[HN Gopher] 37, the median value for the second prime factor of ...
___________________________________________________________________
37, the median value for the second prime factor of an integer
Author : sacrosanct
Score : 151 points
Date : 2023-11-12 18:45 UTC (4 hours ago)
(HTM) web link (grossack.site)
(TXT) w3m dump (grossack.site)
| zelda-mazzy wrote:
| It 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 wrote:
| I don't think that's right. My reading of it was that 37 is the
| median prime among all primes.
| gattilorenz wrote:
| I 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 wrote:
| Original 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 wrote:
| I think that was an intuitive and imprecise way of
| expressing "median value", so that 50% of primes are "up
| to 37"
| NooneAtAll3 wrote:
| did the title change or smth?
| Dylan16807 wrote:
| > 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".
| mg wrote:
| I 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 wrote:
| https://en.wikipedia.org/wiki/Interesting_number_paradox
| mg wrote:
| Well, that is about the question which is the most
| uninteresting number. And the paradox that - whichever number
| it is - this attribute makes it interesting.
|
| But since we are not looking for the most uninteresting
| number, but the most _interesting_ one, we do not have to
| fight with issues of that calibre.
| mikepurvis wrote:
| I 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 wrote:
| Then let's see who manages to write the least interesting
| comment in this thread.
| TeMPOraL wrote:
| Like the Original Sin entered the Garden of Eden through the
| Serpent, _mutable state_ breaks the time-invariant perfection
| of mathematics through the idea of interestingness, since it
| 's clear the least interesting things only remain so _until
| someone notices_.
| swayvil wrote:
| I nominate 120 as "most interesting integer". For obvious
| reasons.
| AnimalMuppet wrote:
| Um, it's not obvious to me. Could you specify?
| InfamousRece wrote:
| The smallest triply perfect number?
| https://oeis.org/A005820
| swayvil wrote:
| So many factors in such a small package, for one.
| jl6 wrote:
| Mathematicians are locked in bitter struggle between the
| Zeroastrians who believe additive identity blesses 0 as truly
| the most foundational and therefore interesting integer, while
| the Unitarians favor multiplicative identity and thus champion
| 1. Uncountable souls have been lost to the conflict between
| these two groups.
| joenot443 wrote:
| Ugh, must we relate every thread about integer fascination
| back to Zerry/Unny zealotry? It's been going on for centuries
| now, nobody's changing their opinion at this point.
| narinxas wrote:
| because numbers have a natural meaning. roughly speaking
| each integer has as many meanings as its value. so 0 and 1
| are just the very beginning.
|
| zero's natural meaning is special, it's the void. sometimes
| the 'variable' stands in as having this meaning, sometimes
| the variable is zero.
|
| one is the easiest simplest to grasp. its meaning is "I",
| ego.
|
| two is duality in the broadest imaginable sense. wars are
| being fought right now over the precise meaning of 3
| ajhurliman wrote:
| This feels closer to a tarot reading than mathematics,
| and I'm here for it
| _0ffh wrote:
| The Tao Produced One;
|
| The One Produced Two;
|
| The Two Produced Three
|
| And The Three Produced All Things.
| Shorel wrote:
| Sounds very close to the firstness, secondness and
| thirdness categories from Charles Sanders Peirce.
|
| (Which I think are nothing but nonsense from a madman =)
| narinxas wrote:
| we can be more precise now:
|
| the three produced the equation, which pretended that all
| things were (could be) equalized into equivalence. they
| called this algebra and parised some monotheistic god;
| they called this magic and GOTO one.
| chaboud wrote:
| I, for one, think zero good will come of it...
|
| Two soon?
| hurryer wrote:
| In modern math all numbers are built on set theory which has
| the empty set as it's foundation.
|
| Empty set is more like 0 than 1. Check mate Unitarians.
| layer8 wrote:
| The empty set is 1 set. Can't have 0 without 1.
| fuzztester wrote:
| The empty set is the set with nothing, i e. _no thing_.
