[HN Gopher] 37, the median value for the second prime factor of ...
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37, the median value for the second prime factor of an integer
Author : sacrosanct
Score : 438 points
Date : 2023-11-12 18:45 UTC (1 days 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"
| zelda-mazzy wrote:
| Im 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 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".
| zelda-mazzy wrote:
| I 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 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.
| function_seven wrote:
| True, but you must still contend with your observational
| influence. Whatever number you crown as "most interesting"
| will become _even more_ interesting for wearing that crown!
|
| So you must take this duty with extreme care and
| thoughtfulness, for when you anoint such a number, your
| action may crystalize it as such for all eternity. Other
| numbers will have to go further to supplant it.
| 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.
| falcor84 wrote:
| The way I understood it, this formulation relies on
| induction on the natural numbers.
| SamBam wrote:
| Because 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?
| 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.
| goodmachine wrote:
| 120 - or twelfty to friends - is a delight.
|
| "The long hundred, also known as the great hundred or
| twelfty,[is the number 120 (in base-10 Hindu-Arabic
| numerals) that was referred to as hund, hund-teontig,
| hundrad, hundrath, or hundred in Germanic languages prior
| to the 15th century, and is now known as one hundred
| twenty, or six score."
|
| https://en.wikipedia.org/wiki/Long_hundred
|
| "120 is highly composite, superior highly composite,
| superabundant,[3] and colossally abundant. With sixteen
| divisors, 120 is the smallest number to have that many
| divisors..."
|
| https://en.wikipedia.org/wiki/120_(number)
| 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 =)
| User23 wrote:
| He was an awfully pragmatic madman. He also discovered
| the universal and existential quantifiers independently
| of Frege, albeit a few years later.
|
| No offense intended, but to the extent his work feels mad
| to you, it might be a "you problem." That said, he never
| did reduce firstness, secondness, and thirdness to unity
| to his own satisfaction either. His work is truly
| tantalizing.
|
| Oh and abduction has had some conceptual influence too.
|
| Sad that the Arisbe fansite got deleted.
| 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.
| smegsicle wrote:
| thats funny i always heard it was an origin with two
| other orthogonal roles
|
| unless youre a dirty filioquist
| mjrpes wrote:
| Perhaps 0 and 1 are the same. If there only exists the
| entity called 1, lacking all properties or relations to
| other entities, what is left? A void called 1, no
| different than 0. The true mystery is: how did plurality
| arise?
| eru wrote:
| > Perhaps 0 and 1 are the same.
|
| Compare
| https://en.wikipedia.org/wiki/Field_with_one_element
| User23 wrote:
| Consider for a moment that Aristotle, and thus most
| likely all the ancients, made a distinction between
| nothing and void.
| chaboud wrote:
| I, for one, think zero good will come of it...
|
| Two soon?
| eru wrote:
| > It's been going on for centuries now, nobody's changing
| their opinion at this point.
|
| Sure, but one side can still out-breed the other.
| 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?
| fuzztester wrote:
| That's a well-known old paradox, or close. Good on you,
| cobber. (AU slang :)
| fuzztester wrote:
| See Russells's paradox:
|
| https://en.m.wikipedia.org/wiki/Russell%27s_paradox
| Akronymus wrote:
| No, because it isn't a set, but rather a class.
| OJFord wrote:
| A concept created at least partially to resolve this
| paradox.
| fuzztester wrote:
| Brilliant. Go to the top of the class, er, set. Oops,
| forgot, sets have no order. List, then.
| fuzztester wrote:
| Ordered list, I mean.
| coldtea wrote:
| You start from 1 empty set though.
|
| If you could start from 0 empty sets, you'd have a point.
| fuzztester wrote:
| >You start from 1 empty set though.
|
| Sez who, zigactly? If you are coldtea, I am coldertea. As
| in, I love my tea colder than you do yours. And starts
| who? Not me.
|
| Don't kid around, dude. I dik around.
|
| >If you could start from 0 empty sets, you'd have a
| point.
|
| What if the set of empty sets has no element? :)
| coldtea wrote:
| Was the first comment under the influence of drugs?
|
| In any case, for the second comment: even if the set of
| empty sets has no elements, it's still ONE set.
| fuzztester wrote:
| >Was the first comment under the influence of drugs?
|
| No, it was not. Regarding this whole subthread in which I
| participated, I did it for some mutual logic-chopping
| fun.
|
| But if you thought there was a chance it was, you
| shouldn't have replied to the second comment, because I
| would have been equally under the influence of the drugs
| for the second, if I had been so for the first comment.
