[HN Gopher] Mathematicians uncover a new way to count prime numbers
___________________________________________________________________
Mathematicians uncover a new way to count prime numbers
Author : nsoonhui
Score : 178 points
Date : 2024-12-13 03:38 UTC (19 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| fruit_snack wrote:
| All this research into prime numbers and for what? (Serious
| question)
|
| Is it that the methods required to do serious research on them
| ends up helping us discover other things?
|
| Is there some deeper truth about the universe hidden in the prime
| numbers?
| whatever1 wrote:
| For the same reason we send probes to outer space. We are
| curious about the universe. There's is something special about
| the prime numbers that we don't understand. Until we do, people
| will have the itch to keep looking.
| ubnvfft wrote:
| Yes, and yes.
|
| Investigating primes is nearly as old as mathematics itself and
| its reasonable to assume other ideas where discovered in the
| hopes of applying them to various problems incorporating prime
| numbers.
|
| From a practical, applied, perspective, "understanding" primes,
| that is making their "hidden" structure a known "truth", would
| either confirm or deny the Riemann hypothesis wherein many
| other conjectures that assume the hypothesis to be true would
| also be "truely" known.
|
| Or from TFA:
|
| > ...In the 19th century, research on these kinds of statements
| led to the development of much of modern number theory. In the
| 20th century, it helped inspire one of the most ambitious
| mathematical efforts to date, the Langlands program. And in the
| 21st, work on these sorts of primes has continued to yield new
| techniques and insights.
|
| > ...Their[the article's sunbjects'] proof, which was posted
| online (opens a new tab) in October, doesn't just sharpen
| mathematicians' understanding of the primes. It also makes use
| of a set of tools from a very different area of mathematics,
| suggesting that those tools are far more powerful than
| mathematicians imagined, and potentially ripe for applications
| elsewhere.
| throwawaycities wrote:
| The Riemann hypothesis makes me feel dumb - not just because
| I can't solve it, no great shame in that - I genuinely get
| lost in amazement and wonderment by the mind that develops a
| function, graphs it, and gleams some insight into numbers.
|
| Something about it I find humbling and makes me think about
| the archetype of mathematicians that lose their minds to
| numbers.
| ykonstant wrote:
| It is mesmerizing, but do note it was not a single mind
| that produced this insight. It was centuries of work. It
| involved, among many others:
|
| 1. Newton and the Bernoulli family developing the theory of
| infinite series and connecting them to discrete sequences,
|
| 2. Wallis developing the first notions of infinite products
| and demonstrating the first non-trivial convergence of
| such,
|
| 3. Euler solving the Basel problem and linking the zeta
| function to the prime numbers (giving a new proof of the
| infinitude of primes),
|
| 4. Gauss and Eisenstein further using Euler's ideas and
| their own unique algebraic insights to understand primes in
| arithmetic progressions, and finally
|
| 5. Riemann taking the zeta function, putting it in the
| complex plane, revealing the unifying theme connecting the
| previous discoveries and making his own fundamentally new
| discoveries with the explicit formula.
|
| And of course the development only accelerated from that
| point on.
| airstrike wrote:
| Thank you for this. I've favorited this comment so that I
| can read on each of these to sate my curiosity. Now I'm
| off to search for accessible resources for these topics
| for those of us non-mathematicians ;-)
| ubnvfft wrote:
| I think once you understand how to apply analytic
| continuation to the problem its relation to primes is
| much more apparent; even without a full understanding of
| the history.
|
| https://en.m.wikipedia.org/wiki/Analytic_continuation
| throwawaycities wrote:
| That's exactly how I begin to put it into context and
| rationalize this kind of work - he was a mathematician so
| this the kind of thing he worked on, and he was also
| working on a body of maths and knowledge.
|
| It's much like physics and the great physics experiments
| throughout history for me, some of them I'd like to think
| I may have been able to develop, but others I just marvel
| at the ingeniousness of the experiments.
|
| Realistically in a vacuum I doubt I'd have even
| identified/defined prime numbers.
| brookst wrote:
| Well why do people study anything? It doesn't have to be
| defended; these people are interested in this topic and
| therefore decided to study it.
|
| There is no master plan; nobody allocates people to these
| problems based on strategic need. It's just interesting.
| card_zero wrote:
| "This interests me" is a hidden moral judgment. Morality is
| all about deciding what to do next. It's right to sometimes
| ask a question about aimlessness. Feeling interested
| motivates us to ignore that question, because it's already
| answered by the feeling. In the stirring of interest is
| concealed an intuitive master plan, which says "I don't know
| where this leads but it feels worthwhile".
|
| Sometimes it's right to drag those intuitive feelings into
| the light and force them to explain themselves, and come up
| with _some_ clue about _in what way_ futzing around with (for
| instance) prime numbers _might_ contribute to all the rest of
| the sprawling web of things we generally value in life. But
| enthusiasm is a precious and wholesome thing, so people
| rarely question it.
