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I wonder if there are 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.)

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I have a déjà vu :) Also, maybe community wiki (~big list)? – efq 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 :) – romkyns Aug 13 '10 at 10:44
4… is a similar question - only the subject of the proof was later confirmed to be true. – romkyns 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 – BCnrd Aug 13 '10 at 14:35

36 Answers 36

The wronskian determinant of $n$ functions which are $(n-1)$ times differentiable is $$W(f_1,\dotsc,f_n)=\det\begin{pmatrix} f_1 & f_2 & \dots & f_n\\ f_1' & f_2' & \dots & f_n'\\ \vdots & & & \vdots\\ f_1^{(n-1)} & f_2^{(n-1)} & \dots & f_n^{(n-1)} \end{pmatrix}.$$

It is clear that if $f_1$, ..., $f_n$ are linearly dependent, $W(f_1,\dotsc,f_n)=0$. For some years, the converse was assumed to be true too, until Peano gave the counterexample: $f_1=x^2$ and $f_2=x\cdot |x|$ are linearly independent, though $W(f_1,f_2)=0$ everywhere. Later, Bôcher even gave counter examples with infinitely differentiable functions.

Bôcher also proved that the converse holds as soon as the functions are analytic. Other conditions are also known for the converse to hold.

Engdahl and Parker describe the history of the wronskian [1]. For a nice proof of Bôcher's result, one can have a look at a paper of Bostan and Dumas [2].

[1] Susannah M. Engdahl and Adam E. Parker. Peano on Wronskians: A Translation, Loci (April 2011), DOI:10.4169/loci003642.
[2] Alin Bostan and Philippe Dumas. Wronskians and linear independence, American Mathematical Monthly, vol. 117, no. 8, pp. 722–727, 2010.

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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.

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The Steiner Ratio Gilbert–Pollak Conjecture was proof 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:

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

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Yes, that thread was mentioned in KConrad's comment of 15 August. – Gerry Myerson 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.

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But this proof was immediately rejected. See 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. – Colin McLarty Dec 13 '14 at 23:40

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).

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protected by François G. Dorais Oct 5 '13 at 17:43

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