A Breakthrough with Fingerprint Numbers - Numberphile 

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.

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.

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.

I accept it.

All I can say about the term "fingerprint number" is that it has Brady's fingerprints all over it. ;-)

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

The sum often diverges, for sets with non zero natural density, for example.

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.

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.

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 🪽

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.