Jared really is an exceptional explainer! There aren't many people who are both brilliant and able to explain the difficult things they understand this clearly. Awesome followup video!
I have a math specific learning disability. Still watch every Numberphile and Numberphile 2 videos. I usually have no idea what’s being discussed but I enjoy the enthusiasm.
Interesting! How was the term 1/(n log n) chosen though? It seems extremely weird to me that the sum over numbers with exactly 10 prime factors would be almost exactly the same as the sum over numbers with exactly 1 billion prime factors (the smallest of which is 2^(1 billion)).
In the previous video, he explained that Erdos proved that the sum over 1/(n*log n) converges for every primitive set. Note that the sum over 1/p (for primes p) blows up to infinity, so you can think of 1/(p*log p) as looking at how a very small adjustment to the decay rate of 1/p can make it converge instead. More generally, logarithmic-type functions crop up a lot in analytic number theory (not just log but stuff like log log log lol!) probably because it's associated with the growth rate of the primes from the Prime Number Theorem. I guess people like this mathematician would have more specific/technical reasons why they find these sums useful, but these are some rough reasons why functions like this tend to appear aiui.
When he points out they proved that the inequalities fail eventually for f(k) he shows a graph where it begins to fail at just f(5), surely that would have been known about hundreds of years ago? What am I missing?
basically the graph was super difficult to compute before Lichtman. to quote the paper: Their [Banks and Martin's] approach is to directly compute the series up to 10^12 ... this approach becomes exceedingly difficult as k grows. This is in part due to the fact that the series for f(N_k), f(N*_k) converge quite slowly. For example, as we shall see, the partial sum up to 10^12 makes up less than half of f(N_4).
Did I catch in the first video that as k tends toward infinity, the fingerprint of f_k converges toward 1? So when Jared defines g_k as just the odds with prime factors count of k, that's just f_k eliminating anything with a prime factor of 2? Maybe not as 2 can be used multiple times I suppose. But it doesn't surprise me if g_k as k tends to infinity tends to 1/2.
It's Jared that's using the term finger print number as if it is accepted. Bradley used the term in the original video and it was new to Jared. If a term is useful and it catches on and becomes "catchy" it's probably a useful term.
I find it interesting that the value that the sum of 1/p log p is in the neighborhood of, but not exactly Fibonacci's number, phi. I'm pretty sure that this is just a coincidental relationship, but it's fun to speculate it there might be a connection afterall
Something less limiting, such as all the multiples of 6, since most of the smaller numbers with 3, 4, or 5 prime factors are divisible by 6. Some of the options would have to skip the first set, since they would exclude primes.
Has something been done to the audio in this video? Jared sounds very odd, like each syllable has been sped up, but the gaps between them also increased. It results in a very artificial sounding staccato-like rhythm.
0:55 "exactly k prime factors" 1:01 "numbers that have exactly 1 prime factor, so in other words the prime numbers". So you're counting prime factors of n with multiplicity, then, rather than prime factors of n? Ω(n)=k, rather than ω(n)=k? If it really is Ω you mean, then another way of saying it is "numbers that are the product of k primes".
I have absolutely no idea what log even is let alone these equations and letters 😂 k prime factors what? I probably shouldn’t even be watching these videos tbh so I will quietly step out of the room
Whenever you encounter a new area of information, there will be a lot of jargon. Step One is: look up each jargon word and figure out what each one means. Then, at least you know what they're talking about. log is short for logarithm by the way.
