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Some time ago I had a chat with a friend (and colleague) about some statement I wanted to prove. I was (and am) sure the statement is true, but couldn't prove it. I described some of my attempts and explained my difficulties.

After a month or so, he came to me and said he thought about it, tried different ideas but nonetheless was unable to prove the result.

In the meanwhile, I was preparing a paper in which I wanted to collect preliminary results, without mentioning the above claim. After the second chat, however, I decided to include the claim we both couldn't prove as a conjecture at the end of the paper. This claim adds something substantial to the paper, I think.

Now, my draft paper was originally intended as authored only by me, but I was motivated to add the final conjecture precisely because my friend (who is skilled mathematician) could not prove it either. I'm curious about how this is perceived in general, so:

Question 1: do you think that co-authorship was in order, in this case?

(By the way, I proposed him a co-authorship, and he refused, so no problem for me). More generally:

Question 2: can you imagine circumstances in which a no-result effort is enough for authorship?

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    $\begingroup$ Based on the information you've given, coauthorship seems like a stretch. In my opinion a statement of gratitude to your friend in the acknowledgements section regarding this conjecture would be the right thing. $\endgroup$
    – Aleksandar Milivojević
    Commented Apr 21, 2022 at 8:40
  • $\begingroup$ Yeah, I tend to agree, and indeed when I proposed co-authorship I thought we could possibly work further on the subject before closing the paper. I added a more general question I'm curious about. $\endgroup$
    – Alessandro Della Corte
    Commented Apr 21, 2022 at 9:10
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    $\begingroup$ There exist problems where 'no result' is all we can hope for. $\endgroup$
    – Roland Bacher
    Commented Apr 21, 2022 at 10:33
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    $\begingroup$ For question 2, in general, if mathematicians agree to work seriously on a problem together, and do, then they all deserve coauthorship even if some of their works were more fruitful than others. This differs from your scenario in that just asking a question doesn't usually constitute such agreement. $\endgroup$
    – Will Sawin
    Commented Apr 27, 2022 at 20:11
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    $\begingroup$ If in the course of working on the problem, your friend amassed evidence towards the conjecture, like special cases, or weaker claims, then these should go in the paper, and coauthorship would definitely be warranted. As it stands, you might mention in the paper, with permission, the politely declined offer of coauthorship based on investigation of the conjecture (I saw this yesterday in a paper, in a slightly different setup). $\endgroup$
    – David Roberts ♦
    Commented Apr 27, 2022 at 21:01

1 Answer 1

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For question 2, consider the following scenario.

There are two mathematicians. Alice chooses a problem and comes up with $N$ possible approaches to solve it. Bob tries $N-1$ of the approaches and can't make them work, and reports this to Alice, who tries the $N$th approach, and succeeds. I think it's clear that for $N$ sufficiently large, Bob deserves coauthorship.

I am not sure exactly what the cutoff is, and it depends on unspecified details, but I think the large $N$ limit is fairly clear.

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  • $\begingroup$ Very sensible. It's crucial that Alice proposed the problem and the $N$ approaches. If instead it was known (for mathematical reasons) that one of them had to work, then Bob in fact had a positive result when excluding some. $\endgroup$
    – Alessandro Della Corte
    Commented Apr 27, 2022 at 20:55
  • $\begingroup$ I like this scenario but in some sense it does not quite address Question 2. I presume Alice put quit a bit more than "no-effort" into coming up with all $N$ of those approaches. $\endgroup$
    – Lee Mosher
    Commented Apr 28, 2022 at 1:52
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    $\begingroup$ @LeeMosher The question is not about a no-effort result but a no-result effort (which Bob did, in this scenario). $\endgroup$
    – Will Sawin
    Commented Apr 28, 2022 at 2:57

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