightly favor odd numbers of factors, even if it moves around a bit along the way.
I know it's a discrete graph, but you can look at the graph @9:16 and consider the area above and under the graph. Saying that that area approaches -infinity as the number approaches infinity would be a weaker version of this statement.
Still not easy to prove, but I would bet that that would be a true statement.
Just subscribed! I've watched you for years on Numberphile, but found your other content to be a bit too long for me to be able to watch in one sitting. This lenght was perfect, more like this would be fabulous! :)
Every prime number has one prime factor, and early on there are a lot of primes. This is probably the driving force behind the function’s initial dive below zero. However, as the primes begin to thin out, so does their impact on the sum.
I knew it was gonna be a mind bender. I knew it would involve some really unique property. I knew it would be fascinating.
I did not know Matt Parker would stick the DJ horn sound effect in.
Ben: *mentions a number over 10^300*
Ben: that a huge number
Googologists: pathetic
At 3:12 : "Conjecture: n is a square if and only if it has an odd number of factors" . It is true , not a conjecture ( 2 minutes on the back of an enveloppe). In fact we may say more : Theorem: n is a k'th power if and only if it has a number of factors congruent to one modulo(k). I am wrong ?
77 has 4 factors: 1, 7, 11, and 77. 4 is congruent to 1 mod 3. But 77 is not a 3rd power.
[10:06 , 11:32] my most favourite part and what i wanted to read/hear on record
So the next challenge would be to find the first number under which positives and negatives cancel each other, if it ever happen.
This would mark a domain of well distributed odd and even numbers if picked up randomly.
Surely 1 is the first exception as it has an even number of prime factors (0)?
Guess this would be filed under "Trivial Solutions"?
Great choice of content and presentation as always Ben!
Excellent video, would love to see more videos by you on similar topics / crazy & unexpected math concepts! Keep it up!
I postulate that there is a prime number around 30'102 that for some odd reason should count as 2 factors on it own. And it's because we count it as only 1 that this conjecture fail.
New fave.
What is the area of that region/running area under the curve as the upper bound increases? It seems doubtful, but does the running area ever become positive?
TIME FOR PÓLYA CONJECTURE 2.0 :
For most integers M up to an integer N (however big), most integers below M have an odd number of prime factors.
So you're basically saying the running sum of the sign of the shown function is always negative. It certainly feels like it but I wouldn't be surprised if it also turns out wrong
This is such a vague conjecture, why even bother with it ?
Best #MegaFavNumbers video for me yet :)
so what i'm curious about is, what would it look like if you were to add up the area underneath the curve of that function up to any given point? that should give some kind of idea of... how much the value of that function tends to be negative or positive for all the previous numbers. and as far as I can tell for everything we've seen in this video up to the ~906 million mark, that area would be very decidedly negative even after you add in that little chunk sticking into the positives. if we graph out the area underneath that curve as its own line, will *that* line, that overall count of how even or odd the prime factors tend to be up to any given point, keep trending into the negatives forever, or will it eventually cross into the positives too?
i have no idea if this question is at all meaningful or relevant to the conjecture or anything else, but it's just the first place my brain goes to and kinda fun to think about!
best MegaFavNumbers video I've seen!
Is the code you used to make those graphs available anywhere?
yeah, i am wondering same
There is an interesting corollary in nature and cosmology to the strong law of small numbers, and this is in the isotropy of the universe which only becomes apparent when viewed across the largest of distance scales. Scales we have only been able to measure in recent decades. It is not necessarily obvious at all that the universe is extremely even in distribution. On scales even up to superclusters the universe looks very lumpy. But pan out far enough and we now know that the distribution of mater is astoundingly uniform.
Great video and thanks to youtube recommendations!
Ok this looks like the chart
I mean exatly the candlesticks you see when trading (long and short)
He nonchalantly says at the beginning that only square numbers have odd number of factors.... wait, what?!.. my mind blown just by that. Heh.. had to pause the video and take some time to digest just that little tid bit.
Ha. It is a nice number fact, and I was a little blown away the first time I realised it. :)
For what it's worth - this vid makes use of the fact: ruclips.net/video/-UBDRX6bk-A/видео.html
Can we have a video of the technical proof that you briefly talked about?
I really like this one
@8:10 is the tanaka you mention any relation to hitomi tanaka?
