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37, the median value for the second prime factor of an integer
- whatever1 3y agoQuestion, has anyone tried Transformer networks for predicting prime numbers?
- matteoraso 3y agoThat would be an incredibly expensive way of generating prime numbers, far more expensive than just checking to see if it has any factors.
- whatever1 3y agoI was just hoping that we can extract some useful patterns from reverse engineering the trained network
- zelda-mazzy 3y agoIt was a bit difficult to grasp at first, but it clicked after realizing it's all primes up to 37, not just 37. Kind of a neat fact, and I enjoyed reading this. Thanks for posting!
- kstrauser 3y agoI don't think that's right. My reading of it was that 37 is the median prime among all primes.
- gattilorenz 3y agoI think you’re both expressing the same concept :) 37 is the median factor, it doesn’t mean that 37 is the actual second prime of 50% of numbers. The OP probably missed “median”, I know I did
- twelvechairs 3y agoOriginal comment said "all primes up to 37" which isn't correct there's no point where 37 is being used as an upper limit
- gattilorenz 3y agoI think that was an intuitive and imprecise way of expressing “median value”, so that 50% of primes are “up to 37”
- zelda-mazzy 3y agoIm going off what the article originally said: "We see that 37 is the prime where roughly half of all numbers have something less than or equal to 37 as their first prime! So we’ve proven that 37 is the median second prime!"
- NooneAtAll3 3y agodid the title change or smth?
- Dylan16807 3y ago> it's all primes up to 37 Can you elaborate on "it" in this sentence? What is "all primes up to 37", when you phrase things in an easier to understand way? I can't tell if you're suggesting the word median is being used in a misleading way? But it's basically the definition of "median 37" that half your numbers are "up to 37".
- zelda-mazzy 3y agoI admit I read it wrong at first, but it made sense to me at the end of the article. So I don't believe median was misleading. My initial thought was, as numbers get larger and larger, the chance that the second most common prime factor is 37 so 50%, which didn't make sense because of how common 3 and 5 are. By the end I realized the function is taking into account the range of prime factors as an aggregate. Now I understand that with enough composite numbers, the chance that their second prime factor is 37 OR BELOW converges to 50%, which is pretty neat.
- mg 3y agoI agree that this makes 37 somewhat interesting. Certainly more interesting than - say - 31. 31 is a prime number too, and therefore somewhat interesting. But for sure not as interesting as 37, which as we just learned, is the median value for the second prime factor of an integer. Any suggestions of integers which are even more interesting? And while we are at it, is there an integer which qualifies to be the most interesting?
- falcor84 3y agohttps://en.wikipedia.org/wiki/Interesting_number_paradox https://en.wikipedia.org/wiki/Interesting_number_paradox
- mikepurvis 3y agoI know it's a joke, but I still feel like it would be more meaningful if it wasn't so generic that it could kind of apply to anything. Like, there's nothing specific to numbers about the idea that the least-"interesting" item in some set is itself interesting for possessing that property.
- mg 3y agoThen let's see who manages to write the least interesting comment in this thread.
- deleted 3y ago[deleted]
- falcor84 3y agoThe way I understood it, this formulation relies on induction on the natural numbers.
- SamBam 3y agoBecause it's numbers, which are sortable, you can ask which is the least member of a set. If a set of positive numbers doesn't have a least member, does the set exist? There's a similar paradox which asks what the smallest number that can't be defined in fewer than thirteen words. There must be some number, right? For instance, 7603201560, or "seven billion six hundred and three million two hundred and one thousand five hundred and sixty" seems to require 16 words. Maybe you can shorten it by removing the "ands," but I can come up with a longer number. But if there were such a number, then couldn't we describe it as "the smallest number that can't be defined in fewer than thirteen words?" And then it just got defined in fewer than thirteen words. So it can't really be a member of that set. So does the set have any members?
- yieldcrv 3y agowhy is this kind of thing interesting? my undergrad math teachers were never able to convey that is there something here to use this knowledge with? like cracking a lotto’s RNG by knowing a second prime probability? that would be interesting to me
- SirYandi 3y agoAn unknown thing becomes a non obvious thing, which makes it interesting. Until it becomes obvious, of course.
- NooneAtAll3 3y agothere are people that think math is a chore - and then there are people that think math is a game for the second kind of people (like me) these "interesting facts" are fascinating even in otherwise uselessness
- ravi-delia 3y agoI mean this is a crazy thing to know about 37. And we know it!
- SonOfLilit 3y agoThe interesting thing is that this feels like a thing that it would be extremely hard to know, and yet we found a way to show it to be true. Many much simpler and more intuitive statements about primes and their distribution are beyond the grasp of modern math to prove or disprove.