| (Don't reply to a druggie.) After all, they were both
| done in one atomic comment transaction.
| fuzztester wrote:
| If I start from 0 empty sets, how do I have _A_ point?
| phkahler wrote:
| >> The empty set is the set with nothing
|
| So it _is_ A set. In fact it 's "the set" that makes it a
| rather special _one_.
| eru wrote:
| No. Set theory is just one possible way you can build up
| numbers and mathematics.
|
| You can just as well build up numbers and the rest of
| mathematics from different foundations. They are all
| equivalent.
|
| Just like running your programs on ARM or on x86 or a
| toaster is all the same, thanks to emulators. (That's
| assuming you have an arbitrary amount of memory, and don't
| mind differences in runtime length.)
| bobbylarrybobby wrote:
| Which is more foundational: the wheel or the axle? Sets
| aren't worth much if you can't take unions, which is where
| 1 comes from.
| 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?
| mikea1 wrote:
| Ah, yes, you are absolutely correct. The _clever form of
| one_ trick did not apply as often as I recalled!
| 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
| plank wrote:
| Ah, but they won the Pi versus Tau competition ;-)
| samcheng wrote:
| Halfasstrians?
| 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.
| awwaiid wrote:
| My spouse often claims that two is the best prime because
| she knows it annoys me. Or perhaps she is truly insane.
| Clearly 2 is the WORST prime, amirite?
| eru wrote:
| Well, it's definitely the primest prime.
| SamBam wrote:
| I've always felt 2 was the most important as well, if only
| because for there to be something (1) there has to be
| nothing (0), and thus the existence of anything defines 2
| states, being and non-being. Without the pair (0, 1) there
| is no distinguishing anything in the universe.
| fallingknife wrote:
| Must be an integer number of souls, though, so even in a
| conflict of infinite time, it won't be uncountable.
| tasuki wrote:
| Surely addition is more foundational than multiplication?
| Multiplication is just repeated addition.
| ak_111 wrote:
| Is this a reference for some fiction? Or you just made it up?
| paulddraper wrote:
| 12
|
| Smallest abundant number
| kmoser wrote:
| 12 is also the largest one-syllable number (at least in
| English).
| 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.
| 8bitsrule wrote:
| Except that it's interesting that prime numbers have led
| _inordinate_ numbers of individuals to devote _uncountable_
| but significants chunks of their lives investigating them.
|
| It's not their fault that they're leftovers when an
| artificial grid is created by crossing-out every multiple of
| 2, 3, 5, or other prime numbers. Yet for some reason this
| seems to give them a fascinating aura for many. Maybe naming
| them 'prime' instead of 'leftover' is to blame.
| 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
| addaon wrote:
| Sure, but 2 is the only even prime, so this is super
| constrained.
| andrewflnr wrote:
| That's the mean.
| saagarjha wrote:
| And it's also the median.
| necovek wrote:
| Depends if you are looking at a mean closed to integers
| or not: one can define a mean without rational or real
| numbers.
| CrazyStat wrote:
| But the median of [1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4]
| is 2. As the individuals observed values collect more and
| more repetitions--as they do when making a list of the
| second prime factor, for example--it becomes increasingly
| unlikely that the median will be between two values, not
| equal to one of the values.
|
| It would be surprising if the median second prime factor
| was not a prime.
| tedunangst wrote:
| But it's not an integer before rounding. 37 is simply the
| nearest number to get approximately 50%.
| 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.
| emmelaich wrote:
| You're perilously close to being a eikositriophile. We try
| not to talk about such things openly fnord.
|
| https://en.wikipedia.org/wiki/23_enigma
| gowld wrote:
| How is that related?
| jkhdigital wrote:
| It is another accident of evolution that we ended up with 12
| finger joints, allowing one to count to twelve with a single
| hand by tapping the thumb on each finger segment in turn.
| necovek wrote:
| Most of your argument seems to stand for 30 instead of 60: 2
| x 3 x 5 = 30, which is a smaller number with mostly the same
| properties as 60.
|
| If you want more divisibility (into quarters, eigths,
| ninths...), you can continue multiplying further, but 60 is
| as much a "historical accident" as 10 is an "accident of
| evolution".
| jameshart wrote:
| 60 is a 'highly composite' number
| (https://en.wikipedia.org/wiki/Highly_composite_number) -
| it is the lowest number with 12 divisors.
|
| 30 has divisors 1, 2, 3, 5, 6, 10, 15, 30 - eight divisors.
| But 24 is lower and also has eight (1, 2, 3, 4, 6, 8, 12,
| 24), making 30 less 'special'.
