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## The purpose of this question is to collect the most outrageous (or ridiculous) conjectures in mathematics.

An outrageous conjecture is qualified **ONLY** if:

### 1) It is most likely false

(Being hopeless is NOT enough.)

### 2) It is not known to be false

### 3) It was published or made publicly before 2006.

### 4) It is Important:

(It is based on some appealing heuristic or idea; refuting it will be important etc.)

### 5) IT IS NOT just the negation of a famous commonly believed conjecture.

As always with big list problems please make one conjecture per answer. (I am not sure this is really a big list question, since I am not aware of many such outrageous conjectures. I am aware of one wonderful example that I hope to post as an answer in a couple of weeks.)

*Very important examples where the conjecture was believed as false when it was made but this is no longer the consensus may also qualify!*

Shmuel Weinberger described various types of mathematical conjectures. And the type of conjectures the question proposes to collect is of the kind:

## On other times, I have conjectured to lay down the gauntlet: “See,

## you can’t even disprove this ridiculous idea."

**Summary of answers** (updated March, 13, 2017):

There are at least as many primes between $2$ to $n+1$ as there are between $k$ to $n+k-1$

A super exact (too good to be true) estimate for the number of twin primes below $n$.

The set of prime differences has intermediate Turing degree.

Hall's original conjecture (number theory).

Tarski's monster do not exist (settled by Olshanski)

All zeros of the Riemann zeta functions have rational imaginary part.

The Lusternik-Schnirelmann category of $Sp(n)$ equals $2n-1$.

The finitistic dimension conjecture for finite dimensional algebras.

The implicit graph conjecture (graph theory, theory of computing)

(From comments, incomplete list) 20. The Jacobian conjecture; 21. The Berman–Hartmanis conjecture 21. The conjecture that all groups are Sofic; 22 The Casas-Alvero conjecture 23. An implausible embedding into $L$ (set theory). 24. There is a gap of at most $\log n$ between threshold and expectation threshold. 25. NEXP-complete problems are solvable by logarithmic depth, polynomial-size circuits consisting entirely of mod 6 gates. 26. Fermat had a marvelous proof for Fermat's last theorem. (History of mathematics).

2Does the Jacobian conjecture qualify? – Steve Huntsman – 2017-01-17T20:03:55.810

Hmm, I think it qualifies as a comment but not as an answer. For an actual answer I would like "most likely false" to represent a large consensus and not a personal view of the answerer. But once I asked the question my view about what qualifies is just one view in the crowd... – Gil Kalai – 2017-01-17T20:12:03.793

2The Berman–Hartmanis conjecture. – T. Amdeberhan – 2017-01-17T20:20:12.753

"In this paper we try to convince the leader that there is no good reason to believe that the Jacobian Conjecture holds. Although there are several arguments in favor of this conjecture, we show that these arguments haven't got the power to justify the statement that the Jacobian Conjecture holds in general." -van den Essen (1997): http://www.seminariomatematico.unito.it/rendiconti/cartaceo/55-4/283.pdf

– Steve Huntsman – 2017-01-17T20:20:38.477Right! I prefer examples where it is still now commonly believed that the conjecture is false and where the proposer proposes the conjecture genuinely suggesting that it is true. But these two requirements may be too harsh. – Gil Kalai – 2017-01-17T20:22:40.550

5There is a fine line between an outrageous conjecture and a bold conjecture. But still I see the spirit of your interesting question. – Joseph O'Rourke – 2017-01-17T20:25:08.323

1It seems to me there is a conflict between "you can't even disprove this ridiculous idea" and "the proposer proposes the conjecture genuinely suggesting that it is true", so it's not clear to me what exactly you're after. – Gerry Myerson – 2017-01-17T21:59:02.647

1

Incidentally, my question at http://mathoverflow.net/q/101821/1946 was asked in the spirit of this question (but it is too recent to qualify for your 2006 requirement).