| brookst wrote:
| It sounds like finding explanations for your interests is
| useful to you, but I don't think that generalizes.
|
| Many people are completely comfortable pursuing interests
| without needing or wanting a logical framework to
| explain/justify.
|
| I enjoy cooking, in the sense that I study and try to
| understand and improve at a technical level. I probably
| could come up with a rationale for why, but I suspect it
| would be post hoc reasoning, so why bother?
| philipov wrote:
| If I told you that all the world's cryptographic security is
| founded on the study of prime numbers, would it be impressive
| enough?
| gpm wrote:
| I'd point you at AES :P
|
| (Not to say that the study of prime numbers isn't hugely
| important to most of cryptography)
| tgv wrote:
| And https://en.wikipedia.org/wiki/Elliptic-
| curve_cryptography
| chr1 wrote:
| Prime numbers and elliptic curves are much more connected
| than one might expect. Each elliptic curve generates a
| function similar to zeta function, and there is a version
| of a Riemann hypothesis for elliptic curves
| https://m.youtube.com/@PeakMathLandscape
| less_less wrote:
| ECC is pretty closely related to the study of prime
| numbers. It might not be built directly on the difficulty
| of factoring, but the theory of how to construct curves,
| how to use them, what's expected to be secure etc goes
| pretty deep.
| adrian_b wrote:
| Actually AES, unlike more ad-hoc block ciphers, is based on
| the theory of finite fields, including GF(8) that is used
| for its non-linear component.
|
| The theory of finite fields is based on the theory of prime
| numbers, because the finite fields are sets of residues
| modulo a prime number or modulo a power of a prime number.
|
| The theory of finite fields is involved in the design of
| many other block cipher functions or secure hash functions
| and also in the design of the most important message-
| authentication methods, like GCM, which is used to
| authenticate this HTML page on the HN site.
|
| So prime numbers are important in most cryptographic
| applications, not only in asymmetric cryptography, like
| Diffie-Hellman or RSA. Prime numbers are used in one way or
| another for the transmission of any HTTPS data packet, not
| only in the key establishment phase of a TLS connection.
| thorel wrote:
| > The theory of finite fields is based on the theory of
| prime numbers, because the finite fields are sets of
| residues modulo a prime number or modulo a power of a
| prime number.
|
| It is note quite correct that the finite field of order
| p^k is the set of residues modulo p^k when k > 1. Instead
| this field is obtained as a splitting field of the field
| of order p (which is the set of residues mod p).
| nostoc wrote:
| AES is kinda useless for securing communications without
| assymetric crypto, unless you want to be sending keys by
| courrier.
| adrian_b wrote:
| With asymmetric crypto, you must also send by courier the
| root certificates (downloading Chrome or Firefox just
| fulfills the role of a courier that is not very
| trustworthy).
|
| There exists absolutely no method of secure communication
| that does not depend on a piece of information that is
| transmitted separately, through a presumed trustworthy
| courier. All the existing methods only attempt to
| minimize the amount of information that must be sent
| through the secure courier.
|
| With symmetric crypto without digital signatures but with
| some kind of Diffie-Hellman, you must send by courier
| only a pre-shared key that is used only for computing
| message-authentication codes that are used only in the
| couple of packets used in a key-exchange algorithm, when
| establishing a secure connection.
|
| Using only symmetric crypto, secure communication can be
| performed in pretty much the same way as with asymmetric
| crypto, by generating fresh random session keys for every
| connection.
|
| The only difference is that the key exchange packets are
| authenticated with a MAC using a pre-shared key, instead
| of being authenticated with digital signatures and a
| chain of certificates going to trusted root certificates.
|
| If for some weird reason one would not want to use a
| Diffie-Hellman variant (e.g. with elliptic curves) to
| protect the session keys, one could use another pre-
| shared key only for encrypting the key-exchange packets.
|
| There are only two advantages for asymmetric crypto, when
| used for secure communication connections.
|
| The first is provided by Diffie-Hellman in any of its
| variants, which ensures perfect forward secrecy, i.e.
| even knowing all the content of some sessions, including
| their secret keys, that does not allow the decryption of
| other sessions. Without Diffie-Hellman, if the pre-shared
| encryption key that is used to protect the key exchange
| packets is captured, all recorded sessions could be
| decrypted. This can be only partially avoided by changing
| that key frequently, which would prevent the decryption
| of past sessions, but not the decryption of future
| sessions.
|
| The second advantage is provided by the authentication of
| the key exchange with digital signatures instead of MACs
| based on pre-shared keys, which is the possibility of
| half authentication, where the server is authenticated
| based on the certificates provided by it, but the client
| is not authenticated, which is the most frequent kind of
| secure communication used on the Internet.
