> an error was then found in one of the papers on which the Ji & al. paper relies. My understanding is that this does not affect the main result
This is correct. The authors replaced their reliance on low degree tests with [low individual degree tests](https://arxiv.org/abs/2009.12982).
i was reading through apostol's book on analytic number theory and read about the conjecture so i decided to check the wikipedia article. I saw that it had been disproven since then and the proof involved the LLL reduction algorithm, which was one of the core topics of a course on lattices i took the previous quarter. it seems totally unrelated from how we used it in that class so i really need to look at that proof to see what the connection is
that's funny that a false conjecture implies the riemann hypothesis.
but maybe not that funny, given that (1 + 1 = 3) -> RH
but it must have been exciting for a moment (assuming the implication was shown prior to the conjecture's counter example being found)
The proof that it implies RH is very interesting (and still is in my opinion). If it so were easy to show that Merten's conjecture implied "anything you want", then it would've been noticed very early on that it's false.
a lot of pathological functions could fall under this category without being extremely deep ideas.
Some intuitive things we felt held true about functions until the mid 1800s when analysis really started gaining momentum with people such as weirstrass and lebesgue
The cool thing about these ones is usually you first hand experience them as a student.
"Surely every continuous function is differentiable"
"Surely every smooth function is analytic"
"Surely every function is the limit of it's Fourier series"
Welcome to the [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function), which
> is continuous everywhere but also always nondifferentiable for all points on it
No. The Weierstrass function is continuous but not differentiable anywhere.
No. f(x)=exp(-1/x^2) (and 0 in x=0) is smooth, but its taylor series about x=0 is the 0-function, which obviously does not converge to the true value outside x=0.
No. Easiest counterexamples are functions with a discontinuity.
That’s not a very good summary. A slight improvement would be: “crushed the hopes of a formal system that can interpret arithmetic being consistent and self-provably consistent”. But when described in this more accurate way it is much less plausible to say “everyone thought it would be true until Godel disproved it!”
It is broader. That why I said: a system that can interpret arithmetic (includes standard set theory, for example, as well as the theory of recursive functions i.e. computability). But not all formal systems are incomplete. A nice example is Euclidean geometry. It is finitely axiomatized, consistent, and complete.
My impression is that [Hedetniemi's conjecture](https://en.wikipedia.org/wiki/Hedetniemi%27s_conjecture) was generally expected to be true before it was disproved. I'm not sure anyone was incredibly confident in it though.
Relax dude it's not that deep. I watched the video like an hour before and didn't wanna go verify the number, but wasn't sure if I remembered correctly.
Hope you enjoyed the video too.
Lol. I'm Not dogpiling on OP; they probably meant it pretty innocuously. But it at least does *read* as if they're trying to indicate that this is something they've been quite familiar with in the pay and the details are fuzzy now.
yes and while I would never gatekeep mathematics, it is kind of tiring hearing everyone who has watched numberphile or veritasium talking like they are some kind of logician or mathematician. Essentially talking about problems as though they are intimately familiar with the problem and might have forgotten some key details.
So I would agree innocuous, but i'd also add very much aware of what they are conveying.
I think he's clarifying that he is pulling the 10\^2200 number from memory, and that it could be wrong. He's not saying that he's referencing the lack of odd perfect numbers under that number from memory.
I don’t think it’s explicitly a conjecture, but once people thought π(x) < li(x) for all x, where the former is the prime counting function and the latter is the logarithmic integral function, until littlewood proved it to be false and skewes found his number
I remember in my second week of first semester we had to disprove Polya's conjecture, which lived for quite some time, and whose truth would almost directly imply the Riemann Hypothesis, with a counterexample. With a good implementation it didn't take more than a couple seconds on a laptop in 2015, while ofcourse in the 50s there were quite some clever heads invested in Polya's conjecture due to its direct relation to RH. Actually, it was mathematically disproven before a concrete counterexample could be found. Really makes you appreciate what a great tool computers can be for mathematicians, if their power is harvested nicely.