| So it doesn 't need 1.
|
| ;)
| layer8 wrote:
| Wouldn't the set _without_ nothing be more fundamental?
| automatic6131 wrote:
| Does the set of sets that exclude themselves exclude
| itself?
| coldtea wrote:
| You start from 1 empty set though.
|
| If you could start from 0 empty sets, you'd have a point.
| User3456335 wrote:
| 0 and 1 are interesting as integers in the same way that a
| blank canvas is interesting as a painting. Very foundational,
| very useful but also very plain.
| mnd999 wrote:
| Pretty interesting if you work in an industry where you use
| them to represent every other number.
| mikea1 wrote:
| In my limited math experience, I really found an
| appreciation of the number one. My favorite part of high
| school algebra was realizing that "solving for x" was often
| just a repeated exercise of multiplying each side of the
| equation by a "clever form of one."
| User3456335 wrote:
| Solving for x often entails multiplying both sides by the
| same number, not necessarily one. Perhaps you're
| referring to simplifying fractions where you do often
| multiply by a clever form of one?
| narinxas wrote:
| not in binary notation they're not
| enriquto wrote:
| We Pragmaticists (some would say "abject pragmaticists")
| advocate for a compromise solution between Zeroastrians and
| Unitarians by taking h=1/2 as the fundamental constant. It
| satisfies several interesting identities involving other
| constants, like i^i=e^(-hp), [?]e^(x^(1/h))dx=p^h,
| [?]n^(-1/h)=hp^(1/h)/3, the celebrated inequality
| (a+b)h>=(ab)^h or the curious fact that 1/h is the smallest
| prime number.
| geraldhh wrote:
| Pragmaticists brought a fraction to an integer competition
| and lost
| narinxas wrote:
| they lost the competition, but in the end it was them who
| paid the prize
| sorokod wrote:
| 0 uncountable?
| anschwa wrote:
| One of my professors would often refer to this struggle as
| taking place between the Nihilists and Unitarians.
| fuzztester wrote:
| You missed the Dualitarians, the Trialitarians, and so on
| through to the Infinitarians ... whoops, forgot Cantor!
|
| https://en.m.wikipedia.org/wiki/Georg_Cantor
| drsopp wrote:
| My favorite is 2: Shared joy is a double joy, and shared
| sorrow is half sorrow.
| paulddraper wrote:
| 12
|
| Smallest abundant number
| resource0x wrote:
| Prime numbers are not especially interesting. E.g. all groups
| of prime size are cyclic. But each composite number has its own
| "character". The "interesting" number among primes is 2. Almost
| every theorem about prime numbers has to consider 2 as a
| special case.
| joe_the_user wrote:
| Much as I appreciate the reference, I think the fact this limit
| exists at all and is an integer is more interesting than what
| the integer is.
|
| A lot of paradoxes and confused ideas begin with "choose an
| integer at random" - what is the average value of number chosen
| between 0 and infinity, for example.
| addaon wrote:
| > and is an integer
|
| How could a median prime factor not be an integer?
| nextaccountic wrote:
| The median value of [1, 2, 3, 4] is 2.5
| layer8 wrote:
| > is there an integer which qualifies to be the most
| interesting?
|
| I'd say that 664571016291591957042161991109590159107314773607
| and 590488317782859198927092718316232684864014739572 qualify,
| which are the big- and little-endian versions of ASCII "the
| most interesting", respectively.
| hurryer wrote:
| 17 is well known as the most random number between 1 and 20.
| User3456335 wrote:
| One other special property prime numbers can have is being
| irregular. And guess what the first irregular prime number is?
| 37. Interesting that it appears so late because irregular prime
| numbers do make up around 41% of all primes. See: https://encyc
| lopediaofmath.org/wiki/Irregular_prime_number#:....
|
| So the most interesting prime number really depends on what you
| consider more interesting. If you prefer the median second
| prime factor, then 37 is the best, but if you prefer the first
| irregular prime number then 37 is also the best. So it all
| depends on your perspective.
