Can anyone cite an example of a mathematics paper that has been retracted?

It is said that on the order of 100,000 new theorems enter the mathematics literature every year. For a number of reasons including hyper-specialization and demands on referee resources it is, in my view, unlikely that all their proofs are correct. Yet it seems no explicit effort is made to clean the literature. False theorms float downstream along with the true ones, available for citation and use in constructing yet further theorems.

Thanks for any insight.

Cheers, Scott

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    $\begingroup$ Corrigenda and errata are published all the time and are easy to find. (It's true that sometimes published theorems are refuted and yet no erratum appears. But your question seems to only ask for one example...) The question in the first paragraph seems trivial (if a published erratum would satisfy you) and the second paragraph is 'subjective and argumentative'. For that reason, I'm voting to close. $\endgroup$
    – HJRW
    Jan 18, 2013 at 11:11
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    $\begingroup$ For one example, google "Daniel Biss". I also agree that the second paragraph is not a question, just a rant. $\endgroup$ Jan 18, 2013 at 11:15
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    $\begingroup$ Also, this question overlaps a lot with mathoverflow.net/questions/35468 $\endgroup$ Jan 18, 2013 at 12:15
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    $\begingroup$ Voted to close for roughly the reasons given by HW. I am not even sure what an "explicit effort" to "clean the literature" should be. All the time papers are studied and thus checked and potential issues raised. Some papers perhaps hardly or not at all, but then most likely those are not used for much else either. $\endgroup$
    – user9072
    Jan 18, 2013 at 12:25
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    $\begingroup$ I voted to reopen: I like the question, though not the way it is formulated. If the author would edit it and ask for a "long list" od retracted papers with an interesting history (for example effect on the subsequent papers that used it), I am sure others would vote to reopen it, and interesting answer would come in mass. $\endgroup$
    – Joël
    May 18, 2013 at 22:51

3 Answers 3


A general existence theorem is proved :

1933 : W. Grunwald, Ein allgemeines Existenztheorem für algebraische Zahlkörper, J. reine angew. Math. 169 (1933), 103–107.

and reproved :

1942: G. Whaples, Non-analytic class field theory and Grünwald's theorem. Duke Math. J. 9, (1942). 455–473.

A counter-example is found :

1948 : S. Wang, A counter-example to Grunwald's theorem, Ann. Math. 49 (1948), 1008–1009.

and the theorem is corrected :

1950 : S. Wang, On Grunwald's theorem, Ann. Math. 51 (1950), 471–484.

twice in the same year :

---- : H. Hasse, Zum Existenzsatz von Grunwald in der Klassenkörpertheorie, J. reine angew. Math. 188 (1950), 40–64.

A quarter of a century later, a simpler proof is given :

1974: J. Neukirch, Eine Bemerkung zum Existenzsatz von Grunwald-Hasse-Wang, J. Reine Angew. Math. 268/269 (1974), 315–317.

but more than half a century later, corrections to the corrections are required :

2007 : W-D. Geyer & C. Jensen, Embeddability of quadratic extensions in cyclic extensions. Forum Math. 19 (2007), no. 4, 707–725.

2011 : P. Morton, A correction to Hasse's version of the Grunwald-Hasse-Wang theorem. J. Reine Angew. Math. 659 (2011), 169–174.

Addendum (2013/05/18)

I'm afraid the above list of errors and corrections might look a bit negative, so let me add a positive note (which will also save you 30,00 € or $42.00 by not having to read it here) :

In 1933, van der Waerden asked in the Jahresbericht : Which quadratic fields can be embedded in cyclic quartic fields ? Solutions were provided by four people, among them Hasse, who generalised the problem to : Under which conditions can a degree-$l$ ($l$ prime) cyclic extension $K_1$ of a number field $K$ be embedded into a degree-$l^n$ cyclic extension $K_n$ of $K$ ?

A. Scholz sent in a "solution" to this problem in 1935 which essentially claimed that the obstructions are purely local in nature. But Hans Richter, a doctoral student of van der Waerden, knew already that there is an exception when $l=2$, so a Scholtz-Richter correction to Scholz's paper was required. In a sense, Richter anticipated not only Wang's counterexample to Grunwald's theorem but also its solution, without mentioning it explicitly as such.

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    $\begingroup$ Would this story be an example of the process that Imre Lakatos described in Proofs and Refutations? If so then it is a natural and extremely productive part of the evolution of mathematics and of science in general. $\endgroup$ May 18, 2013 at 9:15

You have to notice that many of those theorems are dead-ends. They'll either not be used at all, or be superseded by a better one. Corrections sometimes happen, but it looks like pure changes of focus is also a big factor.

How many proofs have there been that given a point and a line, there is a single line parallel to the line going through the point, before non-euclidean geometry settled the matter?

Consider the vast litterature on proving that such and such type of polynomial can be solved by radicals. Galois theory made (almost) all of these obsolete -- be they right or wrong.

The notions of limits, continuity, derivability, etc... had no serious definition for very long, before people started to realize there were problems (like sequence of continuous functions converging to a non-continuous limit) and the $(\varepsilon,\delta)$ definitions were given, and people started to prove more solid results through this framework.

The italian school of algebraic geometry is another example coming to mind, where things were cleaned by a quite radical change in the paradigms of the field.

In fact, one could say that most of mathematics is about trying to get correct theorems out, and clean what is already there through obsolescence, be it gentle(correction) or cataclysmic(not interesting anymore).

As a final remark, I think it is reasonable to expect the recent works in automatic proof-checkers/proof-builders will sparkle a new revolutionary era.


Here are some recent (not famous) examples of papers that have been retracted at the request of the Editor-in-Chief or the Publisher:






I found them via the blog Retraction Watch, which also contains ample discussion about reasons and policies about retracting papers and many more examples, including background information.

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    $\begingroup$ Haha: "In brief an impossible proposition was proved as possible. This is a problematic problem.". $\endgroup$
    – M.B.
    Jan 18, 2013 at 12:09
  • $\begingroup$ Occasionally it happens that papers are accepted largely as a favor to author or friends. This passes unnoticed most of the time, for a host of sociological reasons. More rarely, this may happen with a paper which is wrong and/or meaningless. In such circumstances there is an evident disincentive for anyone involved to rectify matters, especially given how many correct and meaningful papers are never read again either. Concrete examples exist, as do legal reasons for not posting any here $\endgroup$ Jan 18, 2013 at 21:15
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    $\begingroup$ Adam, what legal reasons? If you refer to the recent "Science Fraud" issues, then you have nothing to fear unless you plan on actually accusing mathematicians of fraud. Pointing out that a paper has errors is not the same thing, and should be encouraged. Mistakes happen and they are honest mistakes with no misconduct behind them. $\endgroup$
    – Matt
    Jan 18, 2013 at 23:53
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    $\begingroup$ >Pointing out.... - And most often it is, except occasionally when it isn't. The bottom line is the attitude of the author when faced with the dilemma of whether to engage in a mathematical discussion or to try to change the subject, perhaps by lodging complaints. >Mistakes happen.... - Absolutely, they can be and typically they are, except perhaps when there's a broader purpose involved. "Misconduct" and "fraud" are somewhat stronger words that get people exercised $\endgroup$ Jan 19, 2013 at 3:27
  • $\begingroup$ It's interesting to see that the second and third were accepted apparently because of an impersonation of a referee (so by deliberate fraud) rather than simply being the writings of circle-squarer-type cranks who are tone-deaf to mathematical proof. $\endgroup$ Jun 15, 2020 at 20:03

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