# Widely accepted mathematical results that were later shown to be wrong?

Are there any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time later, possibly even after being used in proofs of other results?

(I realise it's a bit vague, but if there is significant doubt in the mathematical community then the alleged proof probably doesn't qualify. What I'm interested in is whether the human race as a whole is known to have ever made serious mathematical blunders.)

• I have a déjà vu :) Also, maybe community wiki (~big list)?
– M.G.
Aug 13 '10 at 10:41
• @ex Given that I have no other way to earn reputation than by asking questions (my math is a mere long-forgotten-university-level), I'd like at least one extra upvote before marking this CW so I can at least upvote some answers :) Aug 13 '10 at 10:44
• mathoverflow.net/questions/27749/… is a similar question - only the subject of the proof was later confirmed to be true. Aug 13 '10 at 12:16
• The story around the Grunwald-Wang theorem takes the cake on this one, especially Tate's commentary on his reaction to it as a graduate student (but one also has to keep in mind that in those days and earlier, the number of active research mathematicians was a tiny fraction of the number today). See section 5.3 of rzuser.uni-heidelberg.de/~ci3/brhano.pdf Aug 13 '10 at 14:35
• en.wikipedia.org/wiki/Dehn%27s_lemma Aug 15 '10 at 20:35

The Auslander Conjecture states: Every crystallographic subgroup $\Gamma$ of $\mathrm{Aff}(\mathbb{R}^n)$ is virtually solvable, i.e. contains a solvable subgroup of ﬁnite index.

He published an incorrect proof in 1964 of this statement.

In 1983 Fried and Goldman proved Auslander’s conjecture for $n = 3$.

Abels, Margulis and Soifer proved the conjecture for $n\leq 6$ in 2012.

Although it is not my area of expertise, I believe it is considered to be an important open conjecture and has led to active research.

• Sean: More precisely, in this case the proof turned out to be wrong, but the "result" became a major open problem in the field (not a wrong result, as far as we know). Jun 1 '13 at 13:15

No less a mathematician than Kurt Gödel was guilty of claiming to have proved a result that was accepted for decades, and even used by others, before being shown to be wrong. Stål Aanderaa showed that Gödel's argument was incorrect and Warren D. Goldfarb showed that the result itself was false. The claimed result was about the decidability of a class of formulas including equality; see here for more details.

Pontryagin made a famous mistake in A classification of continuous transformations of a complex into a sphere which led him to the false conclusion that the homotopy group $$\pi_{n+2}(S^n)$$ is zero. Later Freudenthal in Über die Klassen den Sphärenabbilgunden I. Grosse Dimensionen and Whitehead in The $$(n+2)^{\text{nd}}$$ homotopy group of the $$n$$-sphere showed with different methods that $$\pi_{n+2}(S^n) \cong \mathbb Z_2$$ and Pontryagin corrected his mistake in Homotopy classification of the mappings of an $$(n+2)$$-dimensional sphere on an $$n$$-dimensional one.

Let me try to explain what Pontryagin got wrong: Let $$f \colon S^{n+2}\to S^n$$ be a smooth map which represents an element of $$\pi_{n+2}(S^n)$$ (in every homotopy class of continouus maps between manifolds there is a smooth representative). Following Sard's Theorem there is an $$x_0 \in S^n$$ which is a regular value of $$f$$, thus $$\Sigma:=f^{-1}(x_0)$$ is a closed $$2$$-dimensional submanifold of $$S^n$$. The normal bundle of $$\Sigma$$ in $$S^{n+2}$$ is trivial and has a natural framing induced by the derivative of $$f$$ and a choice of a basis in $$T_{x_0}S^n$$. Pontryagin defined a map $$\varphi \colon H_1(\Sigma;\mathbb Z_2) \to \mathbb Z_2,$$ where he assigned to every closed curve $$C$$ representing an element of $$H_1(\Sigma;\mathbb Z_2)$$ if the normal bundle over $$C$$ is framed trivially or not (over a circle there are only two homotopy classes of trivializations of the trivial vector bundle since $$\pi_1(SO(n))=\mathbb Z_2$$ provided $$n\geq 3$$). Pontryagin assumed that $$\varphi$$ is a homomorphism and concluded by a surgery argument that every surface $$\Sigma$$ is framed bordant to the 2-sphere $$S^2$$ which would mean that the map $$f$$ is null homotopic (see here for more details and nice pictures!).

