Examples of conjectures that were widely believed to be true but later proved false

Question

It seems to me that almost all conjectures (hypotheses) that were widely believed by mathematicians to be true were proved true later, if they ever got proved. Are there any notable exceptions?

Answer 1

In 1908 Steinitz and Tietze formulated the Hauptvermutung ("principal conjecture"), according to which, given two triangulations of a simplicial complex, there exists a triangulation which is a common refinement of both.

This was important because it would imply that the homology groups of a complex could be defined intrinsically, independently of the triangulations which were used to calculate them.

Homology is indeed intrinsic but this was proved in 1915 by Alexander, without using the Hauptvermutung, by simplicial methods.

Finally, 53 years later, in 1961 John Milnor (some topology guy, apparently) proved that the Hauptvermutung is false for simplicial complexes of dimension $\geq 6$.

Answer 2

Luzin's conjecture was widely believed to be false, until it was proven by Carleson in 1966.

I'm citing from Lennart Carleson's biography: "In 1913 Luzin conjectured that if a function $f$ is square integrable then the Fourier series of $f$ converges pointwise to $f$ Lebesgue almost everywhere. Kolmogorov proved results in 1928 which seemed to suggest that Luzin's conjecture must be false but Carleson amazed the world of mathematics when he proved Luzin's long-standing conjecture in 1966. He explained how he was led to prove the theorem:-

... the problem of course presents itself already when you are a student and I was thinking about the problem on and off, but the situation was more interesting than that. The great authority in those days was Zygmund and he was completely convinced that what one should produce was not a proof but a counter-example. When I was a young student in the United States, I met Zygmund and I had an idea how to produce some very complicated functions for a counter-example and Zygmund encouraged me very much to do so. I was thinking about it for about 15 years on and off, on how to make these counter-examples work and the interesting thing that happened was that I realised why there should be a counter-example and how you should produce it. I thought I really understood what was the background and then to my amazement I could prove that this "correct" counter-example couldn't exist and I suddenly realised that what you should try to do was the opposite, you should try to prove what was not fashionable, namely to prove convergence. The most important aspect in solving a mathematical problem is the conviction of what is the true result. Then it took 2 or 3 years using the techniques that had been developed during the past 20 years or so"

Answer 3

This can perhaps be considered more of a meta-conjecture than a conjecture: Hilbert's program, https://en.wikipedia.org/wiki/Hilbert's_program. The conjecture would be: that set theory (or some set of axioms suitable for doing math) can be proven consistent. Gödel's Incompleteness Theorem disproved this conjecture.

I don't have a reference, but I have the impression that, at the time, Hilbert's program seemed attainable, and Gödel's result came as a surprise.

Answer 4

Littlewood's disproof of the conjecture (maybe of Gauss) that $\text{li}(x) > \pi(x)$.

I think this was widely believed before.

Answer 5

Euler's sum of powers conjecture, if a sum of $k$th powers is a $k$th power, then the sum has at least $k$ terms.

Proposed by Euler in 1769. Counterexample for $k=5$ found in 1966, for $k=4$ in 1986.

Answer 6

Borsuk's conjecture was believed to be true for 60 years till its counterexample was found in 1993 by Jeff Kahn and Gil Kalai.

They constructed an infinite family of counterexamples by using a result of Frankl and Wilson: https://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00398-7/S0273-0979-1993-00398-7.pdf

Here's what Babai and Frankl say about the demise of Borsuk's conjecture in their manuscript "Linear Algebra Methods in Combinatorics":

Dead at the age of 60. Died after no apparent signs of illness, unexpectedly, of grave combinatorial causes. The news of the demise of Borsuk's venerable conjecture (1933) spread like brushfire among combinatorialists in Summer 1992. The disproof, found by Jeff Kahn (Rutgers) and Gil Kalai (Hebrew University), was the hot topic between lectures at conferences (the result came too late to be included on the regular programs). Countless copies of the manuscript traveled over electronic networks, silently crossing oceans and continents at lightening speed. The authors of this book found out about the result in more conventional ways. One of us heard it from Kahn himself while examining Gabi Bollobás's remarkable sculptures at the reception at a meeting in Cambridge, England. By then, in Tokyo, the other author had learned about it in a telephone conversation with a friend in New Jersey.

What Kahn communicated in a few minutes and without the benefit of paper or blackboard was not just the news of the result but also the complete proof. Remarkably, Borsuk's geometric conjecture was disproved in just a few lines, relying on the Frankl-Wilson Theorem (Theorem 7.15), a modular version of the RW theorem.