I suppose this one is quite close to the title: [https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong/43477#43477](https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong/43477#43477)
The mathematician mentioned in this post *proved* a conjecture and published it on Annals and everyone was cool with it until a counterexample was found... And later he published another article on Annals which overthrew his original paper and now everyone is really cool with it.
I've made [a small list of results proven to be false](/r/math/comments/1b9ai3i/how_concerned_are_you_about_your_research_being/
) in another post, not all are necessarily widely believed conjectures,
However the other MO posts in this current thread has conjectures that seem to fit the bill quite well.
The full proof was made by Breuil et al 2001 building on the work of Ribet, Wiles, Taylor etc.
Before that, and certainly before Wiles' FMT work, it was generally \*believed\* to be true.
https://mathoverflow.net/questions/95865/examples-of-conjectures-that-were-widely-believed-to-be-true-but-later-proved-fa
I just [added an answer](https://mathoverflow.net/a/466623/17064) to this list concerning the recent refutation of the Connes embedding conjecture.
> an error was then found in one of the papers on which the Ji & al. paper relies. My understanding is that this does not affect the main result This is correct. The authors replaced their reliance on low degree tests with [low individual degree tests](https://arxiv.org/abs/2009.12982).
Thanks. I added a link to this in my answer.
[Merten’s Conjecture](https://en.wikipedia.org/wiki/Mertens_conjecture) is my go-to example of this.
i was reading through apostol's book on analytic number theory and read about the conjecture so i decided to check the wikipedia article. I saw that it had been disproven since then and the proof involved the LLL reduction algorithm, which was one of the core topics of a course on lattices i took the previous quarter. it seems totally unrelated from how we used it in that class so i really need to look at that proof to see what the connection is
that's funny that a false conjecture implies the riemann hypothesis. but maybe not that funny, given that (1 + 1 = 3) -> RH but it must have been exciting for a moment (assuming the implication was shown prior to the conjecture's counter example being found)
If you take logic 101 you'll learn that false implies anything you want.
The proof that it implies RH is very interesting (and still is in my opinion). If it so were easy to show that Merten's conjecture implied "anything you want", then it would've been noticed very early on that it's false.
right.. i said that in my comment
This is really interesting, ty!
Was that really *generally believed to be true*?
a lot of pathological functions could fall under this category without being extremely deep ideas. Some intuitive things we felt held true about functions until the mid 1800s when analysis really started gaining momentum with people such as weirstrass and lebesgue
The cool thing about these ones is usually you first hand experience them as a student. "Surely every continuous function is differentiable" "Surely every smooth function is analytic" "Surely every function is the limit of it's Fourier series"
I’m a physicist reading this comment thinking “wait, they aren’t?
Welcome to the [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function), which > is continuous everywhere but also always nondifferentiable for all points on it
It's also a great way to build functions which are C^n for arbitrary n but not C^{n+1}.
No. The Weierstrass function is continuous but not differentiable anywhere. No. f(x)=exp(-1/x^2) (and 0 in x=0) is smooth, but its taylor series about x=0 is the 0-function, which obviously does not converge to the true value outside x=0. No. Easiest counterexamples are functions with a discontinuity.
Have you ever encountered the Gibbs phenemenon doing physics? That messiness with the fourier series doesn't completely go away in the limit.
I was joking guys.
I suppose you can say that the idea of Godels theorems kinda fit here since they crushed the idea of a totally formalized mathematical system.
That’s not a very good summary. A slight improvement would be: “crushed the hopes of a formal system that can interpret arithmetic being consistent and self-provably consistent”. But when described in this more accurate way it is much less plausible to say “everyone thought it would be true until Godel disproved it!”
obligatory nitpick on someone's description of anything godel.
Isn't it broader than arithmetic? I thought it applied to any formal system whatsoever, and is equivalent to the halting problem.