Later Pontryagin corrects his mistake here. He shows $$\varphi(C_1+C_2) = \varphi(C_1)+\varphi(C_2) + I(C_1,C_2),$$ where $$I(C_1,C_2)$$ is the intersection number of the two curves $$C_1$$ and $$C_2$$. Thus $$\varphi$$ is a quadratic refinement of $$I$$ and one can associated the Arf invariant $$A(\varphi)$$ to $$\varphi$$ (see Wikipedia). This can be used to enumerated $$\pi_{n+2}(S^n)$$.

According to Branko Grunbaum, An enduring error, Elemente der Mathematik 64 (2009) 89-101, reprinted in Mircea Pitici, ed., The Best Writing On Mathematics 2010, Daublebsky in 1895 found that there are precisely 228 types of collections of 12 lines and 12 points, each incident with three of the others. In fact, as found by Gropp in 1990, the correct number is 229.

The Gropp reference is H Gropp, On the existence and nonexistence of configurations $$n_k$$, J Combin Inform System Sci 15 (1990) 34-48.

Grunbaum has some other examples in this paper, which I may write up for this question.

Another one from Grunbaum's paper:

According to Branko Grunbaum, An enduring error, Elemente der Mathematik 64 (2009) 89-101, reprinted in Mircea Pitici, ed., The Best Writing On Mathematics 2010, Bruckner enumerated 4-dimensional simple polytopes with eight facets, in 1909. But one of Bruckner's polytopes does not exist, according to Grunbaum and Sreedharan, An enumeration of simplicial 4-polytopes with 8 vertices, J Combin Theory 2 (1967) 437-465.

The Steiner Ratio Gilbert–Pollak Conjecture was proved by Dingzhu Du and Frank Hwang in 1990, and published in Algorithmica in 1992. In 2001, a gap of the original proof was found by A.O. Ivanov and A.A. Tuzhilin. And the conjecture remains open now. See: https://doi.org/10.1007/s00453-011-9508-3

According to Branko Grunbaum, An enduring error, Elemente der Mathematik 64 (2009) 89-101, reprinted in Mircea Pitici, ed., The Best Writing On Mathematics 2010, Andreini in 1905 claimed that there are precisely 25 types of uniform tilings of three-dimensional space. But in 1994 Grunbaum showed that the correct number is 28. The reference is B Grunbaum, Uniform tilings of 3-space, Geombinatorics 4 (1994) 49-56.

Grunbaum has yet more examples in this paper, which I may write up for this question.

EDIT: The episode I had in mind turns out to be the work of Robert Coleman repairing a gap in a paper by Manin about the Mordell conjecture over function fields. See comments by KConrad below, giving specific references. Note that this is not about a false result, it is about an accepted proof with a gap that was found 20 to 25 years later and repaired.

Original:Requesting assistance with a memory.

This being community wiki, I will give my vague memory. I think someone who was actually there could tell a good story. I have been searching with combinations of words on google with no success.

Anyhoo, when I was in graduate school at Berkeley in the 1980's, a professor, whom I think was likely Robert Coleman, told us a story about a celebrated result on "function fields" or the function field version of something... The accepted proof was by someone really big, on google I kept running across the name Manin but I am not at all sure about the name. Prof. Coleman decided to present the proof to a class/seminar. Partway through it became clear that the accepted proof just did not work. I have a sense that the class and professor were able to clean up the proof but I have no idea what publication may have come of this. There is also the chance that the seminar did not occur at Berkeley, rather at an earlier job of the professor concerned. Sigh.

So, there are a few ways this story could be filled in. Many MO people are students or postdocs at Berkeley, somebody could walk down the hall and ask Prof. Coleman if that was really him, and if so what actually happened, or ask Ken Ribet, etc.. Again, someone on MO with encyclopedic knowledge of every possible use of the phrase "function field" might be able to say. Or someone very old, yea, verily stricken in years, like unto me.

Finally, note that the title and text of the OP's question disagree a little, and people have posted both "results" that remained false and correct results with incorrect proofs. Also, my memory is really quite good, but I heard this story once and did my dissertation on differential geometry and minimal submanifolds.

