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I am starting to think about a problem, inspired by a series of papers on similiar questions. Is it correct for me to e-mail the professors that wrote these papers, and ask them if they already thought about it? If they did, would it be fair to ask for the ideas they had, and what they tried (of course, assuming they are not currently working on it, and don't plan to publish on their own)?

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I would not email people before thinking about a problem. People are busy, so you need to respect that. Instead I would do the thinking and if you have concrete questions, then you can contact others.

If you want to "brainstorm" with someone, you can try to attend a conference that that person also attends. You can sent an email before, and ask if (s)he wants to set half an hour aside for you to discuss this or that problem. Again it would help if you narrowed the problem down, such that it can be discussed in half and hour and the person can prepare a bit if (s)he wants to.

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Is it correct for me to e-mail the professors that wrote these papers, and ask them if they already thought about it?

There is no "correct" or "wrong" in human interaction. (There may be actions that are legal/illegal, ethical/unethical, moral/immoral. But in this case all these are irrelevant.)

So for your question, I would suggest it is completely legitimate to contact the author. Science is advanced (also) by discourse, and there is nothing wrong in contacting scientists. Many devoted scientists would be happy to receive emails about their work. In your case, you also have a fairly concrete question: "Did you already think/published about related notion X?".

Note: They may choose not to answer you, but that is their problem. It also depends on whether your emails are reasonably comprehensible, to the point, and show potential for progress.

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At the risk of answering your question with a question, do you really want to contact people who don't know you or owe you anything about YOUR problem? What if they have not considered your idea but think you're really on to something and work on it without you? If the problem you have devised is truly unique then you should be proud of that and follow through with it yourself. Instead I would look for a faculty member or someone who you are at least somewhat closer to than the mathematicians you cited and see if they can perhaps mentor you through the problem solving process first. That way, should your idea pan out, you can go to the aforementioned researchers with a valid reason as to why you should be involved in the entire process.

I guess in the event that you are just a passive observer then it would be fine to just go ahead and contact them, but as Maarten stated above, "people are busy", and they may not have the time or energy to respond to questions that are not meticulously developed. Also, even if you inevitably hit a road block where you are unable to go any further on your own, you will learn more from, and get more out of, pursuing the problem as far as you possibly can by yourself.

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