A Short Number Theory Proof

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@filbranden 7 ай бұрын

This is OEIS sequence A165256 "Numbers whose number of distinct prime factors equals the number of digits in the number" It turns out there's 7,812 total numbers in the sequence and the largest one is 9,592,993,410 = 2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 43

@megaing1322 7 ай бұрын

I wrote a small script to analyze this (slightly tricky tbh. Just testing all numbers takes too long for my taste so I had to come up with an at least somewhat clever approach). A few more facts: - There 6 10-digit numbers, 5 of which start with 2x3x...x23, but then plug in the different primes 29, 31, 37, 41, 43, and then additionally to those 5 there is 2x3x...x17x19x29x31 - The larges prime for which we can construct a working number is 467 with 2x3x5x7x467=98,070 - There are exactly three numbers for which the smallest prime factor is 7: 7, 77=7x11, and 91=7x13 . Seven is the largest prime factor that is ever the smallest prime factor in a number - 7,490 out of the 7,812 numbers contain a factor of 2 - 5 numbers contain 2^6, the highest prime exponent in any number: 960=2^6x3x5, 6720=960x7, 73920=6720x11, 87360=6720x13, 960960=73920x13

@omp199 7 ай бұрын

The OEIS really should have specified "in base 10" in the title of the sequence. And if they have included this sequence, are they also going to include the corresponding sequences for _every_ possible base? Are they any more or less interesting than this one? I have never understood why anyone cares about sequences like this: sequences that are specific to the arbitrary base that we happen to have chosen for representing numbers in writing. We're not looking at interesting properties of numbers, at this point. We're just looking at the consequences of an arbitrary choice for representing them.

@clarfonthey 7 ай бұрын

Huh, I think that this actually proves that the statement works for any base too. Since the sequence of primes will eventually reach a prime higher than your base, it doesn't matter even if you've got way more primes available than digits at that point: the number of digits added per prime added will always exceed 1 and then even if it takes a while, you'll end up with more digits than primes. So, for any base N, there are always a finite number of numbers with the same number of digits base N and distinct prime factors.

@NoNameAtAll2 7 ай бұрын

now I wonder about factorial base

@flooofoi 7 ай бұрын

the product of the first n primes is never smaller than the product of the first n+1 numbers which has n+1 digits in factorial base

@inseptus712 7 ай бұрын

@@flooofoi I don't think that's accurate.

@Foxxey 7 ай бұрын

Interestingly enough it's not a coincidence that it breaks after b=10 (the base we are working in). So-called primorial numbers aka. multiplying the first n prime numbers together follow approximately n^n. So if we want n distinct prime factors the smallest example of such number is approximately n^n. Since we can get the number of digits with log_10 we end up with: log_10(n^n) = n. Raising both sides to the 10th power: n^n = 10^n, which gets us to n=10. (This is the only non-trivial point where the graphs meet since x^x > 10^x for larger x.) After that the number of digits is simply too large. I'm quite certain that the turning point "b" for smaller bases would be inaccurate, but it should work for higher ones.

@davidlindstrom4383 7 ай бұрын

This video has my favorite number featured! 6,469,693,230 is fascinating because not only is it the product of all primes less than 30, but the first 3 digits, 2nd 3 digits, and 3rd 3 digits are 2x323, 3x323, and 1x323, respectively, and 2,310 happens to be the product of all primes less than 12! Is that just coincidence?

@ivanerofeev1269 7 ай бұрын

The product of all primes less than 12! is a huge number, btw

@jamieallan2859 7 ай бұрын

One clearly forgets you cannot exclaim when talking with mathematicians

@GhostyOcean 7 ай бұрын

Only in base 10. Choose a different base and it's not as beautiful.

@omp199 7 ай бұрын

Imagine liking a number just because you're from a family of upright apes that happen to have evolved with ten digits.

@pietergeerkens6324 7 ай бұрын

@@omp199 Who are you calling "from a family of upright apes that happen to have evolved with ten digits"? Them's fighting words.

@oldjoec3710 6 ай бұрын

You changed the problem statement in mid-discussion. At 4:30, "we've run into a problem" because the smallest sequence WITH NO REPEATS gives too large and answer. However, at 1:02, you state that repeats are allowed (2x2 x 3). Which rule do we follow? At 4:30, shouldn't we add 2x2 x 3 x 4 x 6 x ... 31 (value about 80 billion) to the list? How about 2x2x2x2 x 3 x 5 x 7 x . . . x 30 (value about 45 billion)? By your original definition of the problem, there ARE infinitely many such numbers, since we can always take out an appropriate number of the the largest factors in the sequential list and replace them with enough 2's or 3' or 5's or 7's to get us to the right order of magnitude.

@nnaammuuss 7 ай бұрын

🙂 so, the next question is, if we do this in d-ary, will the limiting case always be (at least/at most) _d digits?_

@theprof73 7 ай бұрын

Curious that the base 10 limit is 10... is there a pattern with other bases?

@angelodc1652 7 ай бұрын

Might have to get back to you on that one

@angelodc1652 7 ай бұрын

Okay. I found out that 16^16>p(16)# where p(n)# is the product of the first n primes

@bart2019 7 ай бұрын

I'm not sure what KZbin makes of this video, but I got several ads for beauty products (shampoo, nail polish, skin care) before it started. The ad right after it finished was also one of those. Does KZbin associate "prime" with beauty care? I don't know... but it seems that way.

