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Breakthrough a step toward revealing hidden structure of prime numbers
- EMIRELADERO 2y agoThis got me thinking. Imagine this discovery led to a larger breakthrough on prime numbers that allowed easy factorization of large integers and effectively rendered public key cryptography such as RSA ineffective overnight, by allowing anyone with a consumer-grade CPU to crack any production-size key. Does the industry have DR plans for this scenario? Can the big players quickly switch to a different, unbroken encryption system? While it would probably be a heavenly day for jailbreakers, console modders and other "device freedom" types generally, the overall impact would be disastrous and incalculable. Does the industry simply not consider "sudden number theory breakthrough" a possible event?
- apples_oranges 2y agoWe could always switch to symmeric keys (but key exchange would be somewhat problematic). Also elliptic curves crypto doesn't use primes, so even public key/private key crypto would still be around and secure.
- shinycode 2y agoFrom what I remember in my math class where we made those cryptography calculations by hand, the teacher many years ago said that the day we could guess prime numbers it will be a disaster because many cryptographic calculations are based on the premise that we can’t guess prime numbers. I don’t know if that changed ?
- PeeMcGee 2y agoIt's easy to find primes of a given bit length, and it's easy to multiply them together. It's hard to un-multiply a given number (public key) into its unique prime factors (private key).
- mbreese 2y agoBut if we could more easily find primes, the search space for finding those prime factors would be significantly smaller. In my mind, it’s not a question of easy vs hard… it’s a question of fast vs slow. The default algorithm for finding primes is pretty simple, but it takes a lot of math and time. If you reduce the time requirements, then we start to get into trouble.
- JackSlateur 2y agoThe world has moved away from RSA etc to elliptic curves. Not everybody did, through. RSA is no longer the standard, and has not been for many years.
- opyate 2y agoStill, plenty of old stuff was scraped/sniffed under the "store now, decrypt later" methodology.
- milansuk 2y agoTrue. The only solution is to keep your data outside cloud(aka someone else's computer) no matter what encryption you use.
- K0balt 2y agoAlso means it can’t transit the internet. So actually, only on airgapped networks.
- 8372049 2y agoIf we're going to extremes like that, airgapped networks aren't truly safe either
- barelyauser 2y agoAlso, the safest data is the one never sampled into digital format and stored in computer systems.
- mckn1ght 2y agoCould you explain why that is? If I have an airgapped smart home network, someone has to come physically sniff the packets. If it’s only over ethernet, they have to physically plug in. That’s not a scalable attack strategy.
- tommiegannert 2y agoI think this is where the move to elliptic curve comes in, and that seems well on its way. Both in signatures and handshakes (Diffie-Hellman). Perhaps it's not a one minute operation to DR, but I doubt everything would be open if RSA/DH was rendered insecure overnight. I know my SSH keys are a mixed bag right now.
- exe34 2y agodoes the move to elliptic crypto suggest that the people in the know expect prime factorisation to be broken soon?
- staunton 2y agoIt's to do with "if we ever have big/good quantum computers, prime factorization is doable" and with "let's use the new shiny things". On a related note, someone might discover some elliptic curve math tomorrow and your CPU can break all that stuff just as well...
- exe34 2y agoso with my cynic hat on, maybe a bunch of people already have that and that's why we're being moved off the hard stuff.
- staunton 2y agoThe NSA had the option to do something like that when they (via NIST) standardized DES. They chose to standardize a version that's secure against attacks that only they knew at the time, shorten the key length so they can still brute-force it if they really need to, and successfully kept the attack secret until researchers at a foreign university independently discovered it decades later.
- alphager 2y agoYup, that was the older generation. The newer generation used NIST to propagate a backdoored RNG and to weaken several ECC-curves.
- scrapheap 2y agoIf someone found a way to easily factorize large integers easily on consumer grade hardware then it would be very painful as RSA is one of the big public key algorithms. Before you start worrying about it though consider that RSA has held up for 47 years of active cryptanalysis so far - during which time many alternative algorithms have been suggested as being superior, only to be broken a short time later. Also the push to switch to Elliptic-Curve algorithms has been more down to them being easier for computers to use to encrypt/decrypt data. Personally if I had to bet on which public key algorithm will still be around in 10 years time then I'd put my money on RSA.