• Coleman, Manin's proof of the Mordell conjecture over function fields, L'Enseign. Math. 36(1990),393-427. From introduction: "In the process of translating Manin's proof of Mordell's conjecture (1963) into modern language we found a gap. The arguments given by Manin do not suffice to prove Manin's theorem of the kernel. We were able to fill this gap by using those arguments to prove a weaker theorem and combining this with the function field analogue of Siegel's theorem and Manin's ideas to complete the proof of Mordell's conjecture for function fields." This isn't a proof of a false theorem. Aug 16 '10 at 1:02
• Thanks, Keith...I was not sure of much of anything at first except the phrase "function fields" and a location. When I saw the name Coleman I thought, yes, that seems right. Anyway, right, it turns out the OP really wanted proofs of false theorems, so I edited the question title to reflect that, and this episode does not therefore qualify but seems interesting to me...I see, Manin's work was 1963, Coleman's article did not appear until 1990 and Manin's letter to the editor in Izvestiya dates the preprint as 1988. The name "Manin" did not ring any bells even after I saw it on google... Aug 16 '10 at 1:49
• Will, I was involved in a similar story concerning Wiles' proof. At my request, in the fall/winter of 1993 Beilinson agreed to give a few talks (in Moscow) on the 1993 paper. Eventually, he came to the point which he couldn't explain. Unfortunately, in spite of catalyzing the whole process, I didn't attend, so to this day I don't know whether he discovered the notorious gap! Aug 16 '10 at 5:01
• Victor, of course, Wiles. There was a very nice televivion program about that, just called "The Proof," in a series on public TV called NOVA. Evidently the gap appeared when one of the referees ( a chapter each!) I think Nicholas Katz, was going through line by line and demanding that Wiles explain any problems. Then there came a day when the explanations did not work either, but we know the rest. Aug 16 '10 at 22:49
• Apparently the story about Manin's proof of the Mordell conjecture over function fields goes a long way : - 1963 Manin's original paper, - 1990 Coleman's correction, - 1991 Chai publishes 'A Note on Manin's Theorem of the kernel' (Am. J. of Math. 1991) saying "In this note, it will be shown that Manin was right after all", - 2008 Bertrand writes 'Manin’s theorem of the kernel : a remark on a paper of C-L. Chai' (see his homepage) saying "Chai gave two proofs. (...) The first proof concerns a more general situation, but contains a gap". Aug 28 '10 at 12:12

This blog post, the previous one (linked inside) and the addendum caused by reader response, treat exactly this question, including Euler's polyhedra formula from Micah Miller's response.

A recent example: The main theorem of [Masa-Hiko Saito, On the infinitesimal Torelli problem of elliptic surfaces, J. Math. Kyoto Univ. 23, 441-460 (1983). ZBL 0532.14019] has been shown to be incorrect due to a counterexample in [Atsushi Ikeda, Bielliptic curves of genus three and the Torelli problem for certain elliptic surfaces, Adv. Math. 349, 125-161 (2019). ZBL 1414.14004].

This thread on the Italian tradition in algebraic geometry contains some important examples.

• Yes, that thread was mentioned in KConrad's comment of 15 August. Feb 25 '11 at 11:51

I guess one major example is that unique factorisation doesn't always hold in rings of integers of number fields.

Classical attempts at solving Fermat's Last Theorem resulted in moving to cyclotomic fields $\mathbb{Q}(\zeta_n)$ and noting that Fermat's equation factorises to give:

$$\prod_{k=0}^{n-1}(x + \zeta_n^k y) = z^n$$

an equality in the ring of integers $\mathbb{Z}[\zeta_n]$ of such fields.

Lame offered a full proof of FLT along these lines but a crucial assumption was unique factorisation. Kummer was able to provide a counter-example that shows non-unique factorisation for $n=23$ and this spurred off the process of inventing ideals as well as lots of other cool stuff in algebraic number theory.

• But this proof was immediately rejected. See mathpages.com/home/kmath447.htm In short in 1847 Lame' announced his proof of Fermat's Last Theorem. Liouville immediately took the floor to criticize the crucial point. It transpired that Kummer had already (three years before) published a paper showing the failure of unique factorization in some of the relevant fields. Dec 13 '14 at 23:40

Due to the relative obscurity of the journal (and the relatively obvious mistake), this would perhaps rarely count as "widely accepted", but it was used in later publications: E. Habil claimed in https://journals.iugaza.edu.ps/index.php/IUGNS/article/view/1594 :

