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I have found a conjecture in a research article (published in a good journal) on number theory, which is not well known but very reasonable. Let me be clear that, there is no counter-example that vote down the conjecture, rather, its trueness has been proved in some special situation so far. I need this conjecture to develop a tool in order to write a research article. Unfortunately, I am unable to prove the conjecture as of now. Maybe others can prove it after some time.

My question:

Is it a reasonable way to write a research article assuming truth of the conjecture?

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    $\begingroup$ There are lots of excellent papers based on conjectures. $\endgroup$
    – GH from MO
    Commented Feb 13, 2021 at 4:09
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    $\begingroup$ The main thing you want to make sure is that your result is strong or interesting enough that the reader feels it was "worth" assuming the conjecture, and that your explanation of why the conjecture is plausible (whether by citing previous work or your own work) is clear. $\endgroup$
    – Will Sawin
    Commented Feb 13, 2021 at 4:35
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    $\begingroup$ That's quite usual, especially in number theory. $\endgroup$
    – Fedor Petrov
    Commented Feb 13, 2021 at 6:29
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    $\begingroup$ It seems to me a little awkwardly/informally stated. The article itself doesn't assume truth of the conjecture. But the main result of the article (and hence the interest of the article) can assume truth of the conjecture. So, if the conjecture fails, the article is not false, it's just void. By the way it can be useful to clearly separate, in the paper, what relies and what doesn't rely on the conjecture. $\endgroup$
    – YCor
    Commented Feb 13, 2021 at 15:30
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    $\begingroup$ Now whether this can be done (whether a journal should publish it) depends on the conjecture. Whether it's a 150 years old conjecture that it widely expected to hold, with serious evidence, or a conjecture asked 2 years ago in the author's unpublished preprint, makes the picture a little different (of course these are extreme cases). $\endgroup$
    – YCor
    Commented Feb 13, 2021 at 15:31

3 Answers 3

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A research article which assumes the truth of a conjecture also counts (indirectly) as research on the conjecture itself. If you show that certain results follow from a conjecture and those results are later shown to be false, then the original conjecture would have been shown to be false. Another possibility is that the results that you derive could be shown to be equivalent to the original conjecture, in which case independent confirmation of those results will prove the original conjecture. Research of the sort that you are proposing increases the attack surface of the original problem.

Whether or not the result will be publishable is a question which can't be answered in the abstract and is best left to the editor of whatever journal you intend to submit it to.

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As pointed out in the comments, this is common in number theory. Since the OP wants to write a paper, I give some concrete examples. If you google "Assume the Generalized Riemann Hypothesis" you get 4900 results, including theorems of Hecke (1918), Deuring (1933), Mordell (1934), and Heilbronn (1934) all assuming either the Riemann hypothesis or that it's false. Here [PK07] is a more recent paper (published in Number Theory) that assumes the generalized RH. Here [CC15] and here [Ju21] are two other examples, and many more on Google. Hopefully this will help the OP figure out how to write what they want to write.

[PK07] Park and Kwon - Class number one problem for normal CM-fields

[CC15] Carneiro, Chandee, and Milinovich - A note on the zeros of zeta and L-functions

[Ju21] Just - On upper bounds for the count of elite primes

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    $\begingroup$ The difference is that the Riemann hypothesis is a) very famous and b) many people believe it is true. While the conjecture in question is "not well known". $\endgroup$
    – Alexandre Eremenko
    Commented Feb 13, 2021 at 14:20
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    $\begingroup$ @AlexandreEremenko Sure, but the only way a conjecture can get well known is people mentioning it in papers. The opposite phenomenon, of someone making a new conjecture, proving it in a special case or proviiding other evidence, and then deriving a consequence from it, is also fairly common. $\endgroup$
    – Will Sawin
    Commented Feb 13, 2021 at 15:09
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    $\begingroup$ I usually edit 'here' paper links to change them to the names of the linked papers, but, here (!), it seemed like that might disrupt the flow of the exposition. Nonetheless, it seems a shame to have this process subject to link rot, so I edited the names of the papers in at the end. I hope that was all right. // Also, 'including' is a link to the Wikipedia article on the RH. Was that intentional? $\endgroup$
    – LSpice
    Commented Feb 13, 2021 at 23:31
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This is not fundamentally different from the (fairly subtle) question of whether any mathematical result in general is "interesting". Publication typically requires (at least) a result that is both true and interesting. From a logical viewpoint, there is absolutely no problem with proving the truth of a theorem that includes a conjecture in its hypothesis. It is a theorem like any other.

Meanwhile, a major factor in what makes a result uninteresting for publication is if it can be trivially derived from simpler results. This is not an objective characteristic but a time-dependent fact of human knowledge.

In your situation, if and when the conjecture is resolved, we know it will be possible to derive your theorem trivially from a simpler result. Namely, if the conjecture is true, there will be a simpler theorem that omits the conjecture from the hypothesis, from which your theorem will follow trivially. And if the conjecture is false, your theorem will be derivable as a tautology. But I would argue that until we know which obtains, your theorem is not yet uninteresting.

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    $\begingroup$ There's probably a better way to say it than "your theorem is not yet uninteresting." $\endgroup$
    – LSpice
    Commented Feb 13, 2021 at 23:28

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