August Möbius [36] defended Libri, by presenting his former professor's reason for believing that 0^{0}=1 (basically a proof that lim_{x→0+}x^{x}=1). Möbius also went further and presented a supposed proof that lim_{x→0+}f(x)^{g(x)}=1 whenever lim_{x→0+}f(x)=lim_{x→0+}g(x)=0. Of course "S" then asked whether Möbius knew about functions such as f(x)=e^{−1/x} and g(x)=x. (And paper [36] was quietly omitted from the historical record when the collected works of Möbius were ultimately published.)
("His former professor" was Pfaff.)
Now although we know that Sylvester was in the habit of making guesses, and these guesses although often brilliant were not always so,* it would be next to impossible to find a generalisation of his which had no individual instances in support of it.
* See Crelle's Journal, lxxxix, pp. 82–85.
Borchardt was the editor of the journal. Sylvester has authorized Borchardt to withdraw his theorem in Sylvester's name. Final sentence:
Lorsque l'éminent géomètre qui a enrichi de si belles découvertes la théorie des déterminants et l'algèbre des fonctions entières en général, et dont les contributions forment un ornement bien precieux de ce Journal, reviendra sur sa théorie des déterminants composés et qu'il voudra bien destiner pour mon Journal la rectification dont sa formule générale est susceptible, il nous fera connaître, on peut en être sûr, un progrès nouveau que cette branche d'algèbre devra à son initiative.(I don't know if the French original sounds as weird as my attempted translation to English:)Berlin, 4 février 1880.
When the eminent geometer, who has enriched by such beautiful discoveries the theory of determinants and the algebra of entire(?) functions in general, and whose contributions form a very precious ornament of this Journal, will come back to his theory of composite(?) determinants and that he will kindly direct to my Journal the correction to which his general formula is susceptible, he will certainly make known to us a new progress, which this branch of algebra owes to his initiative.
(Note the very polite tone: People did not say that "his formula is wrong", but "his formula is susceptible to a correction". Similarly, when Lhuilier discovered that Euler's formula did not hold in all cases and required additional assumptions he phrased it as follows: [Die Formel] ... "erleidet Ausnahmen."
Voevodsky lists several examples from his own experience:
A fast algorithm for embedding a graph on the torus was required. In 1978 Filotti had published a paper [8] presenting an algorithm for embedding 3-regular graphs on the torus. This was followed by a much expanded version [7] in 1980, which corrected a number of minor errors. This was then followed by papers by Filotti, Miller, and Reif (1979), Filotti and Mayer (1980), Miller (1978), and Djidjev and Reif (1991). These papers (except for Djidjev and Reif [1991]) all used Filotti’s techniques to address embedding and isomorphism problems for graphs of bounded genus.
Filotti’s algorithm is based on the planarity testing algorithm of Demoucron, Malgrange and Pertuiset 1964. We start by pointing out a misconception that Filotti [7] and also Gibbons [Algorithmic Graph Theory, Cambridge University Press, 1985, p. 89] had regarding this algorithm.
A fatal flaw in Filotti’s algorithm [7], which also appears in the algorithm of Filotti, Miller, and Reif (1979), is then described. The algorithm of Djidjev and Reif (1991) is also incorrect, and a fundamental error in it is presented. There appears to be no way to fix these problems without creating algorithms which take exponential time.
...
Juvan, Marinček and Mohar have created a linear time torus embedding algorithm [15] (1995). ... But these approaches are very complex, and very difficult to implement and to ensure the resulting code is correct. However, programming these might either point out further flaws in the reasoning or provide a better understanding of these results.
An attempt is being made by Mohar, Orbanic and Bonnington to create an implementation of [15], but as of July 2006, the program still has bugs. (There is a consensus that the two 24-vertex 3-regular obstructions mentioned earlier are torus obstructions, but the code as of July 2006 failed to recognize this).
Errors in reasoning can be subtle and it is a testament to the difficulties of the embedding problem that an algorithm like Filotti’s [7] can stand for 25 years before the errors are pointed out.
László Fejes Tóth, in his book Lagerungen in der Ebene, auf der Kugel und im Raum from 1953 [Section V.12, p. 151] asserts erroneously that a ruled surface such as a hyperboloid of one sheet can be triangulated with a Hausdorff error ε with only O(1/sqrt(ε)) triangles. This was discovered by Mathijs Wintraecken, see Section 2.4 of his Ph.D. thesis, see also our manuscript Optimal triangulation of saddle surfaces.
D. P. Dobkin, L. Snyder, On a general method for maximizing and minimizing among certain geometric problems, in: 20th Annual Symposium on Foundations of Computer Science (FOCS 1979), 1979, pp. 9–17. doi:10.1109/SFCS.1979.28
From Wikipedia: In 1983, Gorenstein announced that the proof of the classification had been completed, but he had been misinformed about the status of the proof of classification of quasithin groups, which had a serious gap in it. A complete proof for this case was published by Aschbacher and Smith in 2004.