Also see this very recent survey article by Gil Kalai: https://arxiv.org/pdf/1505.04952v1.pdf

Hirsch Conjecture is another possible example. In 2010, Francisco Santos constructed a 43-dimensional polytope of 86 facets with a diameter of more than 43: https://doi.org/10.4007/annals.2012.176.1.7.

Also see this survey by Ziegler: http://www.math.uiuc.edu/documenta/vol-ismp/22_ziegler-guenter.pdf (Wayback Machine)

Answer 7

In complex analysis of one variable, Liouville's theorem says that a bounded entire function is constant. Bernstein (1915-17) proved an analogous result in differential geometry, namely, if the graph of a function $f:\mathbf R^2\to\mathbf R$ of class $C^2$ is a minimal surface in $\mathbf R^3$, then the graph a plane. He then posed the classical Bernstein problem, namely, whether the same result also holds for real functions of $n>2$ variables. In terms of differential equations:

(Classical) Bernstein problem: Let the function $f:\mathbf R^n\to\mathbf R$ of class $C^2$ be a solution of $$\sum_{i=1}^nD_i\left(\frac{D_i f}{\sqrt{1+|D f|^2}}\right)=0.$$ Must $f$ be a linear function?

Recall that a hypersurface in $\mathbf R^{n+1}$ is defined to be minimal if its mean curvature vanishes, where its mean curvature is simply the sum of the principal curvatures (sometimes divided by $n$). Equivalently, the hypersurface is a critical point for the $n$-volume with respect to compactly supported variations. The equation above is the condition that the mean curvature of the graph of $f$ vanishes everywhere.

Part of the importance of the Bernstein problem is that it has a direct bearing on the existence of minimal cones and singularities of minimal hypersurfaces in $\mathbf R^{n+1}$. The answer to the problem was proved to be affirmative in the cases $n=3$ by de Giorgi (1965), $n=4$ by Almgren (1966), and $n\leq7$ by Simons (1968), and apparently there was some hope to extend the result to all dimensions.

However, in 1969 Bombieri, de Giorgi and Giusti constructed a counter-example for $n=8$, which yields a counter-example in each dimension $n>8$ by a standard construction, closing the problem. The complete solution of the Bernstein problem turned out to involve a good deal of geometric measure theory and non-linear analysis.

Answer 8

I believe that Fefferman's disproof in 1971 of the $L^p$ boundedness of disc multiplier for any $p \neq 2$ was considered a great surprise at the time; it showed that the classical result of Bochner and Riesz establishing norm convergence of Fourier series of $L^p$ functions in one dimension failed in two and higher dimensions, if one summed the series in the order of the magnitude of the frequencies (i.e. spherically summed Fourier series). The construction was one of the first applications of Kakeya sets (also known as Besicovitch sets) to harmonic analysis (though there was an earlier paper of Stein and Weiss that also used a related idea). Nowadays, the connection is taken for granted, but it was certainly not obvious at the time of Fefferman's result. (Fefferman himself writes in his paper "... It therefore comes as a surprise, at least to me, that the disc conjecture is false.")

Answer 9

Two widely believed conjectures:

1. The proportion of hyperbolic knots amongst prime knots of $n$ or fewer crossings approaches $1$ as $n$ approaches infinity.
2. The crossing number (the minimal number of crossings of a knot diagram of that knot) of a composite knot is not less than that of each of its factors.

The first conjecture is widely believed because of massive numerical evidence, and because many topological objects that are related to knots are generically hyperbolic- e.g. compact surfaces, various classes of 3-manifolds, closures of random braids...

The second conjecture is also widely believed to be true because of lots of numerical evidence- it is moreover believed that crossing number should in fact be additive with respect to connect sum (so the crossing number of a composite knot should actually be the sum of the crossing number of its components). It was proved for various classes of knots- alternating knots, adequate knots, torus knots, etc.

... and yet ...

Malyutin shows that either Conjecture 1 or Conjecture 2 must be false!!

Update: Malyutin has proven that Conjecture 1 is false. Thus this answer becomes honest, and Conjecture 1 is a conjecture that was widely believed to be true but (much) later shown to be false. arxiv.org/abs/1907.04458

Answer 10

Euler's conjecture about the nonexistence of $n\times n$ Graeco-Latin squares for $n=4k+2$. Disproved for all $k>1$ by the so called Euler's Spoilers Bose, Shrikhande, and Parker.

Answer 11

Lusztig's conjecture in modular representation theory, which would describe the characters of simple $G_1$-modules for $p$ larger than the Coxeter number, was generally believed to be true for a long time. Here $G$ is a reductive algebraic group in characteristic $p$ and $G_1$ is its Frobenius kernel. The conjecture in turn leads to character formulae (or at least, algorithms for calculating characters) for simple representations of $G$.

