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

Question

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.)

Answer 1

I heard that Bott's theorem on the periodicity of the stable homotopy of the unitary group was delayed for some time by an erroneous computation in dimension 10, possibly due to Pontryagin.

Answer 2

Couldn't resist quoting Conway (https://www.maths.ed.ac.uk/~v1ranick/books/dublin.pdf, p. 24) :

For nearly 50 years it has been supposed that the universality problem for quaternary integer-matrix forms had been solved by M. Willerding, who purported to list all such forms in 1948. However, the 15-theorem, which I proved with William Schneeberger in 1993, made it clear that Willerding’s work had been unusually defective. In his paper in these proceedings, Manjul Bhargava [https://www.maths.ed.ac.uk/~v1ranick/books/dublin.pdf, p. 27] gives a very simple proof of the 15-theorem, and derives the complete list of universal quaternaries. As he remarks, of the 204 such forms, Willerding’s purportedly complete list of 178 contains in fact only 168, because she missed 36 forms, listed 1 form twice, and listed 9 nonuniversal forms!

Answer 3

William Shanks's calculation of pi to 707 digits in 1873 seems to have been accepted for 72 years before Ferguson discovered an error in the 528th place.

Answer 4

Verma proved that the multiplicities of all simple modules in a verma module are 1 or 0. When BGG tried to repeat his proof for some other case they found an error. This led to the study of multiplicities in category O etc.

Answer 5

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

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

Very recently, Prasit Bhattacharya and Nitu Kitchloo [The stable Adams conjecture and higher associative structures on Moore spectra, Zbl 07483861] showed that Friedlander's proof of the stable Adams conjecture [Eric M. Friedlander, The infinite loop Adams conjecture via classification theorems for (\mathfrak I)-spaces, Math. Proc. Camb. Philos. Soc. 87, 109-150 (1980) Zbl 0426.55010] is incorrect (and provided two new proofs).

Answer 10

R. V. Gurjar and M. Miyanishi [Affine surfaces with $\bar {\kappa}\leq 1$, Algebraic geometry and commutative algebra, in Honor of Masayoshi Nagata, Vol. I, 99-124 (1988) ZBL0707.14015] gave a classification of smooth affine surfaces of logarithmic Kodaira dimension zero, whose coordinate ring is factorial and has trivial units. G. Freudenburg et al. [``Smooth factorial affine surfaces of logarithmic Kodaira dimension zero with trivial units'', https://arxiv.org/abs/1910.03494] identified a gap in this classification, which was closed in Tomasz Pełka and Paweł Raźny, Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units, ZBL07383176 who classified an additional infinite series of such surfaces.

Answer 11

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.

Answer 12

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.

Answer 13

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].

Answer 14

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].

Answer 15

Not sure how widely known this one is, but I've seen it mentioned on e.g. Wikipedia, so I figured I'd post it.

This paper by Gagola claimed that the commutant of a Moufang loop (the subloop of elements that commute with all other elements) is always normal. This later paper of Griskhov and Zavarnitsine gave a counterexample, of order $3^8$, and mentioned that Gagola had found a hole in his earlier proof.

Answer 16

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

Answer 17

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.

Answer 18

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 $n \geqslant m$."

Answer 19

Recently, Alberto Facchini and Jennifer Parolin A little mistake in a paper by Bob Gilmer on rngs, Zbl 07660836 noticed a minor error in R. W. Gilmer jun.'s classic paper Eleven nonequivalent conditions on a commutative ring, Nagoya Math. J. 26, 183-194 (1966). Zbl 0144.02701.

Answer 20

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).

Answer 21

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).

Answer 22

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.

Answer 23

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.).