- raverbashing 2y agoActually RSA has several "gotchas", so it is not that it has held up but people have managed to work around those gotchas into a working encryption system (Basically your data is not encrypted with RSA, you encrypt a secondary key, send it with RSA but the main encryption is AES see https://en.wikipedia.org/wiki/Transport_Layer_Security#Key_exchange_or_key_agreement https://en.wikipedia.org/wiki/Transport_Layer_Security#Key_e... )
- fharding 2y agoKey exchange is done for speed (symmetric key crypto is way faster than public key) and forward secrecy. It’s not done because RSA is flawed per se. We use DH instead of e.g. ElGamal encryption for the same reasons.
- raverbashing 2y agoYeah it's not so much of a flaw of RSA, but encrypting pure text with it for example is more complicated (and has more caveats with padding, etc) than just encrypting a fixed amount of bytes
- sk5t 2y agoDon’t think this merits an “actually” - using a session key et al. is basic usage and does not bear on the strength of RSA itself.
- keepamovin 2y agoI guess one perspective is finding fast factoring is considered super rare. The story is so many smart people have looked at it, it's probably impossible...for now. But that story may be its own weakness. Anyway, the risk, while just as real as something like the grid being downed by a massive solar storm with multiyear recovery period from stone age due to transformer manufacturing delays and no stockpiles, just seems too minuscule/theoretical to spend much time on - from that point of view. Regarding any plan, I don't know if it's so easy to just switch to ECC, because actual asymmetric encryption with ECC depends on shared secret, which (if you're assuming an unsecured exchange channel due to RSA being broken), is more vulnerable to MITM than RSA. I don't think it's an easy swap out. All that aside, another point of view is RSA is probably already broken, the break is just secret to the codebreaking agencies. It would be very desirable for them to keep their breakthrough secret. That might even involve trying to find ways to suppress any "sudden number theory breakthroughs" hahaha! :)
- atemerev 2y agoThe industry couldn’t even prepare for a bad Crowdstrike update. And yet, it figured things out in a few days or so. The ability to prepare for catastrophic scenarios is overestimated. The ability to survive them is underestimated.
- robertlagrant 2y agoThis would be a lot worse than that. Crowdstrike was bad because everyone lets relatively untested code straight into the Windows kernel - i.e. known incompetence of approach. This would be bad despite massive care taken to have the right approach.
- atemerev 2y agoYes, except there is no “massive care”. If people are OK to install other companies’ rootkits to their critical infrastructure, they will not care about anything else, too.
- digging 2y ago"Some people did X" !== "All people do X"
- robertlagrant 2y agoThe massive care is the algorithm selection process, the careful implementations, and the long-term observation and correction of the performance of the algorithm implementations.
- HeatrayEnjoyer 2y agoCS was a software update. RSA is baked into many silicon circuits and firmware ROMs.
- atemerev 2y agoWell, hardware is replaceable, too.
- golol 2y agoI think someone working in cryptography will worry about a sudden number theory breakthrough that allows for breaking of cryptography as much as a someone working in the energy sector will worry about a sudden physics breakthrough that allows for practically free energy cold fusion energy.
- ertgbnm 2y agoPretty sure that it would require that P=NP if such an event happened. So if factorization was cracked, everything else would be too.
- amelius 2y agoAre you sure about that? And even if problems can be solved in polynomial time, the constants involved can be prohibitively large.
- cvoss 2y agoInteger factorization is an NP problem but is not known to be NP-complete. Therefore, we do not know how to solve all NP problems in P time using a hypothetical P time factorization. P =? NP would remain open.
- zitterbewegung 2y agoMany people in the industry does not think that RSA is crackable due to the assumptions that the Riemann Hypothesis and also the distribution of prime numbers is such a hard problem with a long time of being unsolvable. A possible mitigation for things like websites would be either ECC or even using the quantum resistant encryption systems (the industry would more likely avoid this due to the systems being very prototypical since we have just started researching this). Since old bitcoin wallets can’t be moved off of RSA you can transfer the coins to your wallet and there is no mitigation.
- red_trumpet 2y agoI don't see how proving the Riemann Hypothesis would help cracking RSA? If it helps, couldn't you just assume it is true and start cracking RSA today? If you ever hit a point where it doesn't work then BOOOM: Riemann Hypothesis disproven!