– Joel David Hamkins – 2017-01-17T22:01:05.3279I don't think anyone has disproved the ridiculous ideas that there are only finitely many Mersenne composites, or that all the decimal digits of $\pi$ from some point on are sixes and sevens, or that the partial quotients for continued fractions of real algebraic irrationals are always bounded, but I don't think anyone has proposed any of these ideas genuinely suggesting they are true. – Gerry Myerson – 2017-01-17T22:19:13.167

I don't understand what you mean by point 5. By definition, the negation of any commonly-believed-false statement is a commonly-believed-true statement, isn't it? – Federico Poloni – 2017-01-17T22:32:11.130

19The answers below all look of interest, as does the question. And, to boot, this is Community Wiki. Why not keep it open? – Lucia – 2017-01-17T22:49:23.773

1It depends on the utility of this question. As a short term diversion to appeal to some of the forum community it serves quite well. As part of a database of questions and answers for future reference by the interested-in-mathematics consumer, I think it is too based in opinion and belongs on a blog. If the intent were to set some challenge questions to spur research, then I think the question should be reworded. As it stands now, it isn't much better than an opinion poll. Gerhard "MathOverflow Isn't Question And Opinion" Paseman, 2017.01.17. – Gerhard Paseman – 2017-01-17T23:02:19.833

5@GerhardPaseman I think it is not too uncommon for good mathematicians, working in or near an area, to nevertheless not know about various conjectures. Especially if there has been recent development in tangentially-related areas, this type of list very well might lead to some of these conjectures being refuted. I support keeping the question open. – Theo Johnson-Freyd – 2017-01-17T23:36:34.360

1And besides, I will learn things from reading it! – Theo Johnson-Freyd – 2017-01-17T23:36:51.103

@Theo, in which case, let's rewrite the question to fit both a good intent of the asker and the good intent of MathOverflow. As it is currently written, I am not sure either is achieved. Gerhard "Being Ridiculous Can Serve Research" Paseman, 2017.01.17. – Gerhard Paseman – 2017-01-18T00:30:51.203

1

I feel like a number of famous, "elementary" conjectures, while often believed to be true, have no particular (philosophical) reason to be true, and thus from a cynical perspective might be described as likely false. Examples are the Collatz conjecture (https://en.wikipedia.org/wiki/Collatz_conjecture) and the union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture)

– Sam Hopkins – 2017-01-18T04:56:04.1535What's next - a big list with Trump tweets concerning mathematics? Does this outrageous conjecture of mine count as an example? – Franz Lemmermeyer – 2017-01-18T05:39:21.590

@Sam, I don't know about "philosophical", but I think there are good mathematical reasons for Collatz to be true. – Gerry Myerson – 2017-01-18T05:39:27.290

1I agree with Theo on that. E. g. Socrates was, I think, The asker number one. The way I understand it, the question is about conjectures of Socratic quality. If the things Socrates asked would be only pedagogic challenges and not the things he really burningly wanted to know, he would not be Socrates. And recall what happened to him. By the way there is a Socratic badge here on MO. A golden one ;) – მამუკა ჯიბლაძე – 2017-01-18T05:40:36.183

2This one doesn't count because it status is settled, but from what I understand of the history of mirror symmetry, when the physicists first proposed it, Yau for one initially thought that it was too outrageous to be true. Along similar lines, I believe that Tao initially thought that the phenomenon of compressed sensing couldn't possibly be true, and that the Candes-Romberg-Tao paper was born out of Tao's attempts to find a

disproof. – Timothy Chow – 2017-01-18T21:59:46.0771@TimothyChow The Tao example is a really good one (and with ample documentation), even if it doesn't meet all the conditions of the OP. – Todd Trimble – 2017-01-19T02:36:27.440

1The Kahn-Kalai conjecture (the general one for Boolean functions) almost qualifies here, doesn't it – user36212 – 2017-01-19T23:58:28.360

1Not an expert on this subject, but "all groups are sofic" might potentially qualify here. – Terry Tao – 2017-01-21T22:59:49.977

I'm far from being an expert on this but perhaps Casas-Alvero conjecture is relevant. https://en.wikipedia.org/wiki/Casas-Alvero_conjecture . Although the most outrageous aspect of it is probably the date it was first conjectured (2001 !!!).

