41
$\begingroup$

Years ago I had the pleasure of witnessing Simon Thomas giving a wonderful talk about Martin's conjecture, which I just now fondly remembered reading this question. Even though I am not well-versed in all (or even many) fields of mathematics I find it quite curious that it was the only time I ever heard about two contradictory "official" conjectures. In chit-chat I definitely heard colleagues expressing different sentiments about which way a certain problem might finally be resolved but my guess is that they usually would not have been eager to put it in writing, possibly not even be happy to "go on the record". Also, the problem in question may have been a quite specific one relevant only to a small number of people...

So...my question is: What are other example pairs(triples?) of that kind you are aware of? They ideally should have appeared in writing but I guess that I would be satisfied with conjectures which are ascribed to people by a nontrivial part of the (respective) mathematical (sub)community.

Improve this question
$\endgroup$
15
  • 12
    $\begingroup$ See mathoverflow.net/a/259862/25028 for a prominent example from number theory. $\endgroup$ Commented Aug 14, 2025 at 19:33
  • 10
    $\begingroup$ Here is another example from low dimensional topology: ldtopology.wordpress.com/2018/04/04/… $\endgroup$ Commented Aug 14, 2025 at 19:36
  • 7
    $\begingroup$ @SamHopkins I believe a preprint of Belousov and Malyutin from only a couple of years later shows that the first conjecture is false, as many commenters on that blog post guessed. $\endgroup$ Commented Aug 14, 2025 at 20:14
  • 9
    $\begingroup$ The obvious next question: Are there known triples of conjectures such that (a) only one of them can be true, or (b) at most two of them can be true but not all three? $\endgroup$ Commented Aug 14, 2025 at 23:55
  • 8
    $\begingroup$ A hypothesis is something temporarily assumed to be true so one can reason about what follows if it is true. A conjecture is something that a mathematician believes is true and is willing to stake at least part of their reputation on its truth value. In the Riemann hypothesis case, I believe Riemann had relatively high standards for how much evidence is required before making a conjecture, and his evidence for the hypothesis was somewhat weak (especially in comparison to what we have now). $\endgroup$ Commented Aug 15, 2025 at 14:55

10 Answers 10

Reset to default
11
$\begingroup$

As also noted earlier in a comment by Sam Hopkins, in Malyutin’s On the Question of Genericity of Hyperbolic Knots it is shown that the “genericity of hyperbolic knots” conjecture — namely, that the proportion of hyperbolic knots among prime knots with at most $n$ crossings tends to $1$ as $n \to \infty$ — is incompatible with several standard conjectures, including additivity of the crossing number under connected sum. The latter is one of the oldest open conjectures in knot theory, and the “genericity of hyperbolic knots” conjecture was formulated explicitly in C. Adams’s The Knot Book (although, as far as I know, it had already been widely believed). The genericity conjecture was later disproved: in Hyperbolic knots are not generic it is proved that the proportion of hyperbolic knots does not tend to $1$.

Improve this answer
$\endgroup$
0
34
$\begingroup$

The Eilenberg-Ganea Conjecture and the Whitehead Conjecture can’t both be true. This was shown by Bestvina and Brady in their 1997 paper. As far as I am aware, both conjectures are still open.

Improve this answer
$\endgroup$
1
  • 24
    $\begingroup$ They're still open, but just to be clear it is possible (and in my opinion, even likely!) that they are both false. $\endgroup$ Commented Aug 14, 2025 at 23:21
28
$\begingroup$

The first Hardy-Littlewood conjecture and the second Hardy-Littlewood conjecture were found to be incompatible in joint work by Douglas Hensley and Ian Richards as recounted by Richards in this paper. The official publication of the Hensley–Richards result seems to be D. Hensley and I Richards, Primes in Intervals, Acta Arithmetics 25 (1974), 375-391.

