Proof errors

Wikipedia lists some published "proofs" subsequently discovered to have gaps; often the claimed "theorem" ended up being disproven. But that list is misleading: it focuses primarily on old examples, so it can create or confirm an impression that the problem has disappeared. Another source of old examples is Section 6 of a 1972 survey by Philip J. Davis.

The following list also isn't comprehensive but at least includes some newer examples. The list is of various sources noting (not necessarily for the first time) specific errors in previously published "proofs":

• 2025-01-01 Daniel Litt: summarizing how an attempted proof of a conjecture ran into one wrong recently published paper after another.
• 2024-12-11 Kevin Buzzard: summarizing an error discovered in 2024 in the "proof" of Lemma 8 (which "seems to be false") in a 1976 paper by Norbert Roby; also alluding to errors in a 1978 book appendix on the same topic by Pierre Berthelot and Arthur Ogus.
• 2024-09-29 Peng Wang, Shuping Mao, Ruozhou Xu, Jiwu Jing, and Yuewu Wang, and independently 2024-10-03 Amit Singh Bhati, Michiel Verbauwhede, and Elena Andreeva: pointing out an error in a 2007 "proof" published by David A. McGrew and Scott R. Fluhrer. "We also discuss the impact of these vulnerabilities on XCB-based applications, such as disk encryption, nonce-based encryption, deterministic authenticated encryption and robust authenticated encryption, highlighting the risks due to XCB's failure to achieve STPRP security."
• 2022 Mario Berta, Fernando G. S. L. Brandão, Gilad Gour, Ludovico Lami, Martin B. Plenio, Bartosz Regula, and Marco Tomamichel: pointing out an error in a 2010 "proof" published by Brandão and Plenio. "Hence, the main achievability result of Brandão & Plenio is not known to hold. This puts into question a number of established results in the literature".
• 2021 Richard P. Brent: surveying several flaws in the literature, including some "proof" errors.
• 2019 Leslie Lamport: "George Bergman ... showed me the 51 reviews he had by then written in Math Reviews. Based on those reviews, I judged that exactly 1/3 of the papers he reviewed contained an error [found by Bergman], which I define to mean an incorrect statement in a proof or result that the author believed to be correct." See also 2024 version.
• 2019 Neal Koblitz and Alfred Menezes (also updated after 2019): survey of many known "proof" flaws in cryptography, often outright errors. (This survey also covers a different problem, namely theorems being treated as saying more than they actually do.)
• 2018 Daniel J. Bernstein and Edoardo Persichetti: "This appendix gives counterexamples to [20, Theorem 3.6] and [20, Theorem 3.5]" under the "plausible assumption that there exists some efficient correct deterministic PKE with OW-Passive security". "[20]" is a 2017 paper published by Dennis Hofheinz, Kathrin Hövelmanns, and Eike Kiltz; that paper was updated online in 2021 to include an erratum.
• 2018 Akiko Inoue and Kazuhiko Minematsu: pointing out an error in a 2004 "proof" published by Phillip Rogaway. "We present practical attacks against OCB2, an ISO-standard authenticated encryption (AE) scheme."
• 2017 Daniel Biss: retraction of a 2002 paper.
• 2015 Daniel J. Bernstein: pointing out an error in a 2002 "proof" published by Steven Galbraith, John Malone-Lee, and Nigel Smart.
• 2014 Vladimir Voevodsky: "A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail." Various supporting examples of "proof" errors.
• 2012 Gwern Branwen (also updated after 2012): various error examples, references, and analysis.
• 2009 Daniel Biss and Benson Farb: retraction of a "theorem" published in 2005 in Inventiones Mathematicae; "should be considered an open problem".
• 2008 Theodore P. Hill: pointing out various counterexamples to a 2006 paper published by Steven J. Brams, Michael A. Jones, and Christian Klamler. The history illustrates how broken the post-publication correction process is.
• 2008 Daniel J. Bernstein: pointing out an error in a 1999 "proof" published by Kaoru Kurosawa and Wakaha Ogata. This time the history illustrates how broken the pre-publication reviewing process is.
• 2007 Semyon Alesker: retraction of a "theorem" published in 1999 in Annals of Mathematics. "We do not know if Theorem A is true".
• 2003 Gen Nakamura, Gunther Uhlmann: retraction of a "theorem" published in 1994 in Inventiones Mathematicae. "Unfortunately, we have not been able to prove the global result stated in [NU1]."