14 ms·

Breakthrough a step toward revealing hidden structure of prime numbers

400 points by igitur 2y ago

NiloCK 2y ago
I'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 ago
3 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 ago
Reminds 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 ago
How 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 ago
On 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?
deleted 2y ago
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deleted 2y ago
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samayk60 2y ago
[dead]
SillyUsername 2y ago
And if they crack that, well security is pretty much cracked too...
EMIRELADERO 2y ago
This 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?
- ertgbnm 2y ago
  Pretty 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 ago
    Are you sure about that? And even if problems can be solved in polynomial time, the constants involved can be prohibitively large.
  - cvoss 2y ago
    Integer 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 ago
  Many 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.
throwaway81523 2y ago
This 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 ago
  Discussion: 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 ago
  I 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.
  - munificent 2y ago
    The 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.
    - ghaff 2y ago
      It's dumb but it's also why there's so much use of AI generated images as you say. I'd add that a lot of blog templates essentially require them and the random individual isn'y going to pay for stock imagery.
thom 2y ago
I’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 ago
  Analytic 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.
wood_spirit 2y ago
I’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 ago
  https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral for more reading, Sacks spiral is from 1994.
- zimpenfish 2y ago
  Numberphile[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
samdung 2y ago
[flagged]
TechVoyager42 2y ago
[dead]
- xanderlewis 2y ago
  Begone, LLM slop.
keepamovin 2y ago
People 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 ago
  Ah 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 ago
    They’ve got the fundamental theorem of arithmetic. What more could they want?
    - keepamovin 2y ago
      I think that misses the point which is that the simplicity is overlooked in the common descriptions of primes as "random" or a great "mystery".
      - woopsn 2y ago
        Well, the sequence 0, 2, 4, ... , 2k, ... is indeed simple, can be recovered starting from the value at an arbitrary index (eg the last one announced). As can (3k), (5k), etc... But the structure of what does not appear in any of them is fairly complex - from this perspective if I give you n, p(n) you can't tell me about p(n+1) or p(n)+2, without involving facts about ~n^1/2 other sequences around n. Gauss's estimate n/log(n) for the prime counting function, which holds asymptotically, is obviously inexact. As is the logarithmic integral. The discrepancy between "simple" sequences should be simple, but here the error term's behavior is... hardly that. With respect, this is an epic undertaking. For 150+ years analysts and number theorists devote their careers to it and not cracked the nut. Although there has been great progress. Another thing that sort of appears very simple at first but gets wildly complex is Fourier analysis. It's just a way of writing functions with a trigonometric basis. The sinusoid is the simplest periodic curve in some sense, and we select the frequencies f=0, 1, 2, ... Okay but this is a basis for... what? It's not simple. Another 200 years. The two are connected. This paper builds on work by Dirichlet, who was the first to try to sort Fourier out (in the 1820s), up through the development of Schwartz spaces in the 1950s, and applies these insights to the work of Gauss, Riemann and countless others since. And we still don't understand the structure (eg bounds) of the error term!
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?
- sameoldtune 2y ago
  It’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 ago
Just use 42 everywhere
codeduck 2y ago
[flagged]
- seanhunter 2y ago
  That 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 ago
    I 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
- deleted 2y ago
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- g15jv2dp 2y ago
  Instead 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 ago
  This 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 ago
If 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 ago
Every 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 ago
Something 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 ago
  I recall that Richard Hamming used to also reserve Friday afternoons to deep/big thinking. Sounds wonderful.
  - hennell 2y ago
    Friend 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.

400 points by igitur 2y ago

I'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 ago

3 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 ago

Reminds 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 ago

How 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 ago

On 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?

deleted 2y ago

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samayk60 2y ago

[dead]

SillyUsername 2y ago

And if they crack that, well security is pretty much cracked too...

EMIRELADERO 2y ago

This 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?

ertgbnm 2y ago
Pretty 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 ago
  Are you sure about that? And even if problems can be solved in polynomial time, the constants involved can be prohibitively large.
- cvoss 2y ago
  Integer 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 ago
Many 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.

throwaway81523 2y ago

This 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 ago
Discussion: 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 ago
I 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.
- munificent 2y ago
  The 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.
  - ghaff 2y ago
    It's dumb but it's also why there's so much use of AI generated images as you say. I'd add that a lot of blog templates essentially require them and the random individual isn'y going to pay for stock imagery.

thom 2y ago

I’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 ago
Analytic 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.

wood_spirit 2y ago

I’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 ago
https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral for more reading, Sacks spiral is from 1994.
zimpenfish 2y ago
Numberphile[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

samdung 2y ago

[flagged]

TechVoyager42 2y ago

[dead]

xanderlewis 2y ago
Begone, LLM slop.

keepamovin 2y ago

People 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 ago
Ah 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 ago
  They’ve got the fundamental theorem of arithmetic. What more could they want?
  - keepamovin 2y ago
    I think that misses the point which is that the simplicity is overlooked in the common descriptions of primes as "random" or a great "mystery".
    - woopsn 2y ago
      Well, the sequence 0, 2, 4, ... , 2k, ... is indeed simple, can be recovered starting from the value at an arbitrary index (eg the last one announced). As can (3k), (5k), etc... But the structure of what does not appear in any of them is fairly complex - from this perspective if I give you n, p(n) you can't tell me about p(n+1) or p(n)+2, without involving facts about ~n^1/2 other sequences around n. Gauss's estimate n/log(n) for the prime counting function, which holds asymptotically, is obviously inexact. As is the logarithmic integral. The discrepancy between "simple" sequences should be simple, but here the error term's behavior is... hardly that. With respect, this is an epic undertaking. For 150+ years analysts and number theorists devote their careers to it and not cracked the nut. Although there has been great progress. Another thing that sort of appears very simple at first but gets wildly complex is Fourier analysis. It's just a way of writing functions with a trigonometric basis. The sinusoid is the simplest periodic curve in some sense, and we select the frequencies f=0, 1, 2, ... Okay but this is a basis for... what? It's not simple. Another 200 years. The two are connected. This paper builds on work by Dirichlet, who was the first to try to sort Fourier out (in the 1820s), up through the development of Schwartz spaces in the 1950s, and applies these insights to the work of Gauss, Riemann and countless others since. And we still don't understand the structure (eg bounds) of the error term!

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?

sameoldtune 2y ago
It’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 ago

Just use 42 everywhere

codeduck 2y ago

[flagged]

seanhunter 2y ago
That 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 ago
  I 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
deleted 2y ago
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g15jv2dp 2y ago
Instead 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 ago
This 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 ago

If 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 ago

Every 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 ago

Something 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 ago
I recall that Richard Hamming used to also reserve Friday afternoons to deep/big thinking. Sounds wonderful.
- hennell 2y ago
  Friend 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.