|
| For communication inside a closed environment, i.e. a
| private network, using key exchange authentication based
| on pre-shared keys (but with elliptic-curve Diffie-
| Hellman for protecting the session keys) can be simpler,
| faster and more secure than using digital signatures and
| certificates.
|
| While in the beginning I have used your metaphor about
| sending a pre-shared key or the root certificates by
| courier, the normal mode of transferring pre-shared
| authentication keys is by initial physical pairing (e.g.
| cable connection) of the devices that must be able of
| communicating securely between themselves.
| SAI_Peregrinus wrote:
| Fine, ECC doesn't care much about primes, and is
| asymmetric.
| YetAnotherNick wrote:
| Technically when you say it is based on prime numbers, it is
| based on product of 2 primes.
| nxpnsv wrote:
| Scientific work is too often challenged with this kind of
| question. If all you care about is results you know will happen
| you will never discover anything you don't allready know.
| globnomulous wrote:
| Boorish people dismiss all intellectual work this way, at all
| ages and all skill levels, across the liberal arts and the
| sciences.
| sourcepluck wrote:
| Yes to you and the person you are responding to! And the
| boorishness here is coming from a "tech person" [0], no
| less.
|
| What have the technologically capable people who were the
| ones architecting these systems the past few decades given
| the world: a handful of Big Tech behemoths, with all the
| terrifically negative, stultifying effects that has had.
| The computing world has been willfully fragmented, and the
| landscape is awash with casualties; namely, every person
| out there who is terrified of their computing devices, who
| panics when the first pop-up screen appears.
|
| Which is surely the minority on here, but in the big bad
| world, I would guess it's easily a majority of people.
|
| And then the computer types have the gall to ask what
| number theory has done for humanity..!
|
| The following should go without saying, but let's say it
| anyway: just because tech-y startups continue to attract
| historic levels of investment, doesn't necessarily mean
| that the stuff the tech world produces is in anyway useful
| or special or good or interesting. If you're not going to
| read a book or something on the topic (number theory), at
| least browse a couple of wikipedia articles, or get an LLM
| to summarise it for you, or something.
|
| [0] I'm guessing this entirely from the tone and the forum
| we're on. Please tell me if I'm guessing incorrectly!
| card_zero wrote:
| Dismissing purpose, and living in aimless complacency, is
| _also_ boorish.
|
| I don't want to come off as anti-pig here, pigs are OK.
| Number theory is OK too, it's probably the branch of
| mathematics I dislike least. But it's laudable to
| sometimes ask "what is it all for?", without wanting to
| attack or threaten anybody's occupation. No easy answer
| is available, but it's worth asking.
| ndsipa_pomu wrote:
| > No easy answer is available, but it's worth asking.
|
| Beyond "expanding our knowledge about 'x'", a lot of
| disciplines don't have much of an answer, so I don't see
| the value in asking the question apart from trying to
| dismiss the relevant discipline.
|
| So many of our technological advances have relied on
| chance discoveries (e.g. penicillin), so we can't predict
| ahead of time what the end uses are going to be. This is
| especially the case with maths as it is so abstract, but
| where would we be without Boolean logic, public key
| cryptography, information theory etc.?
| photonthug wrote:
| > > No easy answer is available, but it's worth asking.
|
| > .. trying to dismiss the relevant discipline
|
| Yeah. The question is worth asking yourself, your
| teachers, colleagues, and your friends.. but usually not
| worth asking (or answering!) strangers on the internet.
|
| This type of "but what is it all for!" question has a
| huge asymmetry where it's easy to ask and hard to answer,
| so it comes across as trolling. And if you're talking to
| an adult individual who wants to argue not even about the
| _priorities_ of basic research but the fundamental point
| of _any_ of it, you can be pretty sure any conversation
| about the subject is pointless. People who want to take
| an anti intellectual stance aren't waiting to hear a good
| argument before they change their point of view..
| vbezhenar wrote:
| The resources that public basically "donates" to the science
| are limited and there's always the question of priorities.
| Not just money, but people themselves. This is very valid
| question. One could argue that there's potential in studying
| almost everything: prime numbers, deep space, exotic matter,
| crash particles to each other, discover language structure,
| develop new programming paradigms, model economic behaviour,
| model nuclear winter. But those who giving money, need to
| choose and allocate. And those who want to study prime
| numbers while receiving money from government, inevitable
| need to prove why this particular study has value.
| nxpnsv wrote:
| That validation already happens. I worked as a scientist
| for a decade before I went to industry, getting grants was
| a constant struggle - agencies and politicians don't just
| give out money randomly. One can argue that these processes
| aren't perfect and that some money gets wasted, and that it
| makes for weird incentives - but that's a whole other
| debate.