- tzs 2y agoI think it is the other way around--disproving the RH might break some things. Most mathematicians believe RH is true, and generally when doing industrial number theory people operate under the assumption that RH is indeed true and so if they need to use X to justify something and there is a theorem of the form "if RH is true then X" they use X. Thus a proof of RH is not a problem. It just confirms that what people applying number theory already assumed was correct. A disproof means that those X's might not be true and their use would need to be reassessed.
- AnotherGoodName 2y agoRSA was once 128bits and today has to be 2048bits minimum to be secure because it was essentially broken multiple times. There used to be 128bit rsa encrypting hardware that now doesn’t work at all to protect data due to previous mathematical breakthroughs. The congruence of squares equivalence to factorization demonstrated we need at least 500 bits and then the special number field seive that built on this push it to 1024. The general number field seive pushed it again to 2048. Sure it’s not a log(n) break but it’s been broken. If you look at the complexity analysis of the special vs general number field seive the portion of the exponent going from 1/2 to 1/3 should give you thought. Can it be moved to 1/4? Could it be moved indefinitely to 1/x? The general number field seive is relatively recent. If someone comes up with a similar breakthrough again (and this has happened many times over with rsa) your 2048bit keys won’t be secure just as your 128bit rsa keys from the past are no longer secure.
- neets 2y agoI always assumed in the Anime “Ghost in the Shell Stand Alone Complex” they used, “Barrier Mazes” rather than cryptography for a reason
- dhosek 2y agoI remember telling my high school students that if they found an efficient way to factor large numbers, they would destroy capitalism and a large number of them got very excited about number theory after that.
- jaystraw 2y agoi'm glad they got excited about number theory, but how would improved factoring algorithms destroy capitalism??
- thedangler 2y agoIf someone did find out how to do it, do you think they would make it public?
- devnull3 2y agoIf there is a such a breakthrough then the hackers or even spy agencies will not reveal it. They will instead silently make use of it. It will be essentially a backdoor for them.
- heyoni 2y agoWouldn’t it likely originate from academia? If so you can bet that the work will be published just like this one.
- Retr0id 2y agoIt's hard to know where things are today, but historically, public academia has often been behind the true cutting edge of cryptanalysis. For example, take a look at the history of Differential Cryptanalysis https://en.wikipedia.org/wiki/Differential_cryptanalysis https://en.wikipedia.org/wiki/Differential_cryptanalysis > The discovery of differential cryptanalysis is generally attributed to Eli Biham and Adi Shamir in the late 1980s, who published a number of attacks against various block ciphers and hash functions, including a theoretical weakness in the Data Encryption Standard (DES). It was noted by Biham and Shamir that DES was surprisingly resistant to differential cryptanalysis, but small modifications to the algorithm would make it much more susceptible. > In 1994, a member of the original IBM DES team, Don Coppersmith, published a paper stating that differential cryptanalysis was known to IBM as early as 1974, and that defending against differential cryptanalysis had been a design goal. According to author Steven Levy, IBM had discovered differential cryptanalysis on its own, and the NSA was apparently well aware of the technique.
- heyoni 2y agoThat is both impressive and disappointing. I'm so used to seeing large corporations publishing AI models and other techniques (like Ghidra) that I assumed a finding like that would be disseminated to the public. But you're right, something that could be used to decrypt modern ciphers could very well be kept secret for as long as possible.
- nobodyandproud 2y agoIIRC, Elliptic curve cryptography doesn’t rely on factorization, so there’s already an interim PKC solution in place. I also recall there were many problems with the ECC based algorithms or at least the implementations—something about discrete approximations weakening security? Far beyond my comprehension
- k__ 2y agoThere is also lattice-based cryptography.
- AnotherGoodName 2y agoECC is very closely related though (hidden abelian subgroup problem is the category they both fall under). It’s actually concerning because rsa was broken. The reason we’re not using 128bit rsa keys anymore and instead using 2048bit keys is because rsa was broken by the general number field sieve. We’re now all using ecc to avoid working with very large keys but there’s no mathematical proofs that ecc is anymore difficult. In fact it’s widely believed to be the same problem underneath. That may surprise people. ECC, the thing we rely on, is not proven except by the fact that no one has broken it yet just like rsa was until someone broke it.
- dakiol 2y agoIsn’t this the same as zero-day vulnerabilities? Typically only a bunch of people out there know how to take advantage of such holes, and eventually they get fixed. I guess if public key cryptography gets broken, only a bunch of people would know how to take advantage of it, and eventually it would get fixed.