There are at least two (to my knowledge) conjectured values of $\limsup \frac{p_{n+1}-p_n}{\log^2 p_n}$, depending on which random model of the prime numbers you choose for your heuristic; see e.g. Cramér's conjecture and Terence Tao's blogpost

Improve this answer
$\endgroup$
5
  • 1
    $\begingroup$ I mentioned the first example in a comment but I think this deserves to be posted as a proper answer, so I undeleted this. $\endgroup$ Commented Aug 15, 2025 at 12:28
  • 3
    $\begingroup$ Tao’s blog post covers at least 44 things — where does it cover this $\limsup$? $\endgroup$ Commented Aug 15, 2025 at 14:52
  • 4
    $\begingroup$ @MattF. Conjecture 7 and Prediction 17. $\endgroup$ Commented Aug 15, 2025 at 14:56
  • 6
    $\begingroup$ I don’t think these “two values for lim sup” are in the spirit of the question. It’s not that there would be two competing contemporaneus independent conjectures each of which would imply a different value; rather, the conjecture evolved after some aspects of the original heuristics were found to be inadequate, leading to a refined heuristics. $\endgroup$ Commented Aug 15, 2025 at 17:47
  • 6
    $\begingroup$ Since 1974, and again with the help of (better) computers, the smallest explicit tuple (potential prime number pattern), that will be fatal for either the 1st Hardy-Littlewood conjecture or the 2nd Hardy-Littlewood conjecture, has been located. It is a 447-tuple of width 3159. By the 1st H-L conjecture, primes in this pattern should occur infinitely often, and with the appropriate asymptotic behavior; yet, even a single occurrence of it would violate 2nd H-L conjecture. See Engelsma, K-Tuple Permissible Patterns (web page). $\endgroup$ Commented Aug 16, 2025 at 10:04
23
$\begingroup$

The following came to mind:

It is shown that some plausible conjectures are incompatible:

Improve this answer
$\endgroup$
17
$\begingroup$

This is perhaps not quite what the question is asking for, but it was mentioned in a comment and seems at least tangentially relevant. There are examples of axioms going beyond ZF that contradict each other and yet are both "taken seriously." These axioms are not exactly conjectures in the usual sense of the word, but sometimes people will advocate for them as "basic axioms" to be adjoined to ZF.

  1. The (full) axiom of choice contradicts various other axioms such as the axiom of determinacy and the axiom that "all sets of reals are Lebesgue measurable." Although it is standard mathematical practice to accept the axiom of choice as a basic axiom of mathematics, these other axioms are not necessarily treated as being straightforwardly false, in part because they can be compatible with a weak version of choice.
  2. The existence of measurable cardinals (or even of sharps) is incompatible with $V = L$, the axiom of constructibility. Large cardinal axioms are widely "accepted" by set theorists, but $V=L$ has a few advocates.
  3. Forcing axioms are incompatible with V = Ultimate L (the former imply $2^{\aleph_0} = \aleph_2$ and the latter implies $2^{\aleph_0} = \aleph_1$). There is a nice Quanta Magazine article that gives a nontechnical overview.
Improve this answer
$\endgroup$
4
  • 4
    $\begingroup$ "The (full) axiom of choice contradicts various other axioms..." There's a joke phrasing of this that I have always enjoyed: "The axiom of choice is obviously true, the well-ordering principle is obviously false, and as for Zorn's lemma, who can say?" $\endgroup$ Commented Aug 15, 2025 at 17:43
  • 1
    $\begingroup$ Forcing Axioms and $V = L$ look supsicious to me for the same reason,one just is isolating a corner of the space of consistency, so to speak.This is definitely a worthwhile thing to be doing,but why should the resulting axiom be true? That is not how reality usually behaves. I consider AD fascinating as it seems to decide almost every question while being clearly wrong yet apparently not giving rise to contradictions.I see a certain analogy to physics. We have two theories describing different realms which are at odds with each other.I believe that a more modest approach could have some merit. $\endgroup$ Commented Aug 15, 2025 at 19:37
  • $\begingroup$ To my understanding physicists in quantum gravity are not necessarily trying to conjure up a theory of everything but rather try to understand how reality behaves under certain very specific circumstances and to gain understanding step by step. I believe that this approach would have some merit in set theory too. I find it rather interesting that most ZF consistency results are either about fragments of AD or about choice failing rather ridiculously. Much rarer are proofs of the joint consistency of a consequence of AC and one of AD. $\endgroup$ Commented Aug 15, 2025 at 19:37
  • 1
    $\begingroup$ Would you like an example? Is ZF+"Every set can be orderered linearly."+"Every uncountable set of reals has a perfect nonempty subset." consistent? This is open. $\endgroup$ Commented Aug 15, 2025 at 19:37
14
$\begingroup$