| boxed wrote:
| More money is wasted trying to prevent the money from
| being wasted I would say. The amount of time spend on
| grant applications is crazy.
| ykonstant wrote:
| And even that time pales before the mental toll. When I
| first went about writing a grant application, I thought
| it would be technically easy---I knew the bureaucratic
| details would be tedious, but otherwise straightforward.
| Oh boy, I think it was the most mentally taxing, soul
| crushing work-related task I ever attempted. I failed to
| submit, by the way. My research output also plummeted
| that year.
| boxed wrote:
| Classic "penny wise, pound fooling" crap. We could
| probably be a hundred or more years ahead of where we are
| if we stopped sabotaging ourselves like this...
| boxed wrote:
| This type of expenditure is a rounding error. We spend more
| dollars on charging phones that is then spent on flipping
| videos on TikTok. And TikTok only exists because of this
| type of curiosity based research. Not to mention
| antibiotics, refrigerators, cars, GPS, etc, etc, etc. The
| list is literally too long.
|
| 100% of all progress is science/technology/infrastructure.
| Without it we would be living in caves.
| keepamovin wrote:
| Yes. You shouldn't be downvoted because it's a reasonable
| question and opens things up for fascinating exploration.
|
| I guess there's many things interesting about it, but I see it
| like: prime numbers are the fundamental pattern of magnitudes,
| where the next prime is the first place that no multiples of
| any previous magnitude (prime or not) would ever land on.
|
| In other words, if you took any previous magnitude (ie, any
| number less than that next prime), and copied it over edge to
| edge, the edge would never line up with that next prime.
|
| Because counting and magnitude is so simple and fundamental to
| the space of concepts and even to reality, it's pretty
| fascinating that this extremely simple to describe pattern, is
| nevertheless hard to create a description for that's more
| concise than including all previous primes.
|
| And I think people like finding that kind of 'shorter'
| description, as it indicates a deeper understanding, a new way
| of looking at reality that you didn't see before. And when we
| see that, it will probably be very useful to many other things.
|
| It's fascinating to reflect on all that, and also on how this
| fundamental pattern of magnitudes, their 'self-similar but
| scaling' structure, also relates to the 'compressibility' of
| the number line and information theory.
|
| That's what I think. I think everyone can find their own
| interest in there, there's probably a lot of ways to look at
| it. :)
| lubujackson wrote:
| Some good answers here, but I hate that this has been
| downvoted. It is a valid and reasonable question - we shouldn't
| be downvoting questions like some echo chamber Reddit thread.
|
| Prime numbers are a core and mysterious numeric progression
| that has the unique property of being very taxing to determine
| the factors of any sufficiently large number. This is why they
| are used in cryptography. Investigations into the nature of
| primes has produced many mathematical tools and they have
| bridged many different areas of mathematics together.
|
| But the simplest answer is that prime numbers are a tantalizing
| mystery that is easy for anyone to understand. The deeper you
| dig, the deeper the hole gets and it almost doesn't matter if a
| satisfying "answer" is ever found. Primes are a McGuffin that
| has led to countless discoveries.
| fifticon wrote:
| It depends on which parts of math you feel are 'more real than
| others'. To someone who only just has learned math, the
| increasing counting numbers (1, 2, 3, 466..) are all 'real
| things'. But if you are very cynical about math, you might
| instead argue, that the only 'real' number we have, is the
| number '1'. All those others (2, 466..) are just "applications"
| of that '1'. That is to say, we adopt the shorthand '466',
| because we don't want to write down almost 500 1's each time we
| reference it. (think of it like very bad roman numerals..) In
| this perspective, where we ignore 'ordinary' numbers like 466,
| you might argue that prime numbers are more real, because they
| really 'do' something (that is, construct composite numbers,
| like the example with 1 above.)
|
| You could have a thought experiment of a world, in which we
| never developed arabic numerals or roman numerals, but instead
| did all our math directly on prime numbers. It would be a weird
| world, but still, you might imagine it :-)
| vishnugupta wrote:
| It's research, you don't really ask "for what?". As long as
| some one finds it interesting that's good enough.
| puzzledobserver wrote:
| I am not a mathematician, but here is a motivation I read
| somewhere some years ago.
|
| There are basically two ways to produce big numbers: add two
| small numbers, or multiply two small numbers. You can produce
| all positive integers by starting with zero and repeatedly
| adding one. You can almost do the same thing with
| multiplication too, except for these pesky primes, which are
| somehow atomic. Naturally then, one might ask: (a) How many
| primes are there? (b) How frequently do they occur? (c) Can we
| look at a number and determine whether it is a prime? Now
| consider: Despite being among the oldest of the mathematical
| disciplines, there are still open problems about primes that
| can be explained to high school students.
|
| Also, multiplication and addition are not simply operations
| that are of interest with respect to integers, but similar
| ideas apply to a bunch of other domains too. Polynomials, for
| example. So primality and primality-like ideas are like catnip
| for mathematicians.
