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An open problem that seems to get a lot of attention every once in a while is the amenability of Thompsons group $F$.
The problem seems to generate both proofs and disproofs at a fairly high rate, compared to many other open problems.
What is more, it seems that a big part of these are actually serious attempts by serious mathematicians, rather than the "usual" elementary attempts one sees for the more famous problems.

For examples, see for instance the MO question Is Thompson's Group F amenable? as well as (what as far as I can tell is the newest attempt, but I may have missed some) http://arxiv.org/abs/1408.2188.

Is there something inherent to this problem which causes this, i.e. some aspect that makes so many serious mathematicians convince themselves that they have a solution, and so many other serious mathematicians to take so long to find the errors?

Note that I am specifically not asking about what the errors were in previous attempts, unless there is some general type of error that tends to come up in many of them

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Tobias Kildetoft
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    The sample size seems pretty small here. Nothing that can't be explained by random variation in what problems people think they've solved. – Qiaochu Yuan Nov 13 '14 at 08:35
  • @QiaochuYuan I am not sure I understand what you mean by "what problems people think they have solved". What problems do you mean? – Tobias Kildetoft Nov 13 '14 at 08:42
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    I do not think that there is a good answer to this question. What kind of answer are you looking for? How to measure the difficulty of an open problem if not in terms of failed attempts? Maybe one reason for the number of attempts is that the problem is elementary to state, but I wouldn't call this "inherent to the problem". – Andreas Thom Nov 13 '14 at 08:47
  • I mean consider a random model where each mathematician believes they have solved a random open problem with some probability. It is moderately likely that some open problem, by chance, has the property that somewhat more mathematicians believe they have solved it than other problems. That isn't strong evidence that the problem is particularly special. – Qiaochu Yuan Nov 13 '14 at 08:48
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    @QiaochuYuan But mathematicians thinking they have solved a (quite wellknown) open problem for long enough to actually announce the proof is a fairly rare occurrence, so given the large number of such problems, it seems that it should not take that many to be an unlikely outlier. – Tobias Kildetoft Nov 13 '14 at 09:04
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    See also: http://meta.mathoverflow.net/questions/1971/suitability-about-a-question-on-amenability-of-thompsons-group-f – Todd Trimble Nov 13 '14 at 13:01
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    @QiaochuYuan By way of comparison, one doesn't see at least two people announce proofs that all free group factors are distinct while two more announce proofs that they are all isomorphic – Yemon Choi Nov 13 '14 at 13:32
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    One reason might be that there are some numerical quantities that imply amenability of a group and some of the experimental approximations of these quantities for F seem to hang right near the border. – Benjamin Steinberg Nov 13 '14 at 14:01
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    Many famous open problems probably have some consensus as to which way the answer lies. My feeling is the experts are 50-50 on this one. – Benjamin Steinberg Nov 13 '14 at 14:54
  • There is some "numerical evidence" (that is to say no evidence at all) that Thomson's group is non-amenable: see http://arxiv.org/abs/1312.5722 – ARG Nov 25 '14 at 09:54
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    In my opinion, the question posed by Tobias has replaced the amenability question as the main question of interest. A solution of the original problem that fails to show where the difficulty lay would be quite disappointing. – Matt Brin Jun 16 '15 at 08:46

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For some famous open problems, there are known "obstructions" to resolving them. For example in number theory we have Siegel zeros and the parity problem for sieves, and in computational complexity theory we have naturalization and relativization/algebrization barriers. The obstructions are well known and ideas for surmounting them are scarce, so if a proposed solution appears, experts can often zoom in quickly on where the new idea for dealing with the obstruction must lie. It's harder to be fooled in such cases.

For other open problems, the conceptual territory surrounding the problem is less well understood and mapped out. Instead of a mountain range with only a few well-known passes, the obstacles form a forest with myriad trees among which it is easy to get lost. I know nothing about amenable groups, but if what you say is correct, I would hypothesize that the conjecture falls into the latter category. There are lots of different approaches one could try, and instead of running into the same difficulty over and over again, each new attempt makes a foray into uncharted territory where there is not much past experience to help detect the pitfalls.

Timothy Chow
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    Also amenability has a vast number of equivalent formulations, not all so trivial to prove which leads to many avenues of attack. – Benjamin Steinberg Nov 13 '14 at 17:32
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    I started to write a vague answer but deleted it. The gist of it was as follows: there seem to be two worlds, the "obviously amenable" and the "obviously non-amenable", and many examples one thinks of fall into these classes. Thompson's F belongs to neither. So we have easily applicable sufficient conditions and easily applicable necessary conditions but a big gap in between, and Thompson's F is somehow one of a small number of explicitly described groups that lie in the gap. What I don't know is how it differs from groups in the gap which are nevertheless known to be amenable or non-amenable. – Yemon Choi Nov 14 '14 at 14:15
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    The amenability of Thompson's group does not appear to be an isolated instance of this. The complexity of graph isomorphism is somewhat similar, and led the author of this paper to use the term "disease": http://onlinelibrary.wiley.com/doi/10.1002/jgt.3190010410/pdf

    So now write 'The Thompson Group Amenability Disease' or something like that? And maybe 'The Jacobian Conjecture Disease' while we're at it? What other examples of 'diseases' are out there?

     – DavidLHarden Nov 18 '14 at 23:37
  • @DavidLHarden I am not sure I agree with your comment. The key feature of the problem for Thompson F is that people have made serious claims for and against its amenability. This distinguishes it from problems like Jacobian conjecture, Collatz, RH etc where most failed attempts seek a positive answer – Yemon Choi Nov 19 '14 at 01:29
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    @DavidLHarden Also, see this from Tobias's question: "some aspect that makes so many serious mathematicians convince themselves that they have a solution, and so many other serious mathematicians to take so long to find the errors?" – Yemon Choi Nov 19 '14 at 01:31