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Suppose I work on a difficult open problem, try several directions that haven't been tried before, work for several weeks on these directions, but finally reach a dead end.

Is there a way I can publish my work, so that future researchers know that these directions are futile and won't have to waste several weeks just to get to the same conclusion?

7

I think that one has to distinguish different kinds of failures, and also see if the field is experimental or not.

If you have done experiments and they show that some approach does not work, it is perfectly fine to publish that (although, as is discussed in various places, you often do not get the credit that you deserve for this).

If your work is theoretical, it is important that you can formalize the "dead ends" in a sufficient useful way; for example by proving that objects of a certain kind or with certain properties do not exist or that an (intuitive) lemma is wrong which can be shown by a counterexample.

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