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One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was proven in the paper.

The work was reported on a few seminars, and several experts found the conjectures interesting.

One would like to publish this paper in a refereed journal. The paper was rejected from a certain journal just two days after its submission because "this genre of article does not fit the journal".

QUESTION. Are there examples of publications of this genre in refereed journals?

ADD: The mentioned paper explains the background, states the conjectures, discusses various special cases and consequences, and lists known cases. It is 20 pages long.

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  • $\begingroup$ The thing to do is to wait until the turn of the century, then give a talk at the International Congress of Mathematicians. Well, it worked for Hilbert. $\endgroup$ – Gerry Myerson Sep 11 at 22:14
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If you have numerical evidence in support of the conjecture, the journal of Experimental Mathematics seems to fit the bill:

Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses.

Note that we do value proofs: experimentally inspired results that can be proved are more desirable than conjectural ones. However, we do publish significant conjectures or explorations in the hope of inspiring other, perhaps better-equipped researchers to carry on the investigation. The objective of Experimental Mathematics is to play a role in the discovery of formal proofs, not to displace them.

Several publications in that journal have gotten quite some traction, like:

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Depending upon the content of the paper, you may look at the new Arnold Mathematical Journal (http://www.springer.com/mathematics/journal/40598).

From their site, "Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final" form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold's principle, a general formulation is less desirable than the simplest partial case that is still unknown."

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Supplementing Carlo's answer, I also want to recommend the journal 'Mathematics of Computation'. Several important number-theoretic conjectures were first stated there, supported with numerical data. So if some computation was involved in providing evidence for your conjectures, it seems like a suitable choice.

Here are some notable examples:

  1. "The Riemann Hypothesis and Pseudorandom Features of the Möbius Sequence" by Good and Churchhouse
  2. "A heuristic asymptotic formula concerning the distribution of prime numbers " by Bateman and Horn - This is a conjecture which generalizes Dirichlet's theorem on primes in arithmetic progression.
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  • $\begingroup$ These are presumably very famous papers. There must exist some more recent numerical calculations related to these papers, could you (or anyone else) supply this kind of information? $\endgroup$ – მამუკა ჯიბლაძე Apr 29 '17 at 10:36
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    $\begingroup$ @მამუკაჯიბლაძე Regarding the first conjecture: Nowadays we understand much better the conjecture of Good and Churchhouse. In particular, we realize that it is strongly related to (and probably implied by) Montgomery's Pair Correlation Conjecture, which tells us the gap distribution of spacings between zeroes of the zeta function (it is much deeper than the Riemann Hypothesis). In the following paper of Nathan Ng, Section 4, this connection is discussed briefly: cs.uleth.ca/~nathanng/RESEARCH/mobiusshort.pdf . The Pair Correlation Conjecture was studied numerically by Odlyzko, ... $\endgroup$ – Ofir Gorodetsky Apr 29 '17 at 12:42
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    $\begingroup$ (cont.) ... also in the journal "Mathematics of Computation", see his paper jstor.org/stable/2007890 . Moreover, a function field analogue of the Pair Correlation Conjecture was recently proved by N. Katz, allowing Keating and Rudnick to prove a function field analogue of Good and Churchhouse's conjecture, see their paper here: math.tau.ac.il/~rudnick/papers/antsfs.pdf . $\endgroup$ – Ofir Gorodetsky Apr 29 '17 at 12:44
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    $\begingroup$ (cont.) As for the second conjecture: I do not know enough about the numerical aspect (although Lal studied a very specific case here link.springer.com/article/10.1007%2FBF01951947?LI=true ), but again - a function field analogue of the Bateman-Horn conjecture was essentially settled in the limit of a large finite field. See the work of Entin and the references therein: arxiv.org/abs/1409.0846 . $\endgroup$ – Ofir Gorodetsky Apr 29 '17 at 12:48
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The main contribution of the paper "The McKay conjecture and Galois automorphisms" by Gabriel Navarro is to propose various conjectures. It is published in the Annals of Mathematics.

http://www.ams.org/mathscinet-getitem?mr=2144975

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Yes, there are quite a few papers of this kind. Of course, if proving the implication to known theorems or between the various conjectures is interesting and challenging then this is "ordinary". If the paper contains only a statement of the conjectures then this is indeed more special. There are journals which have short note sections or problem sections appropriate for such papers. (I published a couple of papers with just conjectures in such sections.)

As a general rule, it always makes sense to try other journals, to consult with friends, colleagues and teachers, to put the paper on the arxiv, and to try to improve the paper.

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    $\begingroup$ Thank you. I would curious to see some references to such papers. I can add that the mentioned paper explains the background, states the conjectures, discusses various special cases and consequences, and lists known cases. It is 20 pages long. $\endgroup$ – MKO Apr 28 '17 at 14:01
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    $\begingroup$ "There are papers" -> "There are journals"? $\endgroup$ – David Roberts Apr 29 '17 at 2:33
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    $\begingroup$ Here are 4 such papers. N. Alon, A. Shpilka and C. Umans, On Sunflowers and Matrix Multiplication, IEEE Conference on Computational Complexity 2012, 214-223. Also: Computational Complexity 22 (2013), 219-243; G. Kalai, A new approach to Turan's Problem (research problem) Graphs and Comb. 1 (1985), 107-109;G. Kalai, On the number of faces of centrally-symmetric convex polytopes (research problem), Graphs and Combinatorics, 5 (1989), 389-391. J. Kahn and G. Kalai, Thresholds and Expectation Thresholds, Combinatorics, Probability and Computing 16 (2007), pp. 495-502. $\endgroup$ – Gil Kalai May 1 '17 at 19:22
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The editorial policy of the Ulam Quarterly Journal claims that the journal is "devoted to the publication of original research and open problems in all areas of mathematics."

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This is interesting to me too. In my case, I decided for a simple approach: by posting on the arXiv, here is one such paper. I also created a separate route on my own website. Since then the conjectures have received several attention as you can see at these coordinates where I keep updating new papers proving the claims or certain partial progress. That is one possible choice though, you may still opt to find a home for your work in some journals, such as in Experimental Mathematics as Carlo Beenakker pointed out. Perhaps otheres can name the journals they like suggesting.

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For open problems/conjectures in combinatorial and discrete geometry, there's the journal Geombinatorics focusing on "live mathematics," i.e., research in progress.

One big title is Aubrey de Grey's 2018 "The chromatic number of the plane is at least 5."

The explanation of the journal's topical parameters includes the advice, "We encourage mathematicians in other areas to start similar lively publications." It seems there are some further examples now.

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Such a paper might be appropriate for the Graduate Journal of Mathematics, since a readership of grad students might enjoy a bunch of interesting conjectures. I published a paper there that was mostly a rewriting of results known in computer science, in more mathematical language, plus a list of open problems.

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