log is the "inverse" to exponential. An Exponential is "repeated multiplication" and is expressed with a superscript; or in type followed by a "^" symbol. Examples: since 2^5 = 2 x 2 x 2 x 2 x 2 = 32 i.e. 2 to the power of 5 is 32; thus the equivalent log statement is log_2 32 = 5 or log base 2 of 32 is 5 since 3^4 = 3 x 3 x 3 x 3 = 81 i.e. 3 to the power of 4 equals 81 then in logs: log_3 81 = 4 or log base 3 of 81 equals 4. Now we can do logs with any base and for "pure" math the base chosen is usually the "natural number" and thus the log is the "natural log" (what the natural number equals is irrelevent, but its about 2.71828...; and is expressed as "e" for exponential). For many other contexts natural log is expressed as "ln" as it likely is on your calculator. so the number "log 17" is the natural log of 17 or what power of e equals 17; which is about 2.83321... thus e^2.83321 ~ 17 So what does it mean for a number to have "k prime factors". Well k is a fixed number. lets take 3; or 3 prime factors. These numbers would be 8 (= 2 x 2 x 2), 12 (= 2 x 2 x 3) , 18 (= 2 x 3 x 3) , 20 (= 2 x 2 x 5) , 27 (= 3 x 3 x 3) , 30 (= 2 x 3 x 5), 42 (= 2 x 3 x 7), 44 (= 2 x 2 x 11) , 45 (= 3 x 3 x 5), 50 (= 2 x 5 x 5) , etc notice we count each prime factor (including repeats) and if there are 3 of them its in the list. thus "f_3" = 1/(8 log 8) + 1/(12 log 12) + 1/(18 log 18) + 1/(20 log 20) + 1/(27 log 27) + 1/(30 log 30) + 1/(42 log 42) + 1/(44 log 44) + 1/(45 log 45) + 1/(50 log 50) + .... Thus "1 prime factor" is just the primes themselves, and "2 prime factors" are 4, 6, 9, 10, 14, 15, 21, etc; 3 prime factors I covered etc Thus "f_1" = 1/(2 log 2) + 1/(3 log 3) + 1/(5 log 5) + .... "f_2" = 1/(4 log 4) + 1/(6 log 6) + 1/(9 log 9) + 1/(10 log 10) + .... etc They were doing a lot with "Odds only" which means you exclude the multiplies of 2, thus "1 odd factor" is odd primes so 3 , 5 , 7, 11, 13, 17, etc "2 odd factors" is 9, 15, 21, 25, 33, 35, 39, etc "3 odd factors" is 27, 45, 63 etc so "g_1" = 1/(3 log 3) + 1/(5 log 5) + 1/(7 log 7) + ... "g_2" = 1/(9 log 9) + 1/(15 log 15) + 1/(21 log 21) + ... "g_3" = 1/(27 log 27) + 1/(45 log 45) + 1/(63 log 63) + .... and so on. What they showed was that after a certain point ALL g_k are decreasing. We don't know what the value of k when it for sure starts decreasing is; and its believed to be 1; However it could be g_k decreases to 10^50 then increases for a few g's then decreases from there on out; not saying that is the case but they haven't shown that doesn't happen.
See the previous video at: th-cam.com/video/33YSWaR3kAQ/w-d-xo.html
Jared really is an exceptional explainer! There aren't many people who are both brilliant and able to explain the difficult things they understand this clearly. Awesome followup video!
It’s hard to explain one’s understanding.
I see it both as a sperg and a programmer
It helps that stating these theorems doesn’t require much background knowledge.
I have a math specific learning disability. Still watch every Numberphile and Numberphile 2 videos. I usually have no idea what’s being discussed but I enjoy the enthusiasm.
Have any topics you want help with? I'll make a video on them if you want
Interesting, does it have a name? I've never heard of such thing before.
@@berkeunal5773
I believe it is called dyscalculia. If you search Brady’s Numberphile channel, I believe there is a video on it there.
Beautiful explanation. Brady, your guidance of this interview is masterful.
Interesting! How was the term 1/(n log n) chosen though? It seems extremely weird to me that the sum over numbers with exactly 10 prime factors would be almost exactly the same as the sum over numbers with exactly 1 billion prime factors (the smallest of which is 2^(1 billion)).