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- AnotherGoodName 2y agoThat happened many many times over with rsa! The us government used to restrict export of long rsa keys. At one point much of the world was using 128bit rsa keys but Dixon method had everyone scrambling to use 512bit keys. Then the special number field drive had us all scrambling to use 1024bit keys and the general number field seive again had us scrambling to get to 2048bit keys.l and that really wasn’t that long ago relatively speaking. Check out rsa encryption hardware from the 80s. They are really proud of some of the hardware that can do 512bits! (Useless today) https://people.csail.mit.edu/rivest/pubs/pubs/Riv84.pdf https://people.csail.mit.edu/rivest/pubs/pubs/Riv84.pdf The special and general number field seize complexity statements are a few constants in difference. Look at those constants. Do they seem to be some root limit to you? Is it really that unlikely that there’s not a way to reduce those further making even 2048bit keys useless? You don’t need to ask “what would happen if RSA broke” because those of us who have been through this many times now can straight up tell you. You’ll be scrambling to once more bump up the key size and you’ll be auditing all the potential data leaked.
- simpaticoder 2y agoRSA failures with bit-depth were a matter of degree; a prime number factorization break-through would be a matter of kind.
- AnotherGoodName 2y agoIt’s not log(n) but still a break since we were literally using lower bit strength than was trivially factorable thanks to mathematical advances and to the point of thinking RSA 2048 is safe, well we once thought that about 128bit RSA. If the above pans out like the general number field seive did we may yet need to move the goal posts further. And we really really shouldn’t be surprised if it happens since it’s happened so many times already.
- MobiusHorizons 2y agoI believe this was one of the reasons for the broad adoption of elliptic curve based cryptography. The mechanism is not based on prime numbers, so smaller keys were adequate, and it was hoped that they might avoid future degradation due to prime factorization research. Of course they could still be vulnerable to their own attacks, but it still requires an attacker to expend more resources.
- rainbowzootsuit 2y agoSince Setec Astronomy closed operation, we've been okay.
- BigParm 2y agoDo the mathematicians not prove that this is impossible before we all adopt a cryptosystem? Like I don't think everyone started using RSA on a prayer
- tomtomastom 2y agoNo it's quite hard to prove that
- throwaway81523 2y agoThis is from May and there was a better article in Quanta already discussed here. https://www.quantamagazine.org/sensational-proof-delivers-new-insights-into-prime-numbers-20240715/ https://www.quantamagazine.org/sensational-proof-delivers-ne...
- jhncls 2y agoDiscussion: https://news.ycombinator.com/item?id=40981272 https://news.ycombinator.com/item?id=40981272 There are 6 comments; the last one is clearly the most interesting: a link to a discussion by Terence Tao https://mathstodon.xyz/@tao/112557249982780815 https://mathstodon.xyz/@tao/112557249982780815 Terence Tao also provides links to a presentation by James Maynard and Larry Guth: https://www.ias.edu/video/new-bounds-large-values-dirichlet-polynomials-part-1 https://www.ias.edu/video/new-bounds-large-values-dirichlet-... and https://www.ias.edu/video/new-bounds-large-values-dirichlet-polynomials-part-2 https://www.ias.edu/video/new-bounds-large-values-dirichlet-...
- riidom 2y agoI am a bit disappointed that the article doesn't explain what the introductory illustration about Sack's spiral has to do with any of this.
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- alkyon 2y agoSack's spiral is a variant of Ulam's spiral, he discovered in 1963 using MANIAC II. Edit: https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral
- munificent 2y agoThe somewhat cynical but honest answer is that all articles need some kind of pretty image at the top because when you share a link on any social media platform, the platform looks for an image to use as a thumbnail. If it doesn't find one, it gets posted as just a plain link and almost no one clicks it. This is why Medium requires an image on every post, why every programming blog out there puts random pictures of laptops and coffee cups at the top of their articles, why Unsplash is so popular, and now why AI-generated images at the top of posts are so common. It's dumb.
- wood_spirit 2y agoI’m curious as I hadn’t seen it before and it’s gripping: Is the patterns showing in a polar plot of the prime numbers a recent discovery or is it long known and just used as an illustration? What is it called and what is its history?
- taneliv 2y agohttps://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral for more reading, Sacks spiral is from 1994.