|
| Another reason to like 37 is that it ends in a 7 so that it
| "sounds random" when someone asks for a number and it is better
| than 27 because it is prime. 7 is too low and 17 has
| connotations with bad luck. Although 37 is quite scary too:
| it's not just prime, which is pretty irregular, but it's also
| an irregular prime.
| DevX101 wrote:
| I nominate 60. Babylonian mathematics was sexagesimal (base 60)
| which is a superior mathematical system to base 10. The number
| 60, a superior highly composite number, has twelve factors: 1,
| 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5
| are prime numbers. With so many factors, many common
| mathematical operations are much simpler than in base 10. Base
| 10 has no special mathematical properties apart from as an
| accident of evolution, we primates ended up with 10 fingers.
|
| We still have some vestiges of the babylonian system however,
| 60 minutes in an hour. 360 (6 sixties) degrees in a circle.
| mikewarot wrote:
| Since small prime factor are so useful, why not use 2 3 times
| and 3 2 times along with 5 2*2*2 * 3*3 * 5 =
| 360
|
| 360 is a very handy number, it has a high degree of
| usefulness, when you're circumferential in your calculations.
| aquafox wrote:
| Historically, I'd say 60, because relative to it's size, it has
| many divisors (12), and among them a lot of useful ones (2, 3,
| 4, 5, 10), which is why our time system and trigonometry are
| probably based on it (360 = 6*60; 360 has 24 divisors).
| jvalencia wrote:
| I love the number 27 3^3 2+7 = 3+3+3
| layer8 wrote:
| 37 is also the smallest base (or second int argument) not
| supported by _itoa()_. Surely not a coincidence. ;)
| mort96 wrote:
| The most interesting integer out there must certainly be the
| least interesting integer!
| yieldcrv wrote:
| why 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 wrote:
| An unknown thing becomes a non obvious thing, which makes it
| interesting. Until it becomes obvious, of course.
| NooneAtAll3 wrote:
| there 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 wrote:
| I mean this is a crazy thing to know about 37. And we know it!
| SonOfLilit wrote:
| The 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 wrote:
| I thought it'd be infinite or something. So it's cool to know
| it exists and it's not 1, 2 or infinity.
| Someone wrote:
| I 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.wikipedi
| a.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 wrote:
| What is the association between good passwords and prime numbers?
| politelemon wrote:
| There 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 wrote:
| But don't use a prime number as your password ;-)
| lifeisstillgood wrote:
| forgive 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 wrote:
| it's more that when you filter out 37-and-below, the "rest"
| category as a whole takes ~50% of the numbers
| olejorgenb wrote:
| No, 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 wrote:
| nothing worse then when a good joke goes unnoticed.
| narinxas wrote:
| the tears of clowns...
| narinxas wrote:
| 42 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
| NooneAtAll3 wrote:
| > 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 wrote:
| In 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 wrote:
| > 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.
| raghus wrote:
| Interestingly, 37 also shows up in the Optimal Stopping /
| Secretary Problem.
| nathanfig wrote:
| Was just about to comment this. Wonder if anyone more
| mathematically inclined can see a relationship.
| Someone wrote:
| I 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
| Egidius wrote:
| Reminds 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
| mbfg wrote:
| isn't 37 the solution to the toilet problem, as well? i'm
| supposing the two problems are related.
| User3456335 wrote:
| It'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 wrote:
| See:
|
| https://en.m.wikipedia.org/wiki/37_(number)
|
| Also see:
|
| https://en.m.wikipedia.org/wiki/1
|
| https://en.m.wikipedia.org/wiki/2
|
| https://en.m.wikipedia.org/wiki/3
|
| etc.
|
| Lots of interesting points in those, and not just about math
| nikhilsimha wrote:
| This is one of the best articles I have read in a long long time!
| dataflow wrote:
| This 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.
___________________________________________________________________
(page generated 2023-11-12 23:00 UTC)