- Tyr42 3y agoI thought it'd be infinite or something. So it's cool to know it exists and it's not 1, 2 or infinity.
- Someone 3y agoI don’t think this is _that_ interesting because the probabilities don’t add up to exactly 50%. We have: - the smallest prime factor of less than half of all integers is ≤ 31 - the smallest prime factor of over half of all integers is ≤ 37 So, clearly, the prime p where exactly half of all integers have something ≤ p as their smallest prime factor must lie between 31 and 37 :-) Alternatively, one could argue such a prime doesn’t exist. That, I think, is the better answer. That doesn’t imply there’s no good definition of that median, though because it’s 100% acceptable to have a set of numbers whose median is not an element of that set. https://en.wikipedia.org/wiki/Median#Finite_data_set_of_numbers https://en.wikipedia.org/wiki/Median#Finite_data_set_of_numb...: “If the data set has an even number of observations, there is no distinct middle value and the median is usually defined to be the arithmetic mean of the two middle values.[1][2] For example, this data set of 8 numbers 1, 2, 3, 4, 5, 6, 8, 9 has a median value of 4.5” So i guess the hunt still is on for a good argument as to why a non-prime such as 36 or even a real such as 36.716… should be called the median of the smallest prime factors of all integers (I wouldn’t know whether that exists)
- Obscurity4340 3y agoWhat is the association between good passwords and prime numbers?
- politelemon 3y agoThere shouldn't be a direct association, are you referring to something specific? Choosing a good password is about making it less predictable. Prime numbers do factor into cryptography though, they use prime numbers for key generation and key exchange.
- quickthrower2 3y agoBut don't use a prime number as your password ;-)
- Obscurity4340 3y agoDo passwords eventually break down into some simpler representation tho? Like doesn't some hash function convert it irreversibly into some other number?
- Obscurity4340 3y agoI'm more curious as to why prime numbers are so important to encryption, like what is their role or significance to the process
- spokeonawheel 3y agothere is no such thing as a good password. I have heard of good SSO's though
- Obscurity4340 3y agoWHO hurt you :[
- lifeisstillgood 3y agoforgive my lack of maths to support the intuition, but as we discover ever higher prime numbers the second factor will tend upwards - towards the correct answer of 42?
- NooneAtAll3 3y agoit's more that when you filter out 37-and-below, the "rest" category as a whole takes ~50% of the numbers
- olejorgenb 3y agoNo, this is not an empirical result. The interesting thing is that this is a fixed number (and that it's quite small). ("After all, about half of *all* integers have 2 as their smallest prime factor" is the beginning of the intuition needed I guess)
- mbfg 3y agonothing worse then when a good joke goes unnoticed.
- narinxas 3y agothe tears of clowns...
- narinxas 3y ago42 must mean both 41 and 43 taken together. these two primes are a twin prime above 37. and 29 and 31 both take together mean 30. the twin pair below 37
- deleted 3y ago[deleted]
- NooneAtAll3 3y ago> If we write p_k for the median k-th prime, then they show: log log p_k = ... is this natural log or some base? why not to use ln to keep ambiguity out?
- ravi-delia 3y agoIn analytic number theory we usually only care about growth rates in a very coarse sense- up to scaling by some constant, and asymptotically. Because the log of any base is precisely the same up to constants, it doesn't actually matter. If you look at the expression, to the right of the equals sign is a big O- that's the same big O as the one you might be familiar with in complexity. That being said, in math more generally when we need a concrete log the natural log is pretty much always the way to go- I haven't seen ln in a little while.
- danielam 3y ago> Because the log of any base is precisely the same up to constants i.e., \log_{b_1} n = \frac{1}{\log_{b_2} b_1} \log_{b_2} n where \frac{1}{\log_{b_2} b_1} is the constant fixed for a given choice of `b_1` and `b_2`, i.e., the bases.
- NooneAtAll3 3y agoexcept that in that formula _exact_ "k-b" is used - coarseness is hidden in O(1/sqrt(k)) that follows
- raghus 3y agoInterestingly, 37 also shows up in the Optimal Stopping / Secretary Problem.
- justanotherjoe 3y ago[dead]
- nathanfig 3y agoWas just about to comment this. Wonder if anyone more mathematically inclined can see a relationship.
- Someone 3y agoI don’t know that problem and don’t feel like googling it, but it’s a reasonable guess that’s because 1/e ≈ 0.36787944
- drhagen 3y agoThat seems unlikely. Those numbers only look related in base 10. 1/e is about a third of 1 and 37 is about a third of 100, which is why they look similar in base 10. In base 16, 1/e is about 0.5e2d58 and 37 is 25. And prime numbers are prime numbers regardless of base.