"Let $$(p(n, m))_{n, m \in \mathbb N}$$ be a bounded double sequence of reals, then there are $$n_1 < n_2 < \ldots$$ and $$m_1 < m_2 < \ldots$$ such that the double sequence $$(p(n_k, m_j))_{k, j \in \mathbb N}$$ converges",

which was used in [Asfaw, Teffera M., A proof of Nirenberg conjecture on expansive mappings in Hilbert spaces, ZBL07265518.] to prove an open problem from [Nirenberg, Louis, Topics in nonlinear functional analysis. Notes by R. A. Artino, New York: Courant Institute of Mathematical Sciences, New York University. VIII, 259 p. $6.75 (1974). ZBL0286.47037], but, as V. Kadets mentions in his review, "an easy counterexample to such a `Bolzano-Weierstrass theorem' appears if one takes $$p(n, m) = 0$$ for $$n < m$$ and $$p(n, m) = 1$$ for $$n \geqslant m$$." In 1892, Michel Frolov Equalities of the second and third degree, Bull. Soc. Math. Fr. 20, 69-84 (1892). JFM 24.0176.01. claimed a proof that there are no 7th-order bimagic squares, since there is no series of 7 distinct odd numbers, from 1 to 49, with sum 175 and sum of squares 5775; this is, however, not true (e.g. 1,7,25,31,33,37,41). The non-existence of 7th-order bimagic squares has been only confirmed in 2004 via computer calculations, see http://www.multimagie.com/English/Smallestbi.htm. The PBW theorem of Rinehart [Rinehart, G. S., Differential forms on general commutative algebras, Trans. Am. Math. Soc. 108, 195-222 (1963). ZBL0113.26204] is not true in the claimed generality, as explained by Maakestad in [Maakestad, Helge Øystein, Corrigendum to: “Algebraic connections on projective modules with prescribed curvature”, J. Algebra 463, 281-283 (2016). ZBL1441.14065]. • What is the status of Rinehart's PBW theorem? Is there a claim about where there is a gap in the proof? May 17 at 15:26 • Maakestad's corrigendum derives from the canonical isomorphisms given by Rinehart's theorem a surjective Chern class map; in an example, that would go from$\mathbb Z$to$\mathbb C$, which is impossible. Likely, the theorem is true in the case that$L$in the Lie-Rinehart algebra$\alpha:L\to\mathrm{Der}_R(A)\$ is finitely generated and projective. May 18 at 16:14

Hilbert's sixteenth problem. In his speech, Hilbert presented the problems as:

The upper bound of closed and separate branches of an algebraic curve of degree n was decided by Harnack (Mathematische Annalen, 10); from this arises the further question as of the relative positions of the branches in the plane. As of the curves of degree 6, I have - admittedly in a rather elaborate way - convinced myself that the 11 branches, that they can have according to Harnack, never all can be separate, rather there must exist one branch, which have another branch running in its interior and nine branches running in its exterior, or opposite.

According to Arnold (see his book "What is mathematics?") Gudkov has found 3rd posiible configuration (exist one branch, which has 5 branches running in its interior and 5 branches running in its exterior).

This thread doesn't seem to mention Gauss's first proof of the fundamental theorem of algebra, from 1799. He claimed it as the first really rigorous proof, but it had a topological gap discovered 120 years later by Ostrowski. I think this is pretty famous since I first heard of it in an introductory abstract algebra class. It is discussed a bit in Smale's article about the FTA and complexity theory, here (p. 4).

• I think the question is about "proofs" of statements where it turned out that the statement was wrong. I our case the statement is totally fine, but only the proof was flawed.
– Dirk
Oct 29 '18 at 7:32
• That could be more interesting, but there are several entries in the list for wrong proofs of true statements so I figured this counts. If not, it's ok with me if someone deletes it.
– none
Oct 29 '18 at 8:54
• arxiv.org/abs/1704.06585v1 Dec 14 '19 at 14:45

Very recently, Dobbs A minimal ring extension of a large finite local prime ring is probably ramified, ZBL 07192436. identified an error in the proof that "a separable extension of finite rings is always Galois" (Corollary XV.3 of McDonald, Bernard R., Finite rings with identity, Pure and Applied Mathematics. Vol. 28. New York: Marcel Dekker, Inc. IX, 429 p. (1974). ZBL 0294.16012.).

• Why on earth would one not post this kind of paper on the arXiv??? Sep 5 '20 at 18:32
• It appears that arXiv:math/0606689 is the only one of Dobbs's many papers so far on the arXiv (put there by Kabbaj). Sep 6 '20 at 19:09