Is this just because you can't really have 0 prime factors, so you have to start at 1 prime factor (i.e. the primes), and 1 is odd, so this biases the number of prime factors being odd?
I don't know if I'm right but my idea at the onset was to say : assuming the number of prime factors is perfectly random for all numbers, then it still has to be odd for all primes, and this creates a bias. The numbers that are even powers would pull the other way but they are less dense than the primes.
Petros Adamopoulos yea we are essentially expressing the same idea in different words
1 is the product of 0 primes.
What you say explains the initial downward trend, but doesn't explain later intervals where upward steps predominate over downward steps.
0:53 You have misstated it. But you're in good company, as Wikipedia has misstated it, too. It's not "have an odd number of prime factors", it's "are the product of an odd number of primes". But thanks for going on to explain the correct idea.
Well, yes. The statements are equivalent if you count the repeated prime factors. But your statement is perhaps neater at capturing what we need to count. I liked the reference to the prime *factors* though as an intrinsic property of the number.
What if we put the intuitive idea into a weaker conjecture? The Liouville sum doesn't _always_ stay nonpositive, but does it stay nonpositive _most of the time?_ What if we conjecture that most numbers have an odd number of prime factors... most of the time?
So you're interested in the Liouville sum's sum?
@@rosiefay7283 Pretty much
A great video Ben. More forthcoming?
5:19 " it's kind of
got a self-similar fractally look to it
there's
some interesting consequences from that
um go look up the riemann zeta function
if i see a connection
with the riemann hypothesis in this "
6:20 " and as i mentioned before there is
to do
with the riemann zeta function and look
up the connections on wikipedia or wolfram "
Why is this function continuous? I mean it’s not because it’s not using real numbers. But its values don’t jump. Why?
Another examples of stong law of small numbers where we see issue very very late is 111111111111111... 19 1s is a prime number even though after 11 we didn't have any. Are there more such examples??
23 ones, 317, 1031 and 49081 are prime. It is believed (but not confirmed) that 86453, 109297 and 270343 ones are also prime.
Does the ratio of the area above the axis and below the line vs the area below the axis and above the line have any significance?
I'm curious if primes themselves are counted in the Liouville function, and if so would they always be considered to have 1 factor?
Given any positive or negative bound, is it known is that function breaks it for some arbitrarily large value of n?
Does it cross 0 infinitely many times? Does there exist some mega(fav)number N such that for n>=N it's strictly positive or strictly negative?
Exceptional.
Perfectly explained 🤝
But doesn't the sequence start with positive 1? 1 has 0 prime factors, 0 is even so it would produce a 1, or am I wrong?
You’re right. I guess we have to start at 2 :)
How to properly state conjectures in number theory:
"Most numbers have between log(N) & log(N)+1 prime factors" is bullshit
"For each positive integer, N, let A(N) = the number of integers, k, 1 1/2, then that is what we mean by "Most numbers, N, have between log(N) & log(N)+1 prime factors" ".
This is my favourite too. Was thinking about making a video, but this is easier and probably better.
Sampled at 48khz, 906,150,257 samples would represent roughly 30 minutes of audio. I wonder the Liouville function sounds like?
did you figure that out??
This conjecture reminded me of some talk on Numberphile, and it was Mertens conjecture.
ruclips.net/video/uvMGZb0Suyc/видео.html
They are so much alike!
That one ignored numbers with repeated prime factors didn't it? I wondered at the time what would happen if you included them. Well now I know. Kind of. I guess there's a third sequence that considers the number of distinct prime factors.
If Riemann Hypothesis holds, the Louville Function would even behave exactly like a simple random walk.
But this summatory function looked unlike a random walk. There are long intervals where upward steps are much more common than downward ones, to be followed by long intervals where the reverse happens.
@@rosiefay7283 Wrong, this is a classic example of human misconception of randomness. There is even a statistical test exploiting this ('run' test for randomness). Humans tend to think randomness is somehow connected to perfectly shuffled sequences but in reality randomness also leads to longer sequences in mostly just one direction. Just plot some brownian motions, sometimes they will explode in one direction, sometimes not.
Лучшее видео с MegaFavNumbers.