- gowld 3y agoThat's what PP was saying, in not so many words.
- FabHK 3y ago100/e is close to 37, regardless of base. (That 100 would be chosen as a convenient scaling factor for our notion of “percent” only in base 10 is true, but a separate point.)
- bondarchuk 3y agoNo, that is precisely the point.
- Egidius 3y agoReminds me of "move 37" made by Alpha Go: > Michael Redmond noted that AlphaGo's 19th stone (move 37) was "creative" and "unique". It was a move that no human would've ever made Lee took an unusually long time to respond to the move. An Younggil called AlphaGo's move 37 "a rare and intriguing shoulder hit" but said Lee's counter was "exquisite". He stated that control passed between the players several times before the endgame, and especially praised AlphaGo's moves 151, 157, and 159, calling them "brilliant". https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol
- deleted 3y ago[deleted]
- mbfg 3y agoisn't 37 the solution to the toilet problem, as well? i'm supposing the two problems are related.
- User3456335 3y agoIt's very unlikely that they are related. 1/e which is approximately 0.37 is the solution to the problem you're referring to but the occurrence of 37 depends on the choice of base. It just happens to be the case that in base 10 we find that round(1/e*10^2)=37 but it would be different in other bases. Meanwhile, the median value for the second prime is completely independent of the choice of base.
- fuzztester 3y agoSee: https://en.m.wikipedia.org/wiki/37_(number) https://en.m.wikipedia.org/wiki/37_(number) Also see: https://en.m.wikipedia.org/wiki/1 https://en.m.wikipedia.org/wiki/1 https://en.m.wikipedia.org/wiki/2 https://en.m.wikipedia.org/wiki/2 https://en.m.wikipedia.org/wiki/3 https://en.m.wikipedia.org/wiki/3 etc. Lots of interesting points in those, and not just about math
- deleted 3y ago[deleted]
- spokeonawheel 3y agoI find it very odd that wikipedia exists up to and including 11, but then does not exist at 12. Any reason for this?
- madcaptenor 3y agohttps://en.m.wikipedia.org/wiki/12 https://en.m.wikipedia.org/wiki/12 https://en.m.wikipedia.org/wiki/13 https://en.m.wikipedia.org/wiki/13 https://en.m.wikipedia.org/wiki/14 https://en.m.wikipedia.org/wiki/14 what do you mean?
- fuzztester 3y agoHaven't checked, but a simple reason may be that no one created the pages for some other numbers. After all, it is a volunteer project.
- nikhilsimha 3y agoThis is one of the best articles I have read in a long long time!
- dataflow 3y agoThis doesn't mean there's anything interesting about 37. Rather, the only interesting fact here is that a finite median here exists at all. Once you've established that, it's guaranteed to be some prime number, because we're defining the median of a list to be an element of the list. It just happens to be 37 for this list, but it may as well have been anything else. What could make 37 interesting is if we relaxed the definition of the median to be outside the set itself (which is entirely possible), and yet the limit still, and it still converged to 37 somehow. That would be wild.
- 2-718-281-828 3y agois it trivial that the series of medians for the second prime factor of numbers up to N is a specific number? isn't it just as plausible that this median would increase?
- drexlspivey 3y agoThe median of the first prime factor is 2
- codetrotter 3y agoBut intuitively it’s weird that the median of the second prime factor is not 3, right?
- drexlspivey 3y agoOnly multiples of 6 (1 in 6 numbers) have 3 as their second prime factor so I wouldn’t expect the median to be 3
- deleted 3y ago[deleted]
- eru 3y agoI guess it depends on whether you count multiples of 4 to have 2 as both their second and first prime factor?
- lvass 3y ago>0.000000000000000 of numbers have 2 as their second prime Is the article title correct without explicitly mentioning "non-repeating"?
- p0w3n3d 3y agoAlef 0 numbers have 2 as their second prime
- sophiebits 3y agoThe factors of 12 are 1, 2, 3, 4, 6, 12. The prime factors of 12 are 2, 3.
- kccqzy 3y agoGood nitpick. If you allow repeating primes, then the numbers with 2 as their second prime would be multiples of 4. So a quarter of numbers.
- tgv 3y agoAnd the number of 2s (and 3s) would grow hard, which not favor a finite median.
