[HN Gopher] Big advance on simple-sounding math problem was a ce...
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Big advance on simple-sounding math problem was a century in the
making
Author : isaacfrond
Score : 73 points
Date : 2024-10-15 13:10 UTC (9 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| youoy wrote:
| I will always find these type of explorations fascinating. Number
| theory is so mysterious. I liked these two sentences from the
| article:
|
| > "Mathematics is not just about proving theorems -- it's about a
| way to interact with reality, maybe."
|
| This one I like it because in the current trend of trying to
| achieve theorem proving in AI only looking at formal systems,
| people rarely mention this.
|
| And this one:
|
| > Just what will emerge from those explorations is hard to
| foretell. "That's the problem with originality," Granville said.
| But "he's definitely got something pretty cool."
|
| When has that been a "problem" with originality? Hahah but I
| understand what he means.
| nomilk wrote:
| How do mathematicians come to focus on seemingly arbitrary
| quesrions:
|
| > another asks whether there are infinitely many pairs of primes
| that differ by only 2, such as 11 and 13
|
| Is it that many questions were successfully dis/proved and so
| were left with some that seem arbitrary? Or is there something
| special about the particular questions mathematicians focus on
| that a layperson has no hope of appreciating?
|
| My best guess is questions like the one above may not have any
| immediate utility, but could at any time (for hundreds of years)
| become vital to solving some important problem that generates
| huge value through its applications.
| whatshisface wrote:
| Mathematicians don't care about utility, they care about
| whether a problem is on the boundary of solvability.
| thfuran wrote:
| It's really only the applied mathematicians who tend to care
| much at all about utility. If you ask a bunch of theoretical
| mathematicians why they're working on what they're working on,
| you'll probably get many saying that it's fun or interesting,
| some saying that they needed to publish something, and very few
| answers related to utility at all.
| blessedwhiskers wrote:
| There's something quite interesting about the problems in
| number theory especially. The questions/relationships sometimes
| don't seem useful at all and are later proven to be incredibly
| useful. Number Theory is the prime example of this. I believe
| there's a G H Hardy quote somewhere, about Number Theory being
| obviously useless, but could only find it from one secondary
| source, although it does track with his views expressed in A
| Mathematician's Apology[1] - "The theory of Numbers has always
| been regarded as one of the most obviously useless branches of
| Pure Mathematics."
|
| You can find relationships between ideas or topics that are
| seemingly unrelated, for instance, even perfect numbers and
| Mersenne primes have a 1:1 mapping and therefore they're
| logically equivalent and a proof that either set is either
| infinite or finite is sufficient to prove the other's
| relationship with infinity. There's little to no intuitive
| relationship between these ideas, but the fact that they're
| linked is somewhat humbling - a fun quirk in the fabric of the
| universe, if you will.
|
| [1] https://en.wikipedia.org/wiki/A_Mathematician's_Apology
| vlovich123 wrote:
| > No one has yet found any war-like purpose to be served by
| the theory of numbers or relativity or quantum mechanics, and
| it seems very unlikely that anybody will do so for many
| years.
|
| G.H.Hard. Eureka, issue 3, Jan 1940
| l33t7332273 wrote:
| Less than 100 years later, we stand waiting for nuclear
| bombs guided by GPS to be launched when the authorization
| cryptographic certificate is verified.
| eigenket wrote:
| Nuclear weapons (based on quantum mechanics and special
| relativity) were used less than 6 years after that quote.
| vlovich123 wrote:
| And number theory was critical to breaking enigma. So
| they were all used within 5 years.
| l33t7332273 wrote:
| Was it? My understanding was that they didn't use number
| theoretic approaches.
| 9question1 wrote:
| https://arxiv.org/pdf/math/0702396 is a very thorough answer to
| this question by a very well respected mathematician
| QuesnayJr wrote:
| I think if you are fascinated by primes and look at lists of
| primes, you would eventually notice that twin primes keep
| happening, but they get further and further apart. The first
| time someone published the conjecture that there are infinitely
| many is Polignac in 1849, but I'm sure someone wondered before.
|
| If you are fascinated by primes, then you just want to know the
| answer, independent of any application.
| ykonstant wrote:
| The answer is historical evolution. To you, the problem may
| appear arbitrary, like physicists studying some "obscure
| phenomenon" like the photoelectric effect may have seemed to
| outsiders. But (far, far) behind the scenes, there is a long
| and winding history of questions, answers and refinements.
|
| Knowing that history illuminates the context and importance of
| problems like the above; but it makes for a long, taxing and
| sometimes boring read for the unmotivated, unlike the sexy
| "quantum blockchain intelligence" blurbs or "grand unification
| of mathematics" silliness in pure math. So, few popularizations
| care to trace the historical trail from the basics to the state
| of the art.
| madars wrote:
| It's not arbitrary at all! We know that primes themselves get
| rarer and rarer (density of primes < N is ~1/log(N)), so it is
| natural to ask whether the gaps between them must also
| necessarily increase and, in general, how are they spaced.
| l33t7332273 wrote:
| We know the primes are rich in arithmetic progressions (and
| in fact, any set with positive "upper density" in the primes
| is also rich in arithmetic progressions).
|
| So we do know that there are 100,000,000,000! primes that are
| equidistant from one another, which is neat.
| Ar-Curunir wrote:
| Not everybody is motivated by applications. Caring about
| knowledge for knowledge's sake is a thing, you know.
| andrepd wrote:
| It's the classic Feynman quote (it was about physics, but
| applies to mathematics): it's like sex, it may give us some
| practical results but that's not why we do it ;)
| throw_pm23 wrote:
| With primes I think a general question is to what extent they
| are "random". Clearly, they are defined by a fixed rule, so
| they are not actually random, but they have a certain density,
| statistics, etc. So we can try to narrow down in what sense
| they are like random numbers and in what sense not. Do they
| produce a similar "clustering" or (lack thereof) as if we would
| generate random numbers according to some density, etc. I see
| the twin primes question fitting in such a theme. Happy to
| learn if a number theorist has more insight.
| card_zero wrote:
| That particular one (twin primes) is one of a set of four,
| selected for being "unattackable" in 1912, and still unsolved.
|
| https://en.wikipedia.org/wiki/Landau%27s_problems
| paulpauper wrote:
| I wish the math was explained better . I know the format is
| limited ands it will go over most people's heads but it does not
| do the matter justice.
| _zoltan_ wrote:
| I have an MSc in math but haven't managed to grasp modular
| forms, which is needed for the proof.
|
| If anybody has a learning path or a primer recommendation for
| modular forms (assuming formal math education), I'd be very
| interested in reading more about it.
|
| But as such I doubt it can be more accessible to everybody.
| zyklu5 wrote:
| As a first pass, check out the pop history book called
| Fearless Symmetry by Ash and Gross.
|
| Next, and perhaps I shouldn't be suggesting it, Serre's A
| Course in Arithmetic. Despite it's reputation of terseness it
| is a great book by one of history's great mathematicians and
| worth every sweat and tear spent over its short number of
| pages.
|
| Yet another way is to see the lectures by Richard Borcherds
| (who won the fields for the moonshine conjecture) on youtube.
|
| Finally, since this hacker news check out William Stein's
| Modular Forms: A Computational Approach (pdf online)
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