- zimpenfish 2y agoNumberphile[1] and 3b1b[2] (with a particularly good explanation of why it happens) have done good videos on the prime spiral. [1] https://www.youtube.com/watch?v=iFuR97YcSLM https://www.youtube.com/watch?v=iFuR97YcSLM [2] https://www.youtube.com/watch?v=EK32jo7i5LQ https://www.youtube.com/watch?v=EK32jo7i5LQ
- thom 2y agoI’m both a layman and a simpleton, but seeing Guth’s comments, surely it can’t be a new idea that the fundamental interpretation of primes is something to do with waves and harmonics?
- impendia 2y agoAnalytic number theorist here -- "Fundamental interpretation of primes" is a bit much, but this has been understood for a long time. The short version of the story is - The primes are closely related to the Riemann zeta function, which is more-or-less cobbled out of them; - The Riemann zeta function has a lot more symmetry than one might initially expect, and harmonic analysis is how you prove this; - The (still unproved) Riemann Hypothesis is that the zeta function has still more symmetry beyond what we've been able to prove.
- samdung 2y ago[flagged]
- TechVoyager42 2y ago[dead]
- xanderlewis 2y agoBegone, LLM slop.
- keepamovin 2y agoPeople always think the structure of primes is complex, but it's not really, it's just a recursive structure of the magnitude gaps not landed on by multiples of previous gaps. It doesn't make it easier to "predict" without tracking all prior gaps, but it's not essentially a complex structure. Kind of funny that like such a simple structure is so elusive. Sorta like how the 3n + 1 sequence gives rise to such complexity. Or the logistic map with its parameter above the threshold.
- lmpdev 2y agoAh yes nothing simpler than providing the foundational theory to one of the most rigorous and intellectually intimidating areas of mathematics - number theory /s
- odyssey7 2y agoThey’ve got the fundamental theorem of arithmetic. What more could they want?
- keepamovin 2y agoI think that misses the point which is that the simplicity is overlooked in the common descriptions of primes as "random" or a great "mystery".
- odyssey7 2y agoYes, it’s difficult to predict where such an understanding might lead. If it reframes and redefines all of number theory, then we might call it one component of the foundational theory of number theory. Analogously, if someone proves that P = NP, then that will be great, but the significance of lambda calculus and Turing completeness will remain. If the proof is constructive and practical, we’ll just have to reprioritize and update the list of algorithms we teach to undergrads, issue performance-enhancement updates to some software libraries, and patch any security vulnerabilities. Otherwise, we’ll only need to change a chapter or two in the Theory of Computation courses that universities are increasingly deprioritizing.
- testaccount135 2y ago"they pulled some unorthodox moves to finally break Ingham’s bound" Why is taking methods from other fields an unorthodox move? I come from an engineering background an there it is the common case. The usage of harmonic analysis is a staple in many fields (audio, waves, electrical analysis, statistics) and of course the algorithms are pure math under the hood. If I want to find a reaccuring structure in an underlying system, wouldn't it be normal to try different plotting techniques and choose the one that suits my problem best?
- remus 2y agoUnorthodox is maybe a bit strong, but what they're saying is that it's a novel application of an existing technique from another field. Fairly often you will see big breakthroughs like this in maths, where someone has the insight to see that there are parallels between two seemingly unconnected areas of maths, and you can use ideas from one area to give you insight in to the other area. The tricky bit is that these connections between areas are not usually obvious, so to see the similarity can require a considerable step in understanding.
- sameoldtune 2y agoIt’s kind of silly. Just a reporter reporting. You could say that every discovery in mathematics involves some “unorthodox” move, since the orthodoxy is all that is known so far.