| pdpi wrote:
| Prime numbers are one of the distinguishing features of number
| theory, which means they also show up all over the place in
| anything related to discrete mathematics, which in turn means
| they show up all over the place in computer science.
|
| Any maths-y field of study that has the concept of
| decomposition also has the concept of primality, usually in a
| way that relates to primality in the natural numbers. This
| means anything we learn about prime numbers also extends to
| those other fields of study.
| MattPalmer1086 wrote:
| Prime numbers are like the atoms of numbers. They are
| indivisible and you can make all the other numbers from them.
| So, finding out more about primes translates into tighter
| constraints or proofs in many other (often not obviously
| related) theories.
|
| And, it's beautiful to think about. Maybe huge practical
| innovations might result, or maybe only the pleasure of
| understanding something deep about numbers in the short time we
| exist.
| atoav wrote:
| People are interested in things and like spending their time on
| it, that is called "living". Yet other people make careers out
| of expanding the bounds of knowledge for humanity, often with
| no clear application in mind, this is called foundational
| research.
|
| Sometimes if we are lucky either one of those yields phenomenal
| practical applications, just because some nerd thought there
| was a missing piece in the puzzle and they ought to find it.
|
| I know many nowadays believe that the sole goal of humanity
| ought to be the increase of shareholder value, even if said
| increase is at odds with human survival on this planet. Then
| 99% of us just exist, work our asses off, with little to no
| time spent with our loved ones while leaving the planet and
| humanity in a worse state than previous generations -- and then
| we die.
|
| Was that really it then?
| covofeee wrote:
| For the researchers - because it is fun most likely, and they
| get paid for it. For society, I am glad we live in a society
| where some money is skimmed off for curiousity. But for a
| pratical reasons - this stuff (or some other bet) rears up as
| useful years down the line for something practical. Maybe some
| kind of cryptography or making quantum computing feasible...
| who knows! Imaginary numbers are pretty useful in science, and
| they probably seemed exotic when they first were talked about.
| taneliv wrote:
| Isn't basic research always like that? According to
| Wikipedia[1]: "aim of improving scientific theories for better
| understanding and prediction of natural or other phenomena".
| There is no implied success (it's only an "aim"), or utility,
| beyond that for science itself.
|
| How much we want to support that (financially, socially etc) is
| a question a bit like, how much do we want to support children
| playing. Some disagree such should be supported at all, others
| are indifferent about such, yet others take pride in supporting
| or having supported such. The answer, to both of those
| questions, does have a large effect on how our societies look
| like. However, answering in the affirmative to support does not
| guarantee any positive progress. Likewise, answering in the
| negative, does not prevent progress, or basic research or
| children's play from happening.
|
| [1] https://en.wikipedia.org/wiki/Basic_research
| jojobas wrote:
| Science often discovers and quantifies natural phenomena that
| are useful outside science. Whether pure math dealing with
| gazillion-digit-long primes can be of any use outside of
| satisfying curiosity is unclear.
| ndsipa_pomu wrote:
| Large primes are already useful for encryption - whether
| that would ever need gazillion-digit-long primes is
| questionable.
| zmgsabst wrote:
| We're really bad at handling large, complex structures.
|
| Mathematics dealing with large primes and their complex
| structures is likely to find applications in other complex
| structures, eg in physics or computer science.
|
| Mathematics is modern ontology: even when its self-
| investigation is not directly applicable, the vocabulary
| and semantics developed is often useful for articulating
| other truths.
| taneliv wrote:
| Aren't some modern digital cryptography methods based on
| exactly that?
|
| I do agree on the view that science often discovers useful
| phenomena. What I tried to stress was that basic research
| does not, by definition, aim for such utility. Especially
| with pure math, whether there are any applications for new,
| even groundbreaking discoveries, is often very unclear. And
| when there are, they might be only utilized decades or
| centuries after the initial discovery.
| giorgioz wrote:
| Prime numbers are used in cryptography. Public-private key
| cryptography is based on the fact that is hard to find the
| original two large prime numbers that were multiplied together
| from their result. Example you see written 721.000.165.331 .
| It's hard to calculate that is the product of the two prime
| numbers 730487 * 987013 So if they can calculate bigger prime
| numbers with a new faster method we can know larger prime
| numbers and have safer cryptography. (That's the simplified
| version that I remember as a software engineer)
| skissane wrote:
| > Prime numbers are used in cryptography. Public-private key
| cryptography is based on the fact that is hard to find the
| original two large prime numbers
|
| That's true of one particular - albeit very popular -
| asymmetric cryptosystem, RSA. It isn't a property of
| asymmetric cryptography in general. There are other
| asymmetric encryption schemes which aren't based on the
| hardness of prime factorisation (e.g. DSA, elliptic curves,
| McEliece, NTRU)
| Jevon23 wrote:
| Why bother with research into fundamental physics? Is there
| some deeper truth about the prime numbers hidden in the
| universe?
