How can I know if my conjecture is not lacking mathematical formality?

Question

I'm a teenager and student who came up with his own conjecture. Because I'm not a mathematician and I haven't got the knowledge yet, I would like to know if my conjecture doesn't exceed the limits of the mathematical system. So I am asking if there is any mathematical organisation to which I can send my conjecture and know if it is right or not (as a conjecture). Also, can I trust these organisations? I'm pretty sure that due to my lack of knowledge it won't be correct but you never know. Moreover, can a conjecture exceed the limits of the mathematical system? I'm asking this because I thought that a conjecture isn't a theorem and everybody can say whatever he wants if there isn't a proof about the opposite thing. Last thing, I want to know if I have to send the reason I believe what I believe and made it a conjecture (some sort of proof)? I'm a foreigner so I'm sorry for my English. Thanks in advance!

Answer 1

A couple of things:

• A conjecture is indeed something that one believes is true, but something that hasn't been proven or disproven.
• First you need to convince people that your conjecture is interesting. Just stating why you think something is true is not worth anything. You need to back it up with some evidence. You need to connect the conjecture to an already acknowledged area of mathematics and you need to explain it's importance. It is hard to get famous for doing nothing but stating a conjecture. A conjecture isn't about being able to say something without any evidence.
• I don't think sending your conjecture to a academic journal is such a good idea. Journals mostly deal with provable results. If you have results related to your conjecture, then you could certainly type this up and attach your conjecture. This would nicely give some background to the conjecture.
• Depending on the background material needed to understand your conjecture, you could try to talk to your math teacher. (S)he might not be able to tell you is the conjecture is interesting, but s(he) might be able to point your to someone else.
• You could try to email someone at a nearby university/college. Find the mathematics department's website. Most department websites will have a list with the faculty and their research interests. You could try to send your ideas to this person. Now most likely you are not going to be taken seriously, but I honestly think it is worth a shot.
• Others have already suggested it, but why not post it here? Make a new question where you explain your conjecture. Tell us what is interesting about it and why you believe it is true. Then ask for hints/ideas on how to continue making progress on proving/disproving the conjecture. As you might already have found out, if you just post a question without any context you risk the question being closed. So make sure you phrase it properly.
• If you want to make sure that no one steels your conjecture, you do need to publicly post it. This you can do by using your full name and post it to a personal website. Do remember though that conjectures aren't really worth steeling. If you had a really nice proof, then someone might be willing to take the credit for that. This you can still do by posting it here. Just change your user name to your full name and location.
• A small warning: If you do attach your name to the conjecture and you do advocate it somewhere, you run the risk of being ridiculed if the conjecture isn't interesting. Remember: What goes on the Internet stays on the Internet.

Answer 2

Your conjecture falls into some area of math, say number theory for example. The next step would be to consult literature for any work that might hint it has been proven. You could also post a very specific sub-example of your conjecture, or modify it slightly. For example, suppose your conjecture is that $x^p+y^p=z^p$ has no solutions for $p\geq 3$. You could ask instead, does $x^3+y^3=z^3$ have a solution? That way you're not giving away too much. Think of it in terms of Polya's advice: if you cannot solve a difficult math problem, try solving a smaller, simpler version of it.

Answer 3

So I am asking if there is any mathematical organisation to which I can send my conjecture and know if it is right or not (as a conjecture). Also, can I trust these organisations?

I don't believe there are organizations that solicit (or even openly accept) conjectures, whether to verify if they are true/false, or formatted properly ("a valid mathematical question", if you will). Of course, an internet forum or Question and Answer site is a different story, but the reception is not predictable.

I'm pretty sure that due to my lack of knowledge it won't be correct but you never know. Moreover, can a conjecture exceed the limits of the mathematical system? I'm asking this because I thought that a conjecture isn't a theorem and everybody can say whatever he wants if there isn't a proof about the opposite thing.

I don't know what most of this means. If something is unproven (to the best of our collective knowledge) and not known to be false, that's just the way it is - you shouldn't go around claiming your conjecture is true. You can of course say you believe it's true, it just shouldn't be presented as a fact you personally are simply unable to prove. Likewise, I can say that it seems false to me, or any other qualitative/speculative thing, as long as we're both clear that we're speculating.

There are many conjectures that "exceed the limits of mathematical system" in that nobody has been able to prove them for a long time. Far fewer questions "exceed the limits" by being un-provable (that is, being independent of a set of axioms), but many questions "exceed the limits" simply by being questions that, while pertaining to mathematics, are not really mathematical questions (maybe philosophy of mathematics in disguise, for example), or at least not precisely stated. It's unclear which (if any) of these categories you're concerned with.

Last thing, I want to know if I have to send the reason I believe what I believe and made it a conjecture (some sort of proof)? I'm a foreigner so I'm sorry for my English. Thanks in advance!

Well, since there really is no audience in particular, there are really no rules about how it needs to be presented. That said, if you do present your conjecture, you should tell the (appropriately brief) story behind the conjecture, evidence that it's true for some specific case(s), and an overview of work you've done to attempt verification/a proof. Too much information takes too much effort to process, too little isn't very motivational.