|
| 60 is the lowest highly composite number with 5 (and
| therefore 10) as a factor, which makes it a strong
| attractor when choosing bases for measuring systems.
| necovek wrote:
| Sure, but caring about 5 (and 10) is not what the GP
| comment stressed (actually, quite the opposite). If we do
| care about 10, then 60 is indeed appealing, but also an
| argument for base 10 too -- which they argued against.
|
| "Highly composite" number being used as a base for a
| numbering system does not really strike me as non-
| arbitrary either.
|
| Choosing 60 and saying it makes more sense than, say,
| 2520 (which is also highly composite, but divisible by
| 1-10 too) is pretty arbitrary imho. Yes, my "criteria" is
| similarly arbitrary but focused on "10 does matter" ("I
| want to easily divide this into 2-10 groups"): both are
| too large to have individual names for each digit in a
| supposed number system with base 60 or 2520.
| 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
| joseluis wrote:
| Very pretty. And to relate it somewhat to the topic: 1/27 =
| 0.037 repeating 037 and 1/37 = 0.027 repeating 027 :)
| layer8 wrote:
| 37 is also the smallest base (or second int argument) not
| supported by _itoa()_. Surely not a coincidence. ;)
| rbonvall wrote:
| It's because the characters in the result can be one of 10
| digits or one of 26 letters.
| mort96 wrote:
| The most interesting integer out there must certainly be the
| least interesting integer!
| joewferrara wrote:
| Every prime is odd and 2 is the oddest of them all!
| klyrs wrote:
| Two is the oddest prime.
| doubloon wrote:
| what is interesting to me is that we call imaginary integer
| numbers "Gaussian Integers", but we call the quaternion integer
| numbers "Hurwitz Quaternions" instead of "Hurwitz Integers".
| bandrami wrote:
| I love that there could never be a least interesting natural
| number, because that fact would be interesting. In fact there
| can be no uninteresting natural numbers, because then you could
| list them, and the lowest one would be interesting for that
| fact.
| 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)
| LegionMammal978 wrote:
| As the author says, this isn't the median of any _particular_
| data set, but the limit of the median as the number of points
| goes to infinity:
|
| > "Obvoiusly" there's no uniform distribution on the natural
| numbers, so what does it even mean to choose a "random" one?
| The way the number theorists usually solve this problem is by
| fixing a large number _N_ and looking at the probabilities
| when you pick a random number between 1 and _N_. Then you
| look at the _N_ - [?] limit of these probabilities.
|
| And by using the l2( _p_ ) function, we can find the
| asymptotic proportion that each prime _p_ appears in the data
| set: if the set has _N_ items, then approximately l2( _p_ )*
| _N_ of them will be equal to _p_ , and the actual number will
| asymptotically approach this approximation. Given this, we
| can look at the set in terms of the proportions of its
| elements. We have first, P( _p_ < 37) = l2(2) + ... + l2(31)
| [?] 0.49061; second, P( _p_ = 37) = l2(37) [?] 0.00964; and
| third, P( _p_ > 37) = 1 - P( _p_ < 37) - P( _p_ = 37) [?]
| 0.49975.
|
| Thus, if we sort the set with _N_ data points in ascending
| order, we 'll ultimately see 0.00964* _N_ instances of 37,
| flanked to the left by 0.49061* _N_ values less than 37, and
| to the right by 0.49975* _N_ values greater than 37. For _N_
| odd, the median is defined as the value at the 50% mark in
| the sorted set, which is clearly 37. For _N_ even, the median
| is defined as the mean of the two values surrounding the 50%
| mark; since 0.00964* _N_ - [?] as _N_ - [?], there will
| eventually be at least two instances of 37 around the 50%
| mark, so the median will once again be 37. Thus, for _N_ even
| and odd, the median will always end up at 37.
| ordu wrote:
| Very interesting question. I thought a lot why some math facts
| seem boring to me while others are interesting, sometimes
| exciting.
|
| I believe interesting math facts are interesting because they
| clash with intuition. Like you expect some freedom for a second
| prime factor to be anything it likes, but no, it has a very
| small median.
|
| I misunderstood the claim at first, counted primes in reverse
| order and was really surprised by an idea that second largest
| prime factor is so small and started to think of Eratosthenes
| but continued to read. When I discovered my mistake and then it
| came to Eratosthenes it got almost obvious and not interesting
| really, I didn't even read it when it got techical. Though if
| not my disappointment I'd probably found it interesting,
| because while I believe I could find 37 as a median and prove
| it without any hints, I would need a nudge: I needed to be told
| that median of a second prime factor is a small number.
