8 ms·

Mathematicians uncover a new way to count prime numbers

199 points by nsoonhui 2y ago

fruit_snack 2y ago
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 2y ago
  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.
- khana 2y ago
  [dead]
- ubnvfft 2y ago
  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 2y ago
    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 2y ago
      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 2y ago
        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 2y ago
        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 https://en.m.wikipedia.org/wiki/Analytic_continuation
- brookst 2y ago
  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 2y ago
    "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 2y ago
      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 2y ago
  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 2y ago
    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 2y ago
      And https://en.wikipedia.org/wiki/Elliptic-curve_cryptography https://en.wikipedia.org/wiki/Elliptic-curve_cryptography
      - chr1 2y ago
        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 https://m.youtube.com/@PeakMathLandscape
      - less_less 2y ago
        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 2y ago
      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.
- nxpnsv 2y ago
  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 2y ago
    Boorish people dismiss all intellectual work this way, at all ages and all skill levels, across the liberal arts and the sciences.
    - sourcepluck 2y ago
      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 2y ago
        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.
- keepamovin 2y ago
  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 2y ago
  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 2y ago
  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 2y ago
  It’s research, you don’t really ask “for what?”. As long as some one finds it interesting that’s good enough.
- puzzledobserver 2y ago
  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 2y ago
  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 2y ago
  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 2y ago
  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 2y ago
  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 2y ago
  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 https://en.wikipedia.org/wiki/Basic_research
  - jojobas 2y ago
    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 2y ago
      Large primes are already useful for encryption - whether that would ever need gazillion-digit-long primes is questionable.
    - zmgsabst 2y ago
      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 2y ago
      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 2y ago
  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 2y ago
    > 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 2y ago
  Why bother with research into fundamental physics? Is there some deeper truth about the prime numbers hidden in the universe?
- KWxIUElW8Xt0tD9 2y ago
  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 2y ago
  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/ https://kevinventullo.com/
d-lisp 2y ago
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 2y ago
  The time-on-website usually refers to new page-loads which present a new set of ads.
  - Semaphor 2y ago
    30s ad refreshs are a thing
    - blitzar 2y ago
      Occasionally there are connection issues, best to preload 4 or 5 ads before you load the content - just to be on the safe side.
  - d-lisp 2y ago
    If you give me an array of thousands of adds that would match the target, I could lazy load the add and refresh them once they finish.
DemocracyFTW2 2y ago
> 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 2y ago
  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 2y ago
    Then what about the digits of π? Wikipedia says that "The decimal digits of π appear to be randomly distributed,[a] but no proof of this conjecture has been found"; note [a]: "In particular, π 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 π in its decimal form, and as far as I can remember there are even ways to compute the n-th digit of π without having to compute the preceding ones—which sounds pretty determined to me.
reedf1 2y ago
>> 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 2y ago
  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 2y ago
  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 2y ago
    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 2y ago
      What does "with X" mean in this context?
      - 2y ago
imprime 2y ago
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 2y ago
  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 https://i.ibb.co/s9vYHHk/GAAAAANZ-ANDERS-24-FINAL-Mail.png regards...
Babawomba 2y ago
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 2y ago
  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.
JimmyWilliams1 2y ago
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adgjlsfhk1 2y ago
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 2y ago
  What is the density of a normal set of polynomials? More importantly, how does Szemeredi theorem fit in here?
  - adgjlsfhk1 2y ago
    by this I meant things like proving that for any polynomial (e.g. x^3+1) there are infinite primes of that form (as long as the polynomial doesn't factor), and that their density corresponds to what you'd expect from the prime number theorem.
anthk 2y ago
http://mrob.com/ http://mrob.com/ Concisely: http://mrob.com/pub/math/numbers.html http://mrob.com/pub/math/numbers.html The rest of the site it's amazing too.
evanb 2y ago
> 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 2y ago
  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 2y ago
    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 2y ago
If the twin prime conjecture gets solved we riot
glial 2y ago
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/renaissance-philanthropy-and-xtx-markets-launch-new-9-million-ai-for-math-fund/ 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 2y ago
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.