It is an open question whether or not there can be elliptic curves over $\mathbb{Q}$ of arbitrarily large Mordell-Weil rank. This article, which suggests that the ranks should be uniformly bounded, discusses some of the history and how public opinion seems to have shifted between boundedness and unboundedness.

Improve this answer
$\endgroup$
4
  • 7
    $\begingroup$ See mathoverflow.net/q/325571 for some discussion on "tides turning" more than once on this problem! $\endgroup$ Commented Aug 15, 2025 at 2:56
  • 7
    $\begingroup$ As I commented on another answer: I don't think that the OP's original question is really about "open problems where there have been attempts for or against", but more about cases where two independently formulated conjectures turn out to be mutually exclusive $\endgroup$ Commented Aug 15, 2025 at 14:23
  • 2
    $\begingroup$ @YemonChoi I had read the question as looking for "on the record" examples of "expressing different sentiments about which way a certain problem might finally be resolved". $\endgroup$ Commented Aug 15, 2025 at 15:26
  • $\begingroup$ Reposting a link mentioned in a previous comment so that it appears in the "Linked" questions list: Joe Silverman's answer to "Have the tides ever turned twice on any open problem?" $\endgroup$ Commented Aug 22, 2025 at 21:02
8
$\begingroup$

There are two competing and contradicting conjectures aiming at geometrically characterizing varieties defined over a number a field that admit a potentially dense set of rational points.

A projective variety $X$ defined over a number field $K$ has a potentially dense set of rational points (PD for short) if there exists a finite extension $L \supset K$ such that $X(L)$ is Zariski dense.

In a paper of Harris and Tschinkel the following conjecture was reported (stemming from questions and observations made by Colliot Thélène and Abramovich): a projective variety $X$ has PD if and only if $X$ is weakly special, namely no birational model of an étale cover of $X$ dominates a positive dimensional variety of general type.

Campana proposed a different characterization using his theory of special variety, conjecturing that $X$ has PD if and only if $X$ is special. Here $X$ is special if for every rank 1 saturated coherent sheaf $\mathcal{F} \subset \Omega_X^p$ one has that the Itaka dimension $\kappa(X,\mathcal{F}) < p$. Or, in other words, $X$ does not dominate any Campana orbifold of general type.

Bogomolov and Tschinkel were among the first to construct examples of weakly special varieties that are not special. Hence the two conjectures cannot hold at the same time.

Campana and Păun proved soon after that the complex analytic analogue of the Weakly-Special conjecture does not hold. Similar results were obtained for the function field version. Campana showed that Lang-Vojta conjecture (varieties of (log) general type do not have PD) contradicts the weakly special conjecture. But, at the present, we don't have a non conditional disproof of the Weakly special conjecture, even though it is believed not to be the correct characterization.

More recently a preprint of Bartsch, Campana, Javanpeykar and Wittenberg showed that the Weakly special Conjecture contradicts the abc conjecture (via the orbifold version of the Mordell Conjecture).