|
| I believe that the practice of noticing these confusions and
| resolving it is a basis of a mathematician's mind. As a
| mathematician you need a highly tuned intuition that matches
| real theorems, because you start with intuition, formalize it
| as a theorem and then you look for a proof.
|
| Intuition guesses theorems to prove or to look in literature.
| But if my intuition can be surprised by some fact it means it
| cannot guess it. And probably it cannot guess a lot of other
| related facts.
| spokeonawheel wrote:
| you assume practicality in higher mathematics? thats like
| ancient Babylonians complaining about the impracticality of
| "calculus"
| 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 ;-)
| spokeonawheel wrote:
| there is no such thing as a good password. I have heard of good
| SSO's though
| 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
| drhagen wrote:
| That 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 wrote:
| That's what PP was saying, in not so many words.
| FabHK wrote:
| 100/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 wrote:
| No, that is precisely the point.
| Biganon wrote:
| And thanks to 1/e being 0.367, in Pokemon Go when I catch
| 1000 Pokemon and each of them has a 1 in 1000 chance of being
| shiny, I have a 36.7% chance of having zero shinies among
| them, or approximately a 63% chance of the reciprocate, AKA
| having _at least_ 1 shiny among them.
|
| 1000 is arbitrary of course, but the bigger the number the
| closer to 1/e.
|
| See "Bernoulli trials"
| skykooler wrote:
| It's also the most common number picked when people are asked
| to think of a random number between 1 and 100.
| 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
| spokeonawheel wrote:
| I find it very odd that wikipedia exists up to and including
| 11, but then does not exist at 12. Any reason for this?
| madcaptenor wrote:
| https://en.m.wikipedia.org/wiki/12
|
| https://en.m.wikipedia.org/wiki/13
|
| https://en.m.wikipedia.org/wiki/14
|
| what do you mean?
| 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.
| 2-718-281-828 wrote:
| is 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 wrote:
| The median of the first prime factor is 2
| cmiller1 wrote:
| The sequence of the medians of successive prime factors
| would be interesting
| codetrotter wrote:
| But intuitively it's weird that the median of the second
| prime factor is not 3, right?
| drexlspivey wrote:
| Only multiples of 6 (1 in 6 numbers) have 3 as their
| second prime factor so I wouldn't expect the median to be
| 3
| eru wrote:
| I guess it depends on whether you count multiples of 4 to
| have 2 as both their second and first prime factor?
| NikkiA wrote:
| I'd expect the mode of the second prime factor to be 3,
| but not necessarily the median.
| LanceH wrote:
| > isn't it just as plausible that this median would increase?
|
| Every sixth number will have a factors 2 and 3. (so 3 is the
| second prime for 16.666% of numbers)
|
| Every 10th number will have factors 2 and 5.
|
| Every 15th number will have factors 3 and 5.
|
| Of course then you have to subtract out every 30th number
| which contains a 2, 3 and 5.
|
| They ignored duplicates and it would have been nice if they
| had defined "first" and "second" properly. Also note that a
| prime counts the 2nd factor as "infinite", adding one to the
| "above 37" bucket.
|
| Anyhow, you add those up and it's easy to at least see that
| in any range of numbers it very quickly approaches 50%. Keep
| going to write out the Venn diagram and it happens to come
| out at 37. It's low enough, I assume it could even be done by
| hand.
| LegionMammal978 wrote:
| As we take the limit as _N_ - [?], the list continually grows
| larger. Thus, the number of instances of 37 continually
| increases, so that they occupy an asymptotically constant
| proportion (about 0.963%) of the values in the list; only
| ~49.061% of the values are less than 37, and only ~49.975% of
| the values are greater than 37. After a certain point, for even
| _N_ , there will always be two instances of 37 surrounding the
| 50% mark in the list, so the median will still be exactly 37,
| and not some other value. I wrote up a longer explanation in
| another comment [0].
|
| [0] https://news.ycombinator.com/item?id=38245162
| tromp wrote:
| Out of the integers 1..2N+1, exactly N have smallest prime
| factor 2,
|
| while the remaining N+1 have smallest prime factor > 2.
|
| Thus it is fair to say that the median smallest prime factor
| is 3 rather than 2.
| tromp wrote:
| Oops, that's not correct, since 1 has no prime factor.
|
| We should instead look at the odd-sized range 2..2N, in
| which N elements have smallest prime factor 2, making 2 the
| unique median.
|
| Only in even sized ranges 2..2N+1 is the median shared
| between 2 and 3.