199 points by nsoonhui 2y ago

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 2y ago
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.
khana 2y ago
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ubnvfft 2y ago
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 2y ago
  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 2y ago
    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 2y ago
      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 2y ago
        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 https://en.m.wikipedia.org/wiki/Analytic_continuation
brookst 2y ago
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 2y ago
  "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 2y ago
    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 2y ago
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 2y ago
  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 2y ago
    And https://en.wikipedia.org/wiki/Elliptic-curve_cryptography https://en.wikipedia.org/wiki/Elliptic-curve_cryptography
    - chr1 2y ago
      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 https://m.youtube.com/@PeakMathLandscape
    - less_less 2y ago
      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 2y ago
    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.
nxpnsv 2y ago
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 2y ago
  Boorish people dismiss all intellectual work this way, at all ages and all skill levels, across the liberal arts and the sciences.
  - sourcepluck 2y ago
    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 2y ago
      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.
keepamovin 2y ago
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 2y ago
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 2y ago
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 2y ago
It’s research, you don’t really ask “for what?”. As long as some one finds it interesting that’s good enough.
puzzledobserver 2y ago
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 2y ago
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 2y ago
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 2y ago
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 2y ago
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 2y ago
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 https://en.wikipedia.org/wiki/Basic_research
- jojobas 2y ago
  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 2y ago
    Large primes are already useful for encryption - whether that would ever need gazillion-digit-long primes is questionable.
  - zmgsabst 2y ago
    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 2y ago
    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 2y ago
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 2y ago
  > 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 2y ago
Why bother with research into fundamental physics? Is there some deeper truth about the prime numbers hidden in the universe?
KWxIUElW8Xt0tD9 2y ago
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 2y ago
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/ https://kevinventullo.com/

d-lisp 2y ago

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 2y ago
The time-on-website usually refers to new page-loads which present a new set of ads.
- Semaphor 2y ago
  30s ad refreshs are a thing
  - blitzar 2y ago
    Occasionally there are connection issues, best to preload 4 or 5 ads before you load the content - just to be on the safe side.
- d-lisp 2y ago
  If you give me an array of thousands of adds that would match the target, I could lazy load the add and refresh them once they finish.

DemocracyFTW2 2y ago

> 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 2y ago
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 2y ago
  Then what about the digits of π? Wikipedia says that "The decimal digits of π appear to be randomly distributed,[a] but no proof of this conjecture has been found"; note [a]: "In particular, π 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 π in its decimal form, and as far as I can remember there are even ways to compute the n-th digit of π without having to compute the preceding ones—which sounds pretty determined to me.

reedf1 2y ago

>> 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 2y ago
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 2y ago
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 2y ago
  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 2y ago
    What does "with X" mean in this context?
    - 2y ago

imprime 2y ago

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 2y ago
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 https://i.ibb.co/s9vYHHk/GAAAAANZ-ANDERS-24-FINAL-Mail.png regards...

Babawomba 2y ago

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 2y ago
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.

JimmyWilliams1 2y ago

[dead]

adgjlsfhk1 2y ago

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 2y ago
What is the density of a normal set of polynomials? More importantly, how does Szemeredi theorem fit in here?
- adgjlsfhk1 2y ago
  by this I meant things like proving that for any polynomial (e.g. x^3+1) there are infinite primes of that form (as long as the polynomial doesn't factor), and that their density corresponds to what you'd expect from the prime number theorem.

anthk 2y ago

http://mrob.com/ http://mrob.com/ Concisely: http://mrob.com/pub/math/numbers.html http://mrob.com/pub/math/numbers.html The rest of the site it's amazing too.

evanb 2y ago

> 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 2y ago
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 2y ago
  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 2y ago

If the twin prime conjecture gets solved we riot

glial 2y ago

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/renaissance-philanthropy-and-xtx-markets-launch-new-9-million-ai-for-math-fund/ 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 2y ago

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.