| KWxIUElW8Xt0tD9 wrote:
| I recall many years ago hearing that a mathematician had
| invented something and was very happy about the fact that it
| had absolutely no practical use. I may remember the details
| incorrectly but I believe it was one-way functions -- which are
| used all over the place now in computer security. Someone
| please correct me if I have the details wrong here.
| kevinventullo wrote:
| As a former number theorist turned software engineer, I've
| noodled on connections between algebraic number theory and
| fairly concrete applications here:
|
| https://kevinventullo.com/
| d-lisp wrote:
| I disliked the way this article is written, it reminds me of "we
| had to rollback to the previous UI because the new one was too
| efficient that people spent less time on the website".
| qwertox wrote:
| The time-on-website usually refers to new page-loads which
| present a new set of ads.
| Semaphor wrote:
| 30s ad refreshs are a thing
| blitzar wrote:
| Occasionally there are connection issues, best to preload 4
| or 5 ads before you load the content - just to be on the
| safe side.
| DemocracyFTW2 wrote:
| > the primes aren't random. They're completely determined
|
| I wonder how true this statement is but it probably also relies
| on the understanding of the word 'random'. In the colloquial
| sense, it is certainly true in the sense that (truly) 'random'
| means 'occurring without (discernible) rules'.
|
| However, in a stricter definition 'random' means (if I'm not
| mistaken, hobby-thinker here) just "a given set of numbers S is
| called random with respect to a set of binary tests T when all
| individual procedures in T yield a positive outcome". That is,
| you can only ever a finite set S and test against a finite set T,
| meaning the algorithm that generated your random-looking set S
| can, potentially, always be amended to creep closer to overcome
| the failed tests (and the reverse is also true: one can,
| potentially, always move the goalposts and add another test to T
| to make a set fail that used to pass).
|
| Ultimately, then, randomness--where it is not occurring _eo ipso_
| as in radioactive decay--is always a relative (S WRT T), social
| (people must agree), and finite (can 't test against unseen
| members of S and, per precondition, can't give a rule to cover
| infinity as in 'divisible by eleven') procedure (that may or may
| not practically terminate). In other words, once we settle on a
| given set of tests T to determine what is random, one can
| (potentially) always come up with an algorithm that passes the
| tests, thus looks random though it is determined.
| griffzhowl wrote:
| I think that just shows that those tests for randomness can
| just tell you when something is not random, not when it's
| actually random. There are problems, I think, about giving a
| rigorous definition of random, but I think most would agree
| that if you have an algorithm that predicts with certainty what
| the next element in a sequence will be, then that sequence
| isn't random.
| DemocracyFTW2 wrote:
| Then what about the digits of p? Wikipedia says that "The
| decimal digits of p appear to be randomly distributed,[a] but
| no proof of this conjecture has been found"; note [a]: "In
| particular, p is conjectured to be a normal number, which
| implies a specific kind of statistical randomness on its
| digits in all bases."
|
| So that sounds pretty random to me, yet there are algorithms
| that give you p in its decimal form, and as far as I can
| remember there are even ways to compute the n-th digit of p
| without having to compute the preceding ones--which sounds
| pretty determined to me.
| reedf1 wrote:
| >> But of course, the primes aren't random.
|
| Stunning to me that a staff writer for a science magazine would
| type this sentence without referencing the riemann hypothesis.
| lblume wrote:
| Quanta Magazine tries to make these topics accessible for
| people with low knowledge regarding them, thus mentioning the
| Riemann hypothesis, although adequate and needed for, say, a
| lecture on the topic, would not really help this goal.
| griffzhowl wrote:
| But the primes are obviously not random, independently of the
| Riemann hypothesis - they're determinate consequences of the
| number system. Maybe I missed your point
| graycat wrote:
| In 1933, A. Kolmogorov used H. Lebesgue's _measure_ theory to
| define random variables. With that definition, could have a
| random variable X whose values are only prime numbers and
| such that for each prime number p and for probability measure
| P,
|
| P(X = p > 0),
|
| that is, the probability that X = p is positive.
|
| So, with X, the prime numbers are _random_.
|
| References:
|
| With TeX markup, polished details on measure theory are in
|
| H.\ L.\ Royden, {\it Real Analysis: Second Edition,\/}
| Macmillan, New York, 1971.\ \
|
| Walter Rudin, {\it Real and Complex Analysis,\/} ISBN
| 07-054232-5, McGraw-Hill, New York, 1966.\ \
|
| and polished details on probability theory based on measure
| theory are in
|
| Leo Breiman, {\it Probability,\/} ISBN 0-89871-296-3, SIAM,
| Philadelphia, 1992.\ \
|
| Jacques Neveu, {\it Mathematical Foundations of the Calculus
| of Probability,\/} Holden-Day, San Francisco, 1965.\ \
| ecmm wrote:
| What does "with X" mean in this context?
