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Suppose that a famous mathematician E (long deceased) made a conjecture (C), but no written record of this can be found (either because none exists, or because it exists in a very obscure source). Then, a senior mathematician S, who knows for a fact that (C) was conjectured by E, tells another mathematician X about this problem, and indeed X manages to prove (C). Just to clarify: S is sure about the conjecture, but does not have a publicly available source for it.

This creates a problem: one of the reasons for X's interest in the problem was the question asked by E. Without context, the problem is still interesting, but less so, and it would be better to include a reference to the conjecture whenever communicating the result. Presumably, in verbal communication this is not a problem, since one can simply explain how things are. But what about publishing the result?

I see the following options. Which of these are most widely accepted? Is there any better alternative?

  1. Not mention E at all, simply state the result as is.
  2. Mention that (C) was conjectured by E but not make effort to justify this.
  3. Mention that (C) was conjectured by E, try to use elaborately justify that.
  4. Keep digging for a reference (the real question here is - for how long.).

The question is most natural in mathematics, but I suppose it makes some sense in other disciplines. Feel free to post non-mathematics based answers/comments.

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  • How does S know that the conjecture was made by E? Oct 2, 2014 at 1:31
  • If this make a difference, he saw it in print somewhere, but much search has not revealed a source, which might well be some obscure conference proceedings. My point is that he S is certain that E made the conjecture, and S can be relied upon. Oct 2, 2014 at 1:41
  • Can you ask E to confirm that she conjectured it? Or, if that is not possible (e.g. if E has "left" in the sense of Erdős), can you ask S to try to recall where/when the conjecture was made? Oct 2, 2014 at 1:49
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    "E conjectured C. (S. Personal communication.)" You've given credit where credit is due and cited your source.
    – Bob Brown
    Oct 2, 2014 at 2:36
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    Alternative: S (personal communication) reports a conjecture by E, stating that C holds. May 9, 2016 at 18:52

2 Answers 2

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"Which of these are most widely accepted?" sounds like you're asking a poll-type question, which is not quite kosher for this site. I believe that the situation is sufficiently unusual that there is no "standard" answer. Nevertheless I will say what I think.

First of all I want to express some skepticism of the premise that the information that Conjecture (C) was made by famous, long-deceased professor E is an important to the extent that the work of the paper would be less interesting without it. If Conjecture (C) had been well known to the community and many other people had worked on it and/or written about it, then its provenance would be key information: those who knew about the conjecture would make a clear audience for the paper and would vouch for the prestige and value of the work. But if no one -- or almost no one -- except S has prior familiarity with Conjecture (C), then the information that it is due to professor E becomes more of a piece of trivia / personal motivation for professor X. If the provenance of the conjecture had really been a key motivation for X, then one wonders why X did not verify this before working on the problem.

Second I want to say that I think you are using "knows for a fact" in a way different from the way I would use it. If S is "sure" about the provenance of the conjecture because he saw it in print once but now cannot locate that printed source -- then actually he is not sure, I would say. Often someone thinks they saw a certain piece of information in print, but when they go back to check it turns out they did not remember it accurately. In fact probably every working academic has had this experience more than once. I'm sure S is great, but "S says it, and S can be relied upon" is not really convincing to me no matter who S is. In a professional context, the things that one is "sure" about should be verifiable: if E is a famous mathematician, then it should be possible to get a comprehensive list of their publications, and then with sufficient effort one can look through all of them, including "obscure conference proceedings".

The way I would handle the situation is by first corresponding with S and making sure that S is completely comfortable with being attributed the claim that Conjecture (C) was first made by E: if this gets published, then S may get asked some questions about this. If they are not comfortable I would go with something more mild like mentioning S in the acknowledgments. If on the other hand S really insists that Conjecture (C) is due to E, then that is what X knows and X can put that in the paper in that form, i.e., something like

Conjecture (C) was conveyed to me by S. According to Professor S, the Conjecture was first made in a paper of E. Unfortunately we have not been able to track down a reference."

I would also be prepared to get some questions about this from the editor or referee when you submit the paper.

Finally: have you asked about this on MathOverflow? This would be a very appropriate question on that site: if the paper exists then someone ought to know how to find it, though you may not know nor even know how to find that person. Even if the question does not get answered in this way, you (or X) can still point to it as proof of your due diligence in the matter.

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  • A good answer. It sounds from the question, though, as if the conjecture was unpublished. Perhaps it was mentioned in a talk which S attended and took notes. Or maybe S is in possession of E's unpublished notes, and saw it there. So it could be that evidence of the conjecture exists, but is not in the public record (and so nobody except S might be able to find it). Oct 2, 2014 at 6:38
  • @Nate: What I got from the comments is that (S claims that) S found the conjecture in print. If S learned about it in any other way then including that information in the paper would be a good idea. To my mind, if you e.g. make a conjecture during a talk and then never include it in your (presumably copious) written work, then that says something about how serious you were. Oct 2, 2014 at 13:27
  • Thank you for the answer. Just to clarify, in relation to the first point that you make: I do not mean a major conjecture whose importance is clear to any experts. I mean a (sort of) natural question about elementary objects. One that you ask because it is natural to ask, but do not expect very deep insight from it. In combinatorics/number theory there are quite a lot of those. Oct 2, 2014 at 13:27
  • As for MathOverflow - I haven't tried that, but I will. Oct 2, 2014 at 17:00
  • I agree with the part about the source of the conjecture not being important for the importance of the result (while it can certainly have a big impact on whether it feels like a worthwhile conjecture to work on). On a different note, I do know of at least one conjecture where as far as I know the only places it is in print is in papers dealing with special cases of it (not written by the conjecturer). Oct 2, 2014 at 19:33
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This is a very old question. I would like to add that in my experience, this is actually quite a common occurrence in mathematics, counter to the belief of the previous answer. It's not usually that Famous Mathematician E whispered their conjecture into the ear of Professor S in secret. Rather, that Famous Mathematician E (potentially in an informal context, perhaps not) made a conjecture, and that conjecture---now in a very modified form that fits modern notation and perhaps a more generalized conclusion than was originally postulated---is still referred to as the "E Conjecture" even though it's hard to find an original source (and that original source would not be helpful to the problem at hand).

The solution is simple: find another paper in the literature that names the explicit form of this conjecture (with the right assumptions, etc.) that you want to work with, and cite it as a conjecture in your paper, with something like "E Conjecture (as stated in [reference]).

Examples:

When writing something about the Sato-Tate conjecture, for instance, I was unable to find the original reference for the conjecture: as far as I understand, the conjecture was made by John Tate, to explain computational data recently found by Mikio Sato, about the distribution of Fourier coefficients of modular forms corresponding to non-CM elliptic curves. The statement was later expanded to correspond to all non-CM automorphic forms with trivial nebentypus, and was then proven in 2011. However, it is still referred to, even in the most generalized form, as the "Sato-Tate conjecture" in most of the literature. (even though it is not a conjecture anymore, and the now-proven statement is far more general than what Tate originally envisioned envisioned).

Similarly, suppose you're writing something about the Generalized Riemann Hypothesis for a specific class of L-function, say, Dirichlet L-functions, Rankin-Selberg L-functions, or Symmetric power L-functions; you're unlikely to find a single, first postulation of the conjecture as the "Generalized Riemann Hypothesis for ____ L-functions," even if it is commonly referred to as such in the literature.

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