| sriku wrote:
| I'm curious about the first number to have absolutely nothing
| interesting to be said about it ;)
| xigoi wrote:
| https://en.wikipedia.org/wiki/Interesting_number_paradox
| vouaobrasil wrote:
| In short, it's not really a paradox because in the original
| joke, the assumption is that the first number to have nothing
| interesting about it is itself an interesting property. That
| assumption isn't true when you actually think about it...it's
| actually rather boring.
| antonvs wrote:
| > It just happens to be 37 for this list
|
| What do you mean by "this list" here? It _seems_ to be
| misunderstanding the result.
| jameshart wrote:
| Well sure - the median is just one particular percentile - not
| a magically special one. The existence of a median also implies
| that there are probably limits for every other percentile - or,
| indeed, that for any number, there is some in-the-limit
| proportion of numbers for which the second prime factor is
| greater than that number (and for 37 that happens to be .5).
| madcaptenor wrote:
| Yes, and in small cases you can work out the distribution
| explicitly. For example: - 1/6 of all have their second
| smallest prime factor equal to 3 (that is, they have a factor
| of 2 and a factor of 3) - 1/5 of integers have a factor of 5;
| of those, 1/2 have a factor of 2 or a factor of 3, but not
| both (that is, they're congruent to 2, 3, or 4 mod 6); so
| 1/10 of integers have second smallest prime factor equal to
| 5.
|
| So the proportion of integers with second smallest prime
| factor equal to 5 is 1/6 + 1/10 = 8/30. Thus the first
| quartile of second smallest primes is 5, and similarly for
| any quantile between 1/6 and 8/30.
|
| Don't ask me to work out the third quartile - this
| distribution has a really long tail. In particular I suspect
| the limiting distribution has no mean.
| lvass wrote:
| >0.000000000000000 of numbers have 2 as their second prime
|
| Is the article title correct without explicitly mentioning "non-
| repeating"?
| p0w3n3d wrote:
| Alef 0 numbers have 2 as their second prime
| sophiebits wrote:
| The factors of 12 are 1, 2, 3, 4, 6, 12.
|
| The prime factors of 12 are 2, 3.
| kccqzy wrote:
| Good 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 wrote:
| And the number of 2s (and 3s) would grow hard, which not
| favor a finite median.
| mauricioc wrote:
| An 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
| lvass wrote:
| Sure. I'm wondering what people would respond to blanket
| asking what is the second prime factor of 12, because I would
| say 2, not 3. So 37 being the median here was a bit jarring
| specially since "non-repeating" wasn't even mentioned in the
| article.
| joewferrara wrote:
| Amazing how simple the proof is. 37 is my new favorite prime
| number, haha.
| falseprofit wrote:
| 2.5, the median value for the first prime factor of an integer
| owl57 wrote:
| Probably 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 wrote:
| I'm taking the median of the limiting distribution, and I
| agree that the sequence of medians you describe does not
| converge.
| gowld wrote:
| How 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 wrote:
| Coincidentally, 37 is also the first irregular prime. They're why
| Fermat's last theorem is hard.
|
| https://en.wikipedia.org/wiki/Regular_prime
| magneticnorth wrote:
| Love 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?"
| whatever1 wrote:
| Question, has anyone tried Transformer networks for predicting
| prime numbers?
| matteoraso wrote:
| That would be an incredibly expensive way of generating prime
| numbers, far more expensive than just checking to see if it has
| any factors.
| whatever1 wrote:
| I was just hoping that we can extract some useful patterns
| from reverse engineering the trained network
| XorNot wrote:
| Well, this is just in time for a few birthday card ideas...
| spokeonawheel wrote:
| given 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 wrote:
| Given that RSA primes are usually checked against some
| probabilistic primality test like Miller-Rabin, I would think
| the answer is no
| abbaselmas wrote:
| Sheldon Cooper, the best number 73 (21st prime number), its
| mirror 37 (12th prime number)
|
| https://www.youtube.com/watch?v=HacqfsV7ug0
| abbaselmas wrote:
| Also deep dive in the 73 and 37
| https://www.youtube.com/watch?v=4DQndnAhdxk
| hoseja wrote:
| I really dislike base-10 tricks. They are just cute, rarely
| have any deeper significance.
| mcv wrote:
| I 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 wrote:
| That's the age that Jesus Christ died
| cubefox wrote:
| But what about the arithmetic, geometric or harmonic mean?
| lubujackson wrote:
| This is slightly interesting but I wonder what the median 3rd
| prime is, and 4th etc. Is there some sort of interesting pattern?
| madcaptenor wrote:
| They grow very quickly. The median third and fourth primes are
| 42719 and 5737850066077 - see 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). (It's
| in French.)
| floxy wrote:
| I 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 wrote:
| I'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
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