@maalikserebryakov 7 ай бұрын

Lol youtube is telling you to clean your musty mathematical azz

@MrDannyDetail 7 ай бұрын

I'd guess it's more likely to be getting confused with a certain online retailer's premium shopping service, which you (or someone else using your device or google login) may have happened to have used to purchase that type of products.

@حسينالقطري-ب8ص 7 ай бұрын

Fantastic! Looking forward to seeing more from you.❤

@LogicMasterSteve 7 ай бұрын

How many of those numbers exist? Obviously 6.469.693.230 is the only representative with 10 digits and 10 distinct prime factors. There are already 15 (a surprisingly high number) 9-digit numbers with 9 distinct prime factors. Unfortunately there will be no elegant solution to my question - it is just a matter of (systematic) counting. Which number N of digits will have the most representative numbers with N digits and N distinct prime factors? This is difficult to guess...

@jay_13875 7 ай бұрын

There are several other 10-digit numbers with 10 distinct prime factors. For example 2*3*5*7*11*13*17*19*23*31 = 6,915,878,970 The number 6,469,693,230 is just the smallest 10-digit number you can produce with 10 distinct prime factors.

@MyOneFiftiethOfADollar 7 ай бұрын

Do you mean so far there is no elegant solution? Currently the approach is ad hoc, but Doctor Barker’s video may stir up enough interest to discover a closed form function of n that counts the number n digit numbers with n distinct prime factors. We know C(10)= 1 and C(9)= 15. You are probably right and the question will remain open or possibly for all n greater than some M, C(n)=0

@jay_13875 7 ай бұрын

@@MyOneFiftiethOfADollar I just gave an example that demonstrates that C(10) is at least 2. It would be easy enough to list and check all the possible combinations with a computer program since the 10-digit number can't have any prime factors greater than 43 and it also can't have any prime factors with multiplicity greater than 1. This means there are only `14 choose 10` = 1001 potential candidates (43 is the 14th prime number), most of which will have more than 10 digits. For n=9, it might be trickier since there are a bunch more possible prime factors. But it would be easily solvable for all n with a bit of brute force (since C(n)=0 for all n>10, so you only have to check for n between 1 and 10).

@MyOneFiftiethOfADollar 7 ай бұрын

@@jay_13875 thx, didn’t watch enough of video to even notice C(10)=0 for n >10. I think there is a result that relates to this for composite numbers……

@LogicMasterSteve 7 ай бұрын

@@jay_13875 Thank you for the correction. I was completely missing that one may use prime factors greater than 29. My bad. 🙂 After counting again I am quite sure that C(10) = 6 (there is still not much choice since the multiplicity of all prime factors is one and you have to use the lowest 8 prime numbers 2,3,5,7,11,13,17,19 in order to stay below 11 digits). Of course my claim of C(9)=15 is wrong as well. There are already 15 such numbers with using only prime numbers up to 29...

@Don.Challenger 7 ай бұрын

So what about the more obvious case where number of digits on both sides need to be equal? Positive integers with {the same number of digits} (in base 10) as {[digits of its] distinct prime factors} with one example being: 105 = 3 x 5 x 7 LHS 3 digits RHS 3 digits where your first example would fail: 165 = 3 x 5 x 11 LHS 3 digits RHS 4 digits

@henrikstenlund5385 7 ай бұрын

this is for the 10-system of numbers. What about hexadecimals?

@galinha3167 7 ай бұрын

I love mathematics❤❤❤ i'm from Brazil

@alipourzand6499 7 ай бұрын

Vendredi c'est ravioli & Dr. Barker!

@jonathandyment1444 7 ай бұрын

I am going to stop you here (1:22) and go my way, because the prime factors have to be distinct, but the digits not so (why? it just seems to spoil the art of it). So 11 should be good because it has 1 prime factor and 1 (repeated) digit.

@omp199 7 ай бұрын

Maybe you could make a video about your version of the problem!

@sr6424 7 ай бұрын

Here is an interesting question. If you Google was 1 ever considered prime. You get the answer, it was until around 100 years ago! So if you had made this video then (sorry videos weren’t around then). You would have stopped at a much higher number!

@MyOneFiftiethOfADollar 7 ай бұрын

If 1 was considered prime then 6=3x2 = 3x2x1 = 3x2x1x1 = 3x2x1x1x1 ad infinitum. So by convention 1 is not prime since otherwise the prime factorization would be horribly not unique. And we all know how much mathematicians adore unique objects.

@sebastianmanterfield3132 7 ай бұрын

@@MyOneFiftiethOfADollar unique up to units, as it should be

@MyOneFiftiethOfADollar 7 ай бұрын

@@sebastianmanterfield3132 I believe the development of number theory occurred before abstract algebra where the notion of units was created. Unique Factorization Domains like the Gaussian integers with units 1 -1 i and -i made the notion of uniqueness more interesting I suppose. 1 and -1 are the units of the integers, but we normally only think of positive integers when dealing with fundamental theorem of arithmetic. No such thing as a positive complex number unless it on the real axis.

@samueldeandrade8535 7 ай бұрын

@@sebastianmanterfield3132 unique up to units AND up to order of powers of the prime factors.