- gavagai691 2y ago"Save for Maynard, a 37-year-old virtuoso who specializes in analytic number theory, for which he won the 2022 Fields Medal—math’s most prestigious award. In dedicated Friday afternoon thinking sessions, he returned to the problem again and again over the past decade, to no avail. At an American Mathematical Society meeting in 2020, he enlisted the help of Guth, who specializes in a technique known as harmonic analysis, which draws from ideas in physics for separating sounds into their constituent notes. Guth also sat with the problem for a few years. Just before giving up, he and Maynard hit a break. Borrowing tactics from their respective mathematical dialects and exchanging ideas late into the night over an email chain, they pulled some unorthodox moves to finally break Ingham’s bound." This quote doesn't suggest that the only thing unorthodox about their approach was using some ideas from harmonic analysis. There's nothing remotely new about using harmonic analysis in number theory. 1. I would say the key idea in a first course in analytic number theory (and the key idea in Riemann's famous 1859 paper) is "harmonic analysis" (and this is no coincidence because Riemann was a pioneer in this area). See: https://old.reddit.com/r/math/comments/16bh3mi/what_is_the_big_picture_behind_analytic_number/jzfaku9/ https://old.reddit.com/r/math/comments/16bh3mi/what_is_the_b.... 2. The hottest "big thing" in number theory right now is essentially "high dimensional" harmonic analysis on number fields https://en.wikipedia.org/wiki/Automorphic_form https://en.wikipedia.org/wiki/Automorphic_form, https://en.wikipedia.org/wiki/Langlands_program https://en.wikipedia.org/wiki/Langlands_program. The 1-D case that the Langlands program is trying to generalize is https://en.wikipedia.org/wiki/Tate%27s_thesis https://en.wikipedia.org/wiki/Tate%27s_thesis, also called "Fourier analysis on number fields," one of the most important ideas in number theory in the 20th century. 3. One of the citations in the Guth Maynard paper is the following 1994 book: H. Montgomery, Ten Lectures On The Interface Between Analytic Number Theory And Harmonic Analysis, No. 84. American Mathematical Soc., 1994. There was already enough interface in 1994 for ten lectures, and judging by the number of citations of that book (I've cited it myself in over half of my papers), much more interface than just that! What's surprising isn't that they used harmonic analysis at all, but where in particular they applied harmonic analysis and how (which are genuinely impossible to communicate to a popular audience, so I don't fault the author at all). To me your comment sounds a bit like saying "why is it surprising to make a connection." Well, breakthroughs are often the result of novel connections, and breakthroughs do happen every now and then, but that doesn't make the novel connections not surprising!
- xpil 2y agoJust use 42 everywhere
- codeduck 2y ago[flagged]
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- seanhunter 2y agoThat this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question. - Gauss
- trashtester 2y agoI think Geometric Algebra [1] provide a more natural approach to "imaginary" numbers than Gauss does above. Not only does these algebras give a more intutive understanding of "imaginary" numbers as rotation in a plane (and not simply an alternative R2). They also extend nicely into all sorts of applications in Physics, Machine learning, etc where Lie groups are needed. And there is nothing preventing us from defining e1*e2 as i, and use the regular notation for Complex Analysis where the Group Theory aspects are not needed. [1] https://www.youtube.com/watch?v=PNlgMPzj-7Q https://www.youtube.com/watch?v=PNlgMPzj-7Q
- g15jv2dp 2y agoInstead of expressing your knowledge and superiority by metaphorically rolling your eyes without contributing anything, how about you give a better explanation? Because honestly, I find these two kind-of "okay". And because this is HN, based on past experience, I need to preface with the fact that I'm a professor of math. So no need to start by questioning my knowledge on the topic, just get straight to the point.
- codeduck 2y ago> So no need to start by questioning my knowledge on the topic, just get straight to the point. Undergrad physics, so you are obviously more versed in the field than I am. But, speaking as someone with some small background in this, I would hope that an article on 'science.org' that mentions Gauss and Riemann would go into slightly more detail than i = sqrt(-1). Even a two-liner description of the real and imaginary plane would be an improvement and would possibly motivate people who knew very little about the area into going and researching. The entire article is about possible periodicity in prime numbers - why, then, omit one of the most important things about complex numbers and their relationship to periodic systems? Euler's formula is a beautiful thing, and I say that as a luddite. And as for the harmonic analsys as "something in physics used to separate sounds and their notes" - I mean... that's like saying "Moby Dick" is a book about a whale. Yes, it's technically correct, but there is such a lost opportunity to describe just how all-encompassing Fourier Analysis is and how it naturally ties back to the complex numbers mentioned previously. So, as demanded, here: For inputs, the function takes complex numbers, which are two-dimensional numbers with one coordinate on the real number plane and the other on the so-called "imaginary" plane. Complex numbers are fundamental to the description of many periodic systems such as waves, cycles, orbits etc. harmonic analysis, which is a discipline that originated as the study of the composition of sound but was extended by mathematicians like Taylor and Fourier into a broad system of numerical analysis that is widely used in everything from number theory to neuroscience. It would have taken very little additional effort, but the results would be rather different - showing paths forward rather than walls saying "this is all that there is to this".