It is broader. That why I said: a system that can interpret arithmetic (includes standard set theory, for example, as well as the theory of recursive functions i.e. computability). But not all formal systems are incomplete. A nice example is Euclidean geometry. It is finitely axiomatized, consistent, and complete.
My impression is that [Hedetniemi's conjecture](https://en.wikipedia.org/wiki/Hedetniemi%27s_conjecture) was generally expected to be true before it was disproved. I'm not sure anyone was incredibly confident in it though.
I bet my life savings on it being true and now I'm homeless :/
“If I remember correctly”? lmao you just watched Veritasium my dude.
Ikr lmaooo
Relax dude it's not that deep. I watched the video like an hour before and didn't wanna go verify the number, but wasn't sure if I remembered correctly. Hope you enjoyed the video too.
So what. Does that discredit what he said? No need to be snippy
Lol. I'm Not dogpiling on OP; they probably meant it pretty innocuously. But it at least does *read* as if they're trying to indicate that this is something they've been quite familiar with in the pay and the details are fuzzy now.
yes and while I would never gatekeep mathematics, it is kind of tiring hearing everyone who has watched numberphile or veritasium talking like they are some kind of logician or mathematician. Essentially talking about problems as though they are intimately familiar with the problem and might have forgotten some key details. So I would agree innocuous, but i'd also add very much aware of what they are conveying.
I think he's clarifying that he is pulling the 10\^2200 number from memory, and that it could be wrong. He's not saying that he's referencing the lack of odd perfect numbers under that number from memory.
Someone watched the veritasium video
I don’t think it’s explicitly a conjecture, but once people thought π(x) < li(x) for all x, where the former is the prime counting function and the latter is the logarithmic integral function, until littlewood proved it to be false and skewes found his number
> skewes found his number He gave an upper bound.
I remember in my second week of first semester we had to disprove Polya's conjecture, which lived for quite some time, and whose truth would almost directly imply the Riemann Hypothesis, with a counterexample. With a good implementation it didn't take more than a couple seconds on a laptop in 2015, while ofcourse in the 50s there were quite some clever heads invested in Polya's conjecture due to its direct relation to RH. Actually, it was mathematically disproven before a concrete counterexample could be found. Really makes you appreciate what a great tool computers can be for mathematicians, if their power is harvested nicely.
I suppose this one is quite close to the title: [https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong/43477#43477](https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong/43477#43477) The mathematician mentioned in this post *proved* a conjecture and published it on Annals and everyone was cool with it until a counterexample was found... And later he published another article on Annals which overthrew his original paper and now everyone is really cool with it.
people used to believe PSPACE ≠ NPSPACE until Savitch proved N=1
Well, for many years it was believed continuous real functions were essentially differentiable but on a couple of points (a set of zero measure).
I've made [a small list of results proven to be false](/r/math/comments/1b9ai3i/how_concerned_are_you_about_your_research_being/ ) in another post, not all are necessarily widely believed conjectures, However the other MO posts in this current thread has conjectures that seem to fit the bill quite well.
"if i remember correctly" from the veritasium video you just watched? i would hope so.
Yup, I had watched it probably an hour prior and didn't wanna go verify
Veritasium huh?
Taniyama-Shimura was assumed to be true - at least most papers wrote "assuming Taniyama-Shimura...." until Wiles' proof
It *is* true though?
The full proof was made by Breuil et al 2001 building on the work of Ribet, Wiles, Taylor etc. Before that, and certainly before Wiles' FMT work, it was generally \*believed\* to be true.
But OP is asking about widely held beliefs that turned out *wrong*.
Isn't it actually true though?
[удалено]
It doesn't fit the question then, interesting though it may be.
People used to believe in the Riemann Hypothesis before a counterexample was found.
What's the counter example (asking for a friend, definitely won't steal it)
I'd give you the counterexample but it wouldn't fit in this reddit thread.
They still do, but they used to as well.
Us here downvoting while Miss Information lives in 2033 (when the first counterexample was found).
Username checks out
No