– Saal Hardali – 2017-01-21T23:40:48.920Dear user36212, yes we both regards the conjectures (both for Boolean functions and for graphs) as fairly outrageous and would be very interested in counterexamples. Dear Timothy and Todd, these are good examples and I am certainly not too fussed about meeting all my conditions. – Gil Kalai – 2017-01-22T10:42:01.263

1@GilKalai : It's sort of too late now, but I wonder if another (and possibly better) way to phrase your question would be, what conjectures were regarded as impossibly bold or optimistic at the time they were first made? This would allow for both conjectures that have been settled and conjectures that are still open. It would also allow for bold conjectures that we've gotten used to but that were considered outrageous at first. And it would eliminate statements that nobody has ever believed. – Timothy Chow – 2017-01-22T21:26:23.793

This question reached the sidebar and as such has received a large influx of people who usually don't visit this site. Many conjectures in here are described in ways only mathematicians can understand them. I was wondering if the people who posted an answer already could clarify it in a way that people who aren't well versed in math could understand the conjecture? – Nzall – 2017-01-24T10:19:56.990

Dear @Nzall , what does it mean to reach the sidebar? – Gil Kalai – 2017-01-24T11:15:17.890

@GilKalai It means that the question has had enough activity on this site to be viewed as a "hot network question", which in turn means that it's eligible to appear in the list of questions which appear in the sidebar, beneath the meta highlights and the linked questions. This list is effectively a "best of Stack Exchange", and a lot of people tend to check out the questions on that bar, especially if the title is interesting or (like this question) clickbaity. It means that a lot of users from other exchanges on the network will look at the question and potentially give their input. – Nzall – 2017-01-24T11:26:56.900

@GilKalai The biggest consequence is that the question gets a lot of new readers, votes, comments and potentially answers, many of which haven't had much more than high school or maybe first grade college education on the topic. Making the topic a bit more understandable for those users may introduce them to aspects of mathematics they didn't know exist and may encourage them to look further on the site. – Nzall – 2017-01-24T11:31:34.520

@Nzall, certainly it could be a good idea to add elementary explanations for at least some of the answers. It is not so easy but worth the effort. – Gil Kalai – 2017-01-24T13:37:30.823

Dear Terry and Saal, indeed "all groups are sofic" is a famous conjecture which might be suitable. The Casas-Alvero conjecture strikes me as a good example as well but I dont know anything about it. – Gil Kalai – 2017-01-24T16:24:48.457

Why 2006? It seems rather arbitrary. Perhaps instead "is at least 10 years old" which will allow for more recent "outrageous" answers as time goes on. – Damien – 2017-01-25T03:36:53.823

Hi Damien, I agree with that. I also agree with Timothy's comment regarding unsettled conjectures. I am a little worried that adding settled conjectures would have made the question too board (but those can be mentioned in comments and also in other questions.) – Gil Kalai – 2017-01-25T10:09:29.913

2[Disclaimer: I haven't really done any serious maths since my degree many moons ago, and yes, I arrived here via the sidebar.] I'm surprised noone mentioned Fermat's famous "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain" quote. I guess it's not

technicallya conjecture, but there's an implicit conjecture in there that there exists a marvellous proof which would have been attainable before 1621, and this fits all the criteria of the question, as well as being presumably in keeping with the spirit of the question. – Adam Spiers – 2017-01-26T16:09:34.423Adam, this was indeed a conjecture but it does not fit the question because: a) It was settled (point 2), b) it was not believed to be false (point 1). But I agree that beside the formalities Fermat's conjecture and Fermat's narrow margin claim were quite outrageous! – Gil Kalai – 2017-01-26T17:28:27.913

3@GilKalai No, you misunderstood. I was proposing Fermat's

quoteabout his Last Theorem as an answer to the question, not the Theorem itself. The implicit conjecture that there exists a marvellous proof which would have been attainable before 1621 isnotsettled (point 2), and IIUC is widely believed to be false. – Adam Spiers – 2017-01-29T16:06:05.5371Hmm, I see. This is indeed an outrageous conjecture about mathematics and its history :) – Gil Kalai – 2017-01-29T18:55:58.093

@FranzLemmermeyer I recently made a conjecture that if time goes to infinity trump will make a misspelling on twitter at some point and type abelian instead of a billion. The legend goes that this already happend and that this caused all multiplications at quantum level to be non-commutative, making the universe ridiculously hard to understand. On the other hand there are people thinking that trump tweeting about math is fake news. – M.D. – 2018-07-29T23:47:58.883