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This is quite a general question... apologies if it is too general; I'm just wondering if anyone has any useful tips or advice.

As part my final year university project I've made an algebriac/combinatorial conjecture (I won't say exactly what as I'm not asking for homework help) about something that looks like it might be true... but I also wouldn't be entirely surprised if it was false. I'd like to disprove or prove it, so I've split my time between trying very hard on paper to come up with a proof, while simultaneously computationally searching for counterexamples (the search space is quite large and awkward).

On both fronts I seem to be going nowhere... I haven't identified a single counterexample yet (I have lots that do satisfy the conjecture though), but equally a proof always seems to be just out of reach (there always seems to be one logical step in the way).

Basically this is the first time I've really had to do my "own" maths, so to speak, so I don't really have enough experience or gut instinct to know inuitively whether this is worth pursuing. I fear it's very possible that one of

  1. the counterexample is computationally out of reach;
  2. the proof is beyond my mathematical abilities; or
  3. one them is actually within my reach and I'm about to give up prematurely.

So I'm wondering if people with more experience might be able to share any strategies they use when they find themselves in this situation. Are there indicators/tell-tale-signs I should be looking for to help decide what's more likely to be the case? And does anyone have any advice on how "urn the problem around and start looking at in a different light?

Many thanks.

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    $\begingroup$ Well, not in general, no. If there were a reliable test for when a conjecture was true or not, life would be a lot simpler than it is. Of course, in specific cases it may be possible to say more, but you haven't really given us any information about the problem. $\endgroup$
    – lulu
    Jan 4, 2022 at 15:40
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    $\begingroup$ One thing that sometimes sheds light: try to prove (or disprove) the thing heuristically. That is, make whatever broad assumptions you need to make in order to proceed. Assume, say, that various quantities are as uniformly distributed as they can be. (Again, I have no idea if this is relevant to your particular problem or not). To be sure, this is unlikely to resolve the problem completely, but it might illuminate the issues. $\endgroup$
    – lulu
    Jan 4, 2022 at 15:48
  • $\begingroup$ Related. $\endgroup$
    – J.G.
    Jan 4, 2022 at 16:00
  • $\begingroup$ Unfortunately , when dealing with a new not yet examined problem , you can never be sure whether you have overlooked a relatively simple solution, or whether the problem is extremely difficult or even impossible to solve. There are many very easy seeming open conjectures in number theory , in particular concerning prime numbers. $\endgroup$
    – Peter
    Jan 4, 2022 at 17:46
  • $\begingroup$ Does this answer your question? Maximum amount of time to solve a problem and persistence $\endgroup$ Oct 9, 2023 at 2:04

1 Answer 1

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Congratulations. You're learning how real mathematical research works.

You alternate between believing and doubting. When you are stuck on one you suspect the other and switch modes.

Over time you sense how to choose problems that are goldilocks for you: hard enough to be interesting, not so hard as to be impossible. But it takes time to develop that intuition.

One suggestion for now: talk to people - fellow students, faculty you know. Sometimes explaining your conjecture to a new listener will trigger an insight. Perhaps try explaining to your rubber duck. Or ask about a fragment of your puzzle here on stackexchange.

Finally: you might be able to write an excellent final year project in which you present evidence showing that your conjecture is interesting and difficult.

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