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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 discove

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
    - 2y ago
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/

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
  - 2y ago

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/