They were proven for large $p$ by Andersen-Jantzen-Soergel, with later contributions by others, including an explicit bound by Fiebig. But recently Geordie Williamson proved that the original condition on $p$ is not enough.

See What to do now that Lusztig's and James' conjectures have been shown to be false? for more details.

Answer 12

In introductory functional analysis one learns that every normed linear space with a Schauder basis is separable. The converse of this was a famous question raised by Banach- does every separable Banach space have a Schauder basis? Since almost all known separable Banach spaces had been shown to possess a Schauder basis it was believed that this must be true. But in 1972 Enflo constructed a counterexample to this.

For this achievement of his Enflo was awarded a live goose by Stanislaw Mazur.

See this.

Answer 13

The solution in negative of the isomorphism problem for integral group rings. A counterexample was found by Martin Hertweck:

Hertweck, Martin, A counterexample to the isomorphism problem for integral group rings, Ann. Math. (2) 154, No. 1, 115-138 (2001). jstor, ZBL0990.20002, MR1847590.

Answer 14

There have been multiple conjectures of this type - seemingly motivated, commonly believed, yet false - about the structure of the partial order of Turing degrees of c.e. sets. Two in particular were due to Shoenfield:

• In 1963, he conjectured that, given any finite poset $P$ which embeds (via $f$) into the c.e. degrees preserving $0, 1$, and $\vee$, and $P'\supseteq P$ has the same maximal and minimal elements, and l.u.b.s, as $P$, then $P'$ embeds into the c.e. degrees via an embedding extending $f$.

This was refuted by the construction of a minimal pair of c.e. degrees, that is, a pair of noncomputable c.e. degrees $\underline{a}$, $\underline{b}$ such that no noncomputable set is computable in both $\underline{a}$ and $\underline{b}$.

• Eleven years later, Shoenfield conjectured that all finite lattices were embeddable into the c.e. degrees in a way that preserved 0.

Manuel Lerman counter-conjectured that the lattice $S_8$ was not so embeddable; this was proved by Lachlan and Soare six years later.

The motivation behind both conjectures was the intuition that the c.e. degrees were a nicely behaved structure; in particular, I think it was believed that the poset of degrees c.e. in and above a given $\underline{d}$ should be isomorphic to the poset of c.e. degrees, that the theory of the c.e. degrees is decidable, that the poset is $\aleph_0$-categorical, etc., and all of these turned out to be false.

Answer 15

The dynamical degree of a dominant rational map $f:\mathbb{P}^N\to\mathbb{P}^N$ is defined by the limit $$ \delta(f) := \lim_{n\to\infty} \deg(f^{\circ n})^{1/n}, $$ where $f^{\circ n}$ is the $n$th iterate of $f$. It was conjectured by Bellon and Vialet [1] that $\delta(f)$ is always an algebraic integer, and over the succeeding two decades, this was proven for many classes of maps. But Bell, Diller, and Jonsson [2] recently gave an example of a map on $\mathbb{P}^2$ whose dynamical degree is a transcendental number.

[1] Algebraic entropy, Bellon, M. P. and Viallet, C.-M., Comm. Math. Phys. 204 (1999), 425-437.

[2] A transcendental dynamical degree, Jason P. Bell, Jeffrey Diller, Mattias Jonsson, https://arxiv.org/abs/1907.00675

Answer 16

Mersenne's conjecture on primes is a famous example (although I am not sure how widely it was believed to be true).

Answer 17

Fermat's conjecture that all numbers of the form $ F_{n} : =2^{2^{n}}+1 $ are prime. Euler proved that $ 641\mid F_{5} $ .

Answer 18

The Markus-Yamabe Conjecture in differential equations was posed in 1960. It states that if $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a $C^1$ map such that $f(0)=0$ and the eigenvalues of the Jacobian matrix $Df(x)$ have negative real part for every $x\in\mathbb{R}^n$, then $x=0$ is globally attractive. In the early 1990s, proofs for the $n=2$ case were given, but in 1996 a complicated counterexample in $n=4$ was constructed, and in 1997 a simple polynomial counterexample for $n\geq 3$ was produced.

Answer 19

An example from set theory: My understanding is that it was once widely believed that all reals appearing in canonical inner models of large cardinals (at least up to supercompact cardinals) would be $\Delta^1_3$ in a countable ordinal. This is because it was assumed that linear iterations, the only kind known at the time, would suffice to compare such inner models. This assumption turned out to fail at the level of Woodin cardinals, far below supercompact cardinals. The resulting non-linear iterations (iteration trees) are a basic part of inner model theory today, whereas canonical inner models for supercompact cardinals are still far out of reach.