In the previous video, he explained that Erdos proved that the sum over 1/(n*log n) converges for every primitive set. Note that the sum over 1/p (for primes p) blows up to infinity, so you can think of 1/(p*log p) as looking at how a very small adjustment to the decay rate of 1/p can make it converge instead. More generally, logarithmic-type functions crop up a lot in analytic number theory (not just log but stuff like log log log lol!) probably because it's associated with the growth rate of the primes from the Prime Number Theorem. I guess people like this mathematician would have more specific/technical reasons why they find these sums useful, but these are some rough reasons why functions like this tend to appear aiui.
Jared is using "fingerprint" as if that is the accepted term. Is that the case? Brady came up with "fingerprint number" during the first video.
I accept it.
All I can say about the term "fingerprint number" is that it has Brady's fingerprints all over it. ;-)
When he points out they proved that the inequalities fail eventually for f(k) he shows a graph where it begins to fail at just f(5), surely that would have been known about hundreds of years ago? What am I missing?
basically the graph was super difficult to compute before Lichtman. to quote the paper:
Their [Banks and Martin's] approach is to directly compute the series up to 10^12 ... this approach becomes exceedingly difficult as k grows. This is in part due to the fact that the series for f(N_k), f(N*_k) converge quite slowly. For example, as we shall see, the partial sum up to 10^12 makes up less than half of f(N_4).
got it, thank you @@randomnamegenerator9128
Did I catch in the first video that as k tends toward infinity, the fingerprint of f_k converges toward 1? So when Jared defines g_k as just the odds with prime factors count of k, that's just f_k eliminating anything with a prime factor of 2? Maybe not as 2 can be used multiple times I suppose. But it doesn't surprise me if g_k as k tends to infinity tends to 1/2.
This Jared fellow really butters my bagel
Has the whole world adopted the term finger print number?
Add it to the list of mathematical terms coined by Brady
No, this channel has an annoying tendency to make up "catchy" names. "Anti-prime" for highly composite numbers is one that I particularly dislike.
It's Jared that's using the term finger print number as if it is accepted. Bradley used the term in the original video and it was new to Jared. If a term is useful and it catches on and becomes "catchy" it's probably a useful term.
What is the accepted term, if it is not “fingerprint numbers”?
@@ShankarSivarajan Why is it a bad thing? If it doesn't catch on, no big deal. If it does catch on, it's a bit of fun.
1:00 shoud'nt it be over all prime pwers because all prime powers also have 1 prime factor?
Once of the Numberphiles that went waaaaay over my head. But admittedly I do have mental blocks with some ideas.
That "explicit" pun is pretty funny :p
I find it interesting that the value that the sum of 1/p log p is in the neighborhood of, but not exactly Fibonacci's number, phi. I'm pretty sure that this is just a coincidental relationship, but it's fun to speculate it there might be a connection afterall
5:11 Brady's ghost is visible in the background! 😁
I wonder what the behaviour is like for other infinite subsets of the integers?
The sum often diverges, for sets with non zero natural density, for example.
Checks fingers for K prime factors
I wonder what the results would be over the sets of {0,1,or 2} mod 3. Or over all perfect squares. Or all of the form (2^m) - 1.
Something less limiting, such as all the multiples of 6, since most of the smaller numbers with 3, 4, or 5 prime factors are divisible by 6. Some of the options would have to skip the first set, since they would exclude primes.
That's pretty neat.
GK satisfies the property
general knowledge
I've never been this early before. Cool video!
Has something been done to the audio in this video? Jared sounds very odd, like each syllable has been sped up, but the gaps between them also increased. It results in a very artificial sounding staccato-like rhythm.
That’s just how he speaks. Although I can hear a slight tone in the background like a printer or something that only plays when he speaks.
Redbull was added to keep it under 10 minutes 🪽
Michael Falk
Wow!
Interesting
0:55 "exactly k prime factors" 1:01 "numbers that have exactly 1 prime factor, so in other words the prime numbers". So you're counting prime factors of n with multiplicity, then, rather than prime factors of n? Ω(n)=k, rather than ω(n)=k? If it really is Ω you mean, then another way of saying it is "numbers that are the product of k primes".