Voila! A problem based on this came in the IMEO 2020 paper as Q3 if I remember correctly. There I wondered if this function L(n) ever was positive, and was absolutely convinced that it could never happen. Opened my eyes! I wish this video was made before I gave that test 🙃. Have a look at the AoPS threads for the same to check out that problem. It's pretty interesting.
Question, if it's proven that the Liouville Sum crosses the x-axis an even amount of times, would that then prove that Pólya's conjecture is true after all? Because if it crosses the x-axis an even amount of times it will be negative as x goes to infinity (we start by going negative after all).
Is it proven that there are an infinite number of possible n for which this function is larger than 0? Or could there possibly be a largest number for which it is larger than 0? If so you could still argue that "most numbers" is fulfilled as a criteria. That would also be closer to how I would interpret the initial formulation of the conjecture.
It's actually proven (despite what I say around 10:00) that it crosses between negative and positive infinitely many times. Haselgrove proved this in 1958 in fact. So, unfortunately, even a modified version of the conjecture is false. :(
Even though the polya conjecture is wrong how it was originally stated, isn't it still weird that the liouville function looks that way? I mean if the number of prime factors was completely random, i.e. equally likely even or odd, the liouville function would most probably cross 0 much earlier.
random is not the same as evenly distributed and prime numbers are not evenly distributed.
Asymptotically, do we know if "most" (≥0.5) numbers satisfy the conjecture? Do almost all?
so far have read about:
* 4 comments on marten's conjecture/constant
* 4 comments asking for the python script (not counting my own comment regarding this)
* 3 comments wondering about when the area will be equal to zero
* 2 comments confusing "random" with evenly distributed
... wow, there are many recurring themes in this comment section
Here from the Collatz Conjecture video by Veritasium.
> _"Here from the Collatz Conjecture video by Veritasium."_
same
If the conjecture is about 'most', then if the sum of odd and even is zero, it is already disproven, isn't it? Doesn't have to cross the x-axis into the positive arena for that.
Wonderful video
Strong Russel Crowe in A beautiful Mind vibes in the opening shot! :D
Really really cool!
Any ideas on why this occurs? And I guess some hundreds of people have already created Python codes to see if this happens after more time...
How did you obtain the graph of the Liouville function ?
I wrote Python code to calculate it.
Is the python script for this available?
i want the py script too
But "most" up to N is still vague. Is it 50%, or 66.78% etc...?
exp(10^120) billion years is clearly the best number - that is the Poincaré recurrence time for our universe - it is somewhat sad how few people have an idea how smart Poincaré was.
Checks the first Tree(3) numbers, they all work. then Tree(3)+1 doesn't work.
Yeah, and the crazy thing is, even Tree(3)+1 is basically zero compared to infinity.
Significantly most of the time, most integers have an odd number of prime factors. Done!
how to write such python script? can u post this script on github or gitlab or somewhere like that?
great lighting!
great video! Awesome
Just to be annoying, wouldn't the first case be n=1? Since 1 has zero prime factors? I know it is a corner case, so it doesn't matter all that much. Unless you somehow argue 0 has and odd number of prime factors? :p
so how can this conjecture be put to use?
Mathematically, we know that if Polya's conjecture is true then the Riemann hypothesis is also true.
Outside of math? I dunno. But I'll tell you now that if you go into math expecting every to turn single idea into laser guns and warp drives, then you're in for a rough time.
Nice one
I wonder if the cumulative sum of that sequence would always be negative. It certainly seems that way.
how far in excess was it … oh wait it must have been 1
Hello, could you please share your python script?
i want the py script too
Ok now let's find the smallest counter-example to Mertens conjecture !
We already have
@@arnouth5260 Nope
Восьмияче́йник sure, no explicit counterexample has been found, but we know for a fact it exists between 10^16 and 10^(7.91*10^39). Seeing how huge these are the exact number will probably not be found anytime soon.
@@arnouth5260 Yes but I asked for the smallest counter-example :D
Actually the smallest example is -1. So myehh
Ben u r cool I love ur talks (edit: and sings haha)
You could very much reduce the amount of computation required by choosing say every 5th or 10th number, and selectively doing all computation for areas when it gets close to zero. I can think of other ways to reduce the amount too. You see, the trick here is to Compute Smart 😀
hey, can u share the scipt? or your version of it?
Thanks!
Commenting for those metrics!