| graycat wrote:
| Uh, Joe, you have random variable X. What is its value?
|
| Sam, just a minute. Let me _draw a sample_. Got one: X =
| 7.
|
| But, Joe 7 is not very surprising or interesting. Is
| there anything else?
|
| Sam, sure, one more minute. Got one!
|
| Joe, well, then, WHAT is it????
|
| Sam, sorry, it has 2^12345 digits and will take a while
| to print it out; this morning I have a coffee shop
| meeting with Susan; and I don't want to miss Susan, cute,
| pretty, sweet, smart, darling, adorable, precious ...,
| and single!
|
| Exercise: Show that there is a random variable Y with the
| same _distribution_ as X and such that X and Y are
| _independent_ random variables. I.e., knowledge of X
| tells us nothing about Y.
|
| The intuitive concept of _random_ is closer to
| _unpredictable given even everything else_ , that is,
| what probability theory defines as _independent_.
|
| There are more details in Royden, Rudin, Breiman, and
| Neveu. To preview, there is a non-empty set Omega with a
| collection F of subsets that form a _sigma algebra_ and a
| _measure_ P on F. Then _random variable_ X is a
| _measurable_ function from Omega to the set of all prime
| numbers. So, for some point w in Omega and function X,
| X(w) is a prime number. Can think of w as a _trial_.
|
| Uh, this morning I'm working on some Rexx code, so here I
| can't reproduce or compete with the references by Royden,
| Rudin, Breiman, and Neveu.
| graycat wrote:
| Edit:
|
| Replace
|
| P(X = p > 0)
|
| with
|
| P(X = p) > 0
| ecmm wrote:
| https://en.wikipedia.org/wiki/Riemann_hypothesis?oldformat=t.
| ..
| wruza wrote:
| _oldformat=true_
|
| I'm curious what this does, cause it seemingly changes
| nothing (tested on mobile, with "desktop version" too).
| YetAnotherNick wrote:
| Riemann hypothesis(conjecture) doesn't prove much for prime,
| other than a tight bound on the prime counting function.
|
| Most just mean related Riemann's explicit formula for prime
| when they link Riemann hypothesis and prime number.
| bmacho wrote:
| Weird. It is obviously true. Also it is redundantly explained
| in the very next half sentence:
|
| > But of course, the primes aren't random. They're completely
| determined,
|
| Also this whole thread that this post started is so stupid.
| imprime wrote:
| Prime numbers, in their spirit, are like decomposing a problem
| into independent smaller problems. It is a search for a divide
| and conquer algorithm. So when people learn how to decompose and
| put together new problems there is a source for new knowledge.
| One of the most trivial examples is how if you can factorize a
| polynomial into factors you have a simple way to solve for the
| roots of that polynomial by solving the smaller problem of
| finding the roots of the factors and the union of all roots is
| the root of the initial polynomial. In that example the three
| parts: decompose, solve subproblems and finally compose the final
| solution are clearly exposed.
|
| Nowadays we are wondering if LLM are going to be the next prime
| numbers. The question is if solving the language problem is going
| to provide us the key for solving the AGI problem. We still don't
| know what is the equivalent of a prime for a LLM, that is the
| smaller independent part that allow it to express some knowledge,
| the pieces could be embeddings or the topology of the layers or
| some new insight.
|
| Some more random ideas, just like Norvig see python as a more
| practical Lisp, the basic ideas from prime numbers impregnate a
| great part of mathematics. You can be really far from the root
| but the principles are always with you, primes are a second
| nature, you have internalized all their properties and
| dynamically you learn to see primes like features in many fields
| (prime ideals, spectrum of a ring, points in commutative
| algebra).
|
| The problem of how many primes (in relation to positive integers)
| is like wondering whether a given decomposition exists in a
| general sense that could allow us to solve the general problem.
| So few primes implies that the theory could solve some kind of
| problems but not many. What are LLMs number?, that is the
| question if solving the language problem will allow us to solve
| some very general problems, like a good approximation to AGI. The
| problem of whether LLMS will open the way to AGI could be the
| next Riemann hypothesis once we succeed in defining what is a
| prime number in relation to a LLM.
|
| We are trying to prove escalating laws for how LLM improve when
| increasing the number of parameters, this is like trying to guess
| the convergence of an infinite serie by using a finite sum. The
| analogous of a Riemann hypothesis could be defining certain kind
| of LLM and conjecturing if it could obtain AGI pass some
| threshold for the number of parameters.
|
| Edited: Sorry for being overly verbose this mornig!