↑
All kwat ki game me
... say what ...
Yees
Cyborg voice 🙂
This is unironic, but ironic.
This is irrational, but rational
I have absolutely no idea what log even is let alone these equations and letters 😂 k prime factors what? I probably shouldn’t even be watching these videos tbh so I will quietly step out of the room
Whenever you encounter a new area of information, there will be a lot of jargon. Step One is: look up each jargon word and figure out what each one means. Then, at least you know what they're talking about. log is short for logarithm by the way.
I remember watching these videos before i could even add fractions together
log is the "inverse" to exponential. An Exponential is "repeated multiplication" and is expressed with a superscript; or in type followed by a "^" symbol.
Examples: since 2^5 = 2 x 2 x 2 x 2 x 2 = 32 i.e. 2 to the power of 5 is 32; thus the equivalent log statement is log_2 32 = 5 or log base 2 of 32 is 5
since 3^4 = 3 x 3 x 3 x 3 = 81 i.e. 3 to the power of 4 equals 81 then in logs: log_3 81 = 4 or log base 3 of 81 equals 4.
Now we can do logs with any base and for "pure" math the base chosen is usually the "natural number" and thus the log is the "natural log" (what the natural number equals is irrelevent, but its about 2.71828...; and is expressed as "e" for exponential). For many other contexts natural log is expressed as "ln" as it likely is on your calculator.
so the number "log 17" is the natural log of 17 or what power of e equals 17; which is about 2.83321... thus e^2.83321 ~ 17
So what does it mean for a number to have "k prime factors". Well k is a fixed number. lets take 3; or 3 prime factors. These numbers would be 8 (= 2 x 2 x 2), 12 (= 2 x 2 x 3) , 18 (= 2 x 3 x 3) , 20 (= 2 x 2 x 5) , 27 (= 3 x 3 x 3) , 30 (= 2 x 3 x 5), 42 (= 2 x 3 x 7), 44 (= 2 x 2 x 11) , 45 (= 3 x 3 x 5), 50 (= 2 x 5 x 5) , etc
notice we count each prime factor (including repeats) and if there are 3 of them its in the list.
thus "f_3" = 1/(8 log 8) + 1/(12 log 12) + 1/(18 log 18) + 1/(20 log 20) + 1/(27 log 27) + 1/(30 log 30) + 1/(42 log 42) + 1/(44 log 44) + 1/(45 log 45) + 1/(50 log 50) + ....
Thus "1 prime factor" is just the primes themselves, and "2 prime factors" are 4, 6, 9, 10, 14, 15, 21, etc; 3 prime factors I covered etc
Thus "f_1" = 1/(2 log 2) + 1/(3 log 3) + 1/(5 log 5) + ....
"f_2" = 1/(4 log 4) + 1/(6 log 6) + 1/(9 log 9) + 1/(10 log 10) + ....
etc
They were doing a lot with "Odds only" which means you exclude the multiplies of 2,
thus "1 odd factor" is odd primes so 3 , 5 , 7, 11, 13, 17, etc
"2 odd factors" is 9, 15, 21, 25, 33, 35, 39, etc
"3 odd factors" is 27, 45, 63 etc
so "g_1" = 1/(3 log 3) + 1/(5 log 5) + 1/(7 log 7) + ...
"g_2" = 1/(9 log 9) + 1/(15 log 15) + 1/(21 log 21) + ...
"g_3" = 1/(27 log 27) + 1/(45 log 45) + 1/(63 log 63) + ....
and so on.
What they showed was that after a certain point ALL g_k are decreasing. We don't know what the value of k when it for sure starts decreasing is; and its believed to be 1; However it could be g_k decreases to 10^50 then increases for a few g's then decreases from there on out; not saying that is the case but they haven't shown that doesn't happen.
4th
He is explicitly not an English major
first