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Big advance on simple-sounding math problem was a century in the making
by isaacfrond
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.
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.
That particular one (twin primes) is one of a set of four, selected for being "unattackable" in 1912, and still unsolved.
As it happens, the problem the article is about (primes of the form n^2+1) is also one of that set of four (Landau's problems).
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.
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.
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.
> 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
Less than 100 years later, we stand waiting for nuclear bombs guided by GPS to be launched when the authorization cryptographic certificate is verified.
Nuclear weapons (based on quantum mechanics and special relativity) were used less than 6 years after that quote.
And number theory was critical to breaking enigma. So they were all used within 5 years.
Was it? My understanding was that they didn’t use number theoretic approaches.
My bad. You’re right. It was group theory and cryptanalysis. Number theory comes in later in the 1970s for public key cryptography (1976 publicly, early 1970s at GCHQ). So the military work on it really started in the late 1960s.
https://arxiv.org/pdf/math/0702396 is a very thorough answer to this question by a very well respected mathematician
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.
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.
If you wonder my mathematicians are so obsessed with the prime numbers, you can think of them as one of the most fundamental mathematical patterns—arising from the interplay of addition and multiplication—that we are just barely skimming the surface of understanding.
You could almost equally ask why physicists are so obsessed with elementary particles.
Mathematicians don't care about utility, they care about whether a problem is on the boundary of solvability.
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.
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.
Not everybody is motivated by applications. Caring about knowledge for knowledge's sake is a thing, you know.
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 ;)
And most programmers don't do either :)
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.
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.
A standard textbook reference is
Diamond, Shurman, A First Course in Modular Forms, Springer GTM vol. 228.
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)
Crafted by Rajat
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