| comicabout wrote:
| Quote: 'Prime numbers, in their spirit, are like decomposing a
| problem into independent smaller problems. So when people learn
| how to decompose and put together new problems there is a
| source for new knowledge.'
|
| A (reupped) _comic_ (in _german_ ) maybe if you like (I think
| they don't like hotlinkng): https://i.ibb.co/s9vYHHk/GAAAAANZ-
| ANDERS-24-FINAL-Mail.png
|
| regards...
| Babawomba wrote:
| let's not ignore the practical side. Algorithms for studying
| primes drive advances in computing, machine learning, and data
| science. Cryptography literally depends on them. Plus, big
| unsolved problems like the Riemann hypothesis could completely
| reshape number theory.
|
| Green and Sawhney's work is especially exciting because it shows
| how tools from one field--Gowers norms--can unlock progress in
| another. That kind of cross-disciplinary insight is where
| breakthroughs happen. And yeah, it's fair to question funding
| priorities, but basic research has given us antibiotics, GPS, and
| even computers. Without it, we'd still be in caves.
| math_dandy wrote:
| Cryptanalysis relies on deep conjectural heuristics in analytic
| number theory. These conjectures becoming theorems wouldn't
| affect cryptanalysis at all, because their validity is already
| baked in. If, however, any of these conjectures turn out to be
| false, there would be ramifications.
| adgjlsfhk1 wrote:
| This is a huge result! It seems like it could be a non-trivial
| amount of the way towards solving the general type of problem of
| proving that the primes show up with expected density for most
| "normal" sets of polynomials.
| LPisGood wrote:
| What is the density of a normal set of polynomials?
|
| More importantly, how does Szemeredi theorem fit in here?
| anthk wrote:
| http://mrob.com/
|
| Concisely:
|
| http://mrob.com/pub/math/numbers.html
|
| The rest of the site it's amazing too.
| evanb wrote:
| > There are infinitely many primes that can be formulated by
| squaring two whole numbers and adding them together. [...] By
| insisting that one of the numbers you're squaring be odd, perhaps
| [...] makes the problem much harder.
|
| Does it? For any number a, a^2 = a (mod 2), and primes greater
| than 2 are all odd, so if a prime p = a^2 + b^2, doesn't one of a
| or b have to be odd? Reducing mod 2, p = 1 (mod 2), a^2 + b^2 = 1
| (mod 2), a + b = 1 (mod 2), so either a = 0 (mod 2) and b = 1
| (mod 2) or vice-versa?
|
| edit:
|
| If Euler proved infinitely many such primes exist then "With this
| in hand, Green and Sawhney proved Friedlander and Iwaniec's
| conjecture: There are infinitely many primes that can be written
| as p^2 + 4q^2." makes no sense without a further condition of p
| or q, let (in my notation) a=p be odd and b=2q be even.
|
| Now having finished the article, I think this was just sloppy
| writing, and the actual accomplishment is related to the post-
| perhaps clause: one of p or q has to itself be a perfect square?
| Anyway, I have very little certainty about what was actually
| accomplished from reading this article.
| masfuerte wrote:
| There is a further condition on _p_ and _q_. They both have to
| be prime. The article states this very clearly, though it may
| have been updated?
| feoren wrote:
| They state that condition when they introduce the _p^2 +
| 4q^2_ condition, but at the point that GP quoted ( "one must
| be odd"), they had only referred to them as "numbers" and
| "whole numbers". So it's not clear whether the article
| considers _p_ and _q_ being prime as a condition on _p^2 +
| q^2_ or not. GP 's point is valid.
| kouru225 wrote:
| If the twin prime conjecture gets solved we riot
| glial wrote:
| If anyone here is working on similar problems, I just wanted to
| flag this recent announcement of a new $9M funding pool:
|
| https://renaissancephilanthropy.org/news-and-insights/renais...
|
| > Proposals should be aligned with one of the following
| categories:
|
| > Production grade software tools: AI for auto-formalization,
| proof generation, synthesis of verifiable code, and more
|
| > Datasets: Open-source collections of theorems, proofs, and math
| problems Field building: Textbooks, courses, and resources to
| grow the AI-for-math community
|
| > Breakthrough ideas: High-risk, high-reward approaches to AI-
| driven math research
| eh_why_not wrote:
| In the recent past, I've accepted that all titles are not
| informative anymore. But that there was hope that the subtitles
| were actually informative (i.e. the subtitle was the real title,
| and the title was the clickbait).
|
| In this article, neither is informative.
|
| And even after several paragraphs in, you don't know what the
| general area of the proof is. Just meandering long-winded story-
| telling.
|
| If anyone of you authors/editors of this magazine are here;
| please, for the love of all that's holly, put the crux of the
| matter at the top and then go off to tell your beautiful,
| humanized, whatever... story.
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