Answer 20

Polya conjecture was proved to be false in 1958.

Answer 21

My impression is that Stanley's partionability conjecture and his depth conjecture (which was shown to imply the partionability conjecture in 2008) were both believed to be true, until Duval, Goeckner, Klivans, and Martin found a counterexample to the partionability conjecture in 2015. See this AMS survey article: https://www.ams.org/journals/notices/201702/rnoti-p117.pdf. This fits with a few other examples already mentioned (the Hauptvermutung, the Hirsch conjecture, ...) which warn us that although simplicial complexes/polytopes may appear to be intuitively simple objects, they can in fact be extremely complicated.

Answer 22

Let $k \geq 3$ be fixed. Ramsey's theorem says that if $n$ is sufficiently large and we color the edges of the complete graph $K_n$ red or blue, there must be at least one monochromatic $K_k$. As it turns out, it's not just "at least one" but many: An averaging argument shows that as $n \rightarrow \infty$ a positive fraction of all the $\binom{n}{k}$ copies of $K_k$ in our coloring must be monochromatic.

This leads to a natural follow-up question: How few copies can we get? If we consider all $2$-colorings of $K_n$, which one (asymptotically) minimizes the number of monochromatic copies of $K_k$?

This was first studied for the case $k=3$ (monochromatic triangles) by Goodman, who in 1959 gave an explicit answer asymptotic to $\frac{1}{4} \binom{n}{3}$. The fraction $\frac{1}{4}$ has a natural interpretation here -- if we color randomly, this is the expected fraction of monochromatic triangles. Three years later, Erdős observed that the random coloring gives an upper bound of $2^{-\binom{k}{2}+1}$ on the minimum fraction of monochromatic $K_k$, and said it "seems likely" this was asymptotically optimal.

By 1980 Burr and Rosta conjectured that something even stronger was true: For any fixed graph $H$ the asymptotic way to minimize monochromatic copies of $H$ was just to color randomly. It wasn't until 1989 that Sidorenko gave a counterexample to the Burr-Rosta conjecture (a triangle with a pendant edge) and Thomason disproved Erdős's original conjecture by giving a coloring with significantly fewer monochromatic $K_4$ then random.

It is still an open question to determine the optimal coloring to minimize monochromatic $K_4$, and also still an open question to determine for which graphs the Burr-Rosta conjecture is true (such graphs are termed "common" in the literature).

Answer 23

The Hedetniemi's conjecture was proposed in 1966. It stated that the chromatic number of the tensor product of two graph $G$ and $H$ is equal to the minimum of the individual chromatic numbers of the graphs $G$ and $H$. This conjecture was disproved by giving an explicit counterexample, by Yaroslav Shitov in 2019 in this paper published in Annals of Mathematics.

Answer 24

The longest-standing one of the sort is the "conjecture" that the parallel postulate can be proved using Euclid's first four postulates. I know that it is a far-fetched understanding of "conjecture". But, after all, it is something that people tried to prove by doing it for more than two thousand years, believing that it is true.

Answer 25

Igor Pak wrote an entertaining blog post on the topic of counterexamples, where he listed several of the conjectures mentioned here in the answers, as well as others, such as the general Burnside problem and Tait's conjecture. He also points out that there is a Wikipedia page listing disproved conjectures.

Answer 26

Two examples from lattice theory: is every lattice with unique complements distributive? [no] is every distributive algebraic lattice isomorphic to the lattice of congruences of a lattice? [no] See http://www.ams.org/notices/200706/tx070600696p.pdf

Grätzer, George, Two problems that shaped a century of lattice theory, Notices Am. Math. Soc. 54, No. 6, 696-707 (2007). ZBL1286.06001, MR2327971.

Answer 27

The Permanent-on-Top conjecture was proved false after about 50 years. It was stated in 1966, which conjectured that the largest eigenvalue of the Schur power of a positive definite Hermitian matrix is the permanent of that matrix. It was similar to the result of Schur that the smallest eigenvalue of the Schur power matrix of the same positive definite Hermitian matrix is the determinant of the matrix.

Answer 28

• Here is one which I learnt from this answer here, on the asymptotic behavior of $\log (n) - \frac{n}{\pi(n)}$ (Legendre conjectured it tended to something other than the correct limit $1$).

Answer 29

How about the Pythagorean tenet that all numbers are rational?