- nyc111 2y ago“This left a small but unsettling possibility that many zeros could be hiding out right at three-quarters.” Ok, but if zeros there are found some mathematicians may as well call them “trivial zeros.” Can there be an objection to that?
- seanhunter 2y agoThis is way above my paygrade, but trivial zeros of the zeta function are at the negative even integers (ie they are of the form s = -2n for some natural number n) because that's what Riemann said in his paper where he made the conjecture[1] This equation now gives the value of the function ζ(s) for all complex numbers s and shows that this function is one-valued and finite for all finite values of s with the exception of 1, and also that it is zero if s is equal to a negative even integer. I don't think people get to retcon some other kind of zero into being trivial. [1] https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-...
- fredgrott 2y agoIf you plot the Gauss and Riemann curves in a specific space you see something more magical.... To see what I am talking about as in trivial and non-trivial zeros see this wikipedia animation https://en.wikipedia.org/wiki/File:Riemann3d_Re_0.1_to_0.9_Im_1_to_51.ogg https://en.wikipedia.org/wiki/File:Riemann3d_Re_0.1_to_0.9_I... Basically, it implies that there is another relationship between real and imaginary numbers we have not yet stumbled upon.... And,this has implications upon finding the gravity theory as Riemann math is involved in quantum mechanics.... Strange science that primes is or might be involved in gravity theory....
- hyperbolablabla 2y agoEvery time I hear about James Maynard it really solidifies my opinion that he's one of those once in a generation geniuses. He's already contributed so much to prime number theory, it really feels like there might be a proof of the Riemann Hypothesis within my lifetime.
- timmb 2y agoSomething inspiring about this: "In dedicated Friday afternoon thinking sessions, he returned to the problem again and again over the past decade, to no avail."
- eismcc 2y agoI recall that Richard Hamming used to also reserve Friday afternoons to deep/big thinking. Sounds wonderful.
- hennell 2y agoFriend of mine worked used to block off his friday afternoons for 'weekly review'. Which was part big thinking, part end of week nap, and mostly avoiding colleagues who had tricky tasks 'needed first thing monday' they had forgotten to bring up before.
- NiloCK 2y agoI've been fascinated by this question since I learned the sieve of eratosthenes as a kid. The meta logic of it is so simple: Primes are specifically the numbers that are left over after the structured numbers (composite) ones are removed. Everything - [structured numbers] = [ chaos? the abyss? some meta structure? ]
- igtztorrero 2y ago3 years ago, somebody post on HN, an animation about prime numbers, it was beautiful looking how prime numbers show a pattern, it looks like the image in this article
- gxs 2y agoReminds me of a story where some egghead friend of mine had a friend that was a researcher at a state school in California. In his research, he found something like getting unenriched uranium to react (please excuse my complete lack of familiarity with the subject). Apparently some government agency stepped in, classified his research and asked him to start. Makes me where else this might have happened - there must be some interesting stuff out there.
- huyvanbin 2y ago> “At first sight, they look pretty random,” says James Maynard, a mathematician at the University of Oxford. “But actually, there’s believed to be this hidden structure within the prime numbers.” What would the pattern of primes hypothetically look like? Is there expected to be some kind of closed form formula? If the Riemann hypothesis were proven, what would be the next step to understanding the distribution? Or is the proof itself expected to hold this answer?
- RIMR 2y agoHow is this any different from Sach's original work from 2003? https://naturalnumbers.org/sparticle.html https://naturalnumbers.org/sparticle.html The organized patterns of primes and composites was an understood feature of the Sack's Spiral since the day he published his findings online.
- markjspivey 2y ago"analyze this for hidden underlying structure or emergent properties" https://chatgpt.com/api/content/file-HFFSXBEAtdR1fbum5ZCEloge https://chatgpt.com/api/content/file-HFFSXBEAtdR1fbum5ZCElog...
- Aachen 2y ago"missing or invalid access token"
- 6gvONxR4sf7o 2y agoOn a slight tangent, this line makes me think about aspects of automated provers that I don’t even know if we’ve begun thinking about: > “It’s a sensational breakthrough,” says Alex Kontorovich, a mathematician at Rutgers University. “There are a bunch of new ideas going into this proof that people are going to be mining for years.” Frequently, a proof of a thing is less interesting as a way to bring rigor than it is as a new way to look at a thing. I wonder if there’s been any work on that side of things in automated mathematics?
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- SillyUsername 2y agoAnd if they crack that, well security is pretty much cracked too...