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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