References:

  • Finn Bartsch, Frédéric Campana, Ariyan Javanpeykar, and Olivier Wittenberg. The weakly special conjecture contradicts orbifold mordell, and thus abc. ArXiv preprint:2410.06643, 2024.
  • Fedor Bogomolov and Yuri Tschinkel. Special elliptic fibrations. In The Fano Conference, pages 223–234. Univ. Torino, Turin, 2004.
  • Frédéric Campana. Orbifolds, special varieties and classification theory. Ann. Inst. Fourier (Grenoble),54(3):499–630, 2004.
  • Joe Harris and Yuri Tschinkel. Rational points on quartics. Duke Math. J., 104(3):477–500, 2000.
  • Erwan Rousseau, Amos Turchet, and Julie Tzu-Yueh Wang. Nonspecial varieties and generalised Lang–Vojta conjectures. Forum of Mathematics, Sigma, 9:1–29, 2021.
Improve this answer
$\endgroup$
6
$\begingroup$

I don't know if things have moved on since it was active but this question: Is Thompson's Group F amenable? discusses the amenability of Thompson's Group F. There have been claimed proofs both that it is amenable and that it is not so I suppose both statements have been conjectures for some meaning of the term.

Improve this answer
$\endgroup$
3
  • 10
    $\begingroup$ I guess there are other instances like this, of big open questions in an area, where there is no consensus about the truth (maybe the existence of a complex structure on $S^6$ would be another example). But these sort of go against what I take to be the implicit thrust of this question, which is a search for conjectures that are not obviously related and for which their mutual incompatibility is a nontrivial result. $\endgroup$ Commented Aug 15, 2025 at 14:12
  • 9
    $\begingroup$ I like this problem very much (in the sense of a spectator enjoying the thrill from a safe distance) but I don't think it quite fits the intended purpose of the question, for more or less the same reason that Sam Hopkins has articulated. $\endgroup$ Commented Aug 15, 2025 at 14:21
  • $\begingroup$ OK. Fair enough! $\endgroup$ Commented Aug 18, 2025 at 8:45
5
$\begingroup$

The function $s(n)$ is defined, for integers $n\ge2$, to be the sum of all the proper divisors of $n$, that is, all the divisors other than $n$ itself. E.g., $s(6)=1+2+3=6$.

The aliquot sequence of $n$ is the sequence $n,s(n),s(s(n)),s(s(s(n))),\dotsc$.

The Catalan-Dickson Conjecture holds that all aliquot sequences are bounded.

The Guy-Selfridge Conjecture holds that most sequences for even $n$ are unbounded.

Catalan's statement, dated 18 April 1888, can be found at https://www.aliquot.de/literatur/catalan.pdf It can also be found in Catalan's Mélanges Mathématiques, https://archive.org/details/mlangesmathm00catauoft/page/n7/mode/2up?view=theater page 240. A citation for Dickson is Theorems and tables on the sums of divisors of a number, Quart. J. Math., v. 44, 1913, pp. 264-296.

Guy and Selfridge have several papers on aliquot sequences. I'm not sure where their conjecture appears.

I thank Lola Thompson for reminding me of this pair of contradicting conjectrures, in a talk she gave at a recent conference.

Improve this answer
$\endgroup$
5
$\begingroup$

Lovász conjecture – Every finite connected vertex-transitive graph contains a Hamiltonian path. While László Babai expressed opposite conjecture. (Quote from Pak, Igor; Radoičić, Radoš (2009), "Hamiltonian paths in Cayley graphs")

In a survey article [B, §3.3], Babai is sharply critical of the Lovász conjecture: “In my view these beliefs only reflect that Hamiltonicity obstacles are not well understood; and indeed, vertex-transitive graphs may provide a testing ground for the power of such obstacles.” Babai conjectured that for some $c > 0$, there exist infinitely many Cayley graphs without cycles of length $\geq (1-c)n$. Clearly, Babai’s conjecture contradicts the Lovász conjecture. In a different direction, it it [sic] worth noting Thomassen’s work [Th] suggesting that there might be only finitely many counterexamples.

Improve this answer
$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.