- mauricioc 3y agoAn important part of reading mathematics is mentally filling in the "correct" interpretation for terms. The optimal amount of detail typically depends on the target audience. Part of that is because making notation fully specified in English is typically cumbersome: If they had written "as their second non-repeating prime", someone could come along saying "shouldn't it say 'second-smallest' or 'in increasing order' as well?". In papers, the use of such terms would typically be accompanied by a formal definition using more precise (but not fully precise, see the talk below) mathematical notation. I mention this not to be pedantic, but because inferring the "invisible" part of mathematics (when it can be done) instead of asking humans to do it explicitly tends to be a big usability improvement in theorem provers. Andrej Bauer has a nice talk about it at https://www.youtube.com/watch?v=wZSvuCJBaFU https://www.youtube.com/watch?v=wZSvuCJBaFU
- joewferrara 3y agoAmazing how simple the proof is. 37 is my new favorite prime number, haha.
- falseprofit 3y ago2.5, the median value for the first prime factor of an integer
- owl57 3y agoProbably not. We are talking about some function of first N numbers and its limit when N approaches infinity. There is more than one reasonable way to clarify "first prime factor" and "first N numbers", but I think for all of these reasonable definitions the median diverges, having different subsequential limits for even and odd values of N.
- falseprofit 3y agoI'm taking the median of the limiting distribution, and I agree that the sequence of medians you describe does not converge.
- gowld 3y agoHow so? 1 has no first prime factor so must be excluded along with 0. 2 is always >=50% of the head sequences of the rest: 2 3 2 5 2 7 ...
- zmgsabst 3y agoCoincidentally, 37 is also the first irregular prime. They’re why Fermat’s last theorem is hard. https://en.wikipedia.org/wiki/Regular_prime https://en.wikipedia.org/wiki/Regular_prime
- magneticnorth 3y agoLove it when an article so clearly explains the answer to my first burning question - in this case, "How in the world do you prove that?"
- XorNot 3y agoWell, this is just in time for a few birthday card ideas...
- spokeonawheel 3y agogiven this knowledge, does this have implications that RSA would twice as crackable? Given casting a wide net, you could assume one of the prime factors is 37 and just try it against the pub key
- free_bip 3y agoGiven that RSA primes are usually checked against some probabilistic primality test like Miller-Rabin, I would think the answer is no
- abbaselmas 3y agoSheldon Cooper, the best number 73 (21st prime number), its mirror 37 (12th prime number) https://www.youtube.com/watch?v=HacqfsV7ug0 https://www.youtube.com/watch?v=HacqfsV7ug0
- abbaselmas 3y agoAlso deep dive in the 73 and 37 https://www.youtube.com/watch?v=4DQndnAhdxk https://www.youtube.com/watch?v=4DQndnAhdxk
- hoseja 3y agoI really dislike base-10 tricks. They are just cute, rarely have any deeper significance.
- mcv 3y agoI once heard of a (tongue-in-cheek) list of interesting facts about numbers, and I think it listed 37 as the first uninteresting number, but I guess that was wrong.
- hugodan 3y agoThat’s the age that Jesus Christ died
- cubefox 3y agoBut what about the arithmetic, geometric or harmonic mean?
- deleted 3y ago[deleted]
- lubujackson 3y agoThis is slightly interesting but I wonder what the median 3rd prime is, and 4th etc. Is there some sort of interesting pattern?
- madcaptenor 3y agoThey grow very quickly. The median third and fourth primes are 42719 and 5737850066077 - see https://oeis.org/A284411 https://oeis.org/A284411 . If p_k^* is the median kth prime then we have log log p_k^* = k - C + O(1/sqrt(k)) where C is a constant around 0.59483; that is, p_k^* grows like exp(exp(k)). This is "Corollaire 1.2" of the paper that the post links to (https://tenenb.perso.math.cnrs.fr/PPP/p_k%28n%29.pdf https://tenenb.perso.math.cnrs.fr/PPP/p_k%28n%29.pdf). (It's in French.)
- floxy 3y agoI wonder what the growth rate of the mean of the second prime factor is? I guess I'd expect it to grow without bound, but maybe pretty slowly?
- thrusong 3y agoI've been obsessed with this number for most of my life. It's like I see it on every clock I look at, or on every license plate in front of me. There are random things about it like channel 37 has significance: https://en.wikipedia.org/wiki/Channel_37 https://en.wikipedia.org/wiki/Channel_37
- nuc1e0n 3y ago37 is one of the "scary" large numbers on countdown. The others being 12, 62 and 87.
- drudru 3y agoNo mention of 37 signals? Maybe the aliens are trying to test our math skills?
- rlupi 3y agoIs there any interesting theorem or result from l-functions or modular forms that covers this result? I am learning about them and I find the topic fascinating. I discovered them watching https://www.peakmath.org/quest-for-f1 https://www.peakmath.org/quest-for-f1 videos on youtube. Also worth exploring: http://lmfdb.org http://lmfdb.org