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I am working on a problem where a particular estimate/conjectural inequality appears to hold very robustly in extensive numerical experiments, but I cannot find a viable proof route. The overall statement is relatively minor but has implications for the larger project.

I am looking for community best practices on what to do next in this situation, especially when one has limited resources as an amateur mathematician (no large compute cluster, not part of a research group, etc.).

Concretely, I would appreciate advice on questions like:

  • What kinds of numerical experiments are considered most informative for mathematicians: stress tests, worst-case searches, parameter sweeps, plotting normalized quantities, fitting to plausible main terms, etc.?
  • What standards are typical for reporting computations responsibly (reproducibility, error control, interval arithmetic, certified computation), and where does one draw the line between “suggestive” and “compelling” evidence?
  • When (if ever) is it appropriate to attempt to publish when one of the propositions is empirically supported but not fully proved? (Making clear that the proposition is conjectural)

I am specifically looking for methodological guidance and references to established practices. Any pointers to papers/books on certified computation in analytic number theory (or adjacent areas) and on good “conjecture hygiene” would also be welcome.

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    $\begingroup$ You could follow the examples of papers published here: tandfonline.com/journals/uexm20/… depending on the relevance of your conjecture and your computations, you could publish your results there too $\endgroup$ Commented Feb 26 at 0:34
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    $\begingroup$ There are journals that publish experimental results (like the Journal of Experimental Mathematics), but it is unusual to publish work that is purely experimental. In general, it is not easy to publish papers that don't prove something. There are papers that are meant to state conjectures and assemble evidence for them. For instance, I wrote one here: arxiv.org/abs/1208.3216. However, I would discourage someone from writing a paper like that if they don't already have a track record of proving theorems since it will probably be hard to get it published. $\endgroup$ Commented Feb 26 at 1:12
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    $\begingroup$ But "prove something" doesn't always mean "prove what you want to prove". I think a paper which states an appealing conjecture and verifies many special cases numerically can be a very nice paper. And if you can get a preprint version in front of the right experts, they may prove your conjecture! That's what happened for us with arxiv.org/abs/1607.00047v1 . In our case, Luke Pebody proved our conjecture after seeing arXiv v1, but I think it would have made a good paper even if he hadn't. $\endgroup$ Commented Feb 26 at 2:26
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    $\begingroup$ @jpvigneaux The board of Experimental Mathematics resigned en masse a couple of years ago and many of them are now with the Journal of Experimental Mathematics. $\endgroup$ Commented 2 days ago
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    $\begingroup$ Since you ask about "papers/books on certified computation in analytic number theory (or adjacent areas)", keep in mind that analytic number theory is notorious for having had conjectures supported by a lot of numerical data turn out to be wrong. $\endgroup$ Commented 10 hours ago

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I don't think that there's any one-size-fits-all advice that applies to all cases. The right approach will depend on the specific conjecture. However, a few things can be said.

  1. Especially if this is a new conjecture, don't focus just on numerical evidence. Approach it like any other conjecture. Can you prove special cases? If so, can you explain what prevents your proof from generalizing? Can you generalize the conjecture itself (a plausible but false generalization may be interesting)? Can you relate it to known conjectures? Can you give a heuristic argument for why it should be true (besides the numerical evidence)? Can you find an alternative formulation of the conjecture that is equivalent to the original conjecture, but not obviously so?
  2. If you have carried out step 1 to the best of your ability, it may suggest which numerical explorations are the more interesting ones. For example, maybe there's some special case that seems just barely out of reach, or maybe there's some region of the parameter space that seems like the most likely candidate for counterexamples.
  3. Think hard about efficient algorithms for carrying out your computation. Generally, people will be more interested in a new idea you have for an efficient algorithm than they will be in how big a computer you have.
  4. In terms of writing up your results, there are several MO questions about documenting computer-assisted work, e.g., Computer calculations in a paper and other questions that are linked to it.
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