Worry about stealing of research ideas

You need to consider what role you want the math professor to take: are you looking for a co-author, a collaborator, a consultant, or a service provider.

Since you lack background in one area and are working in a place that does not prioritize publications, I would consider bring the individual on as a co-author. If that is your decision you need to provide the co-author with everything you are planning on doing so that he/she can make an informed decision. This obviously puts you at risk of being scooped. This then leads to the question of how to minimize the chances of being scooped. The first thing to remember is most researchers do not want to get into a publish first race. Scooping tends to be accidental and a result of two people independently starting the same/similar projects. You can minimize your chances by contacting an established person who is known to play "nice". The second thing you can do is to bring the research along as far as you can without the collaborator. This means they will be starting from so far back that it is unwise to enter the race.

This is a common in most fields, I am not a mathematician and hear it occassionally in my field as well. The simple answer is that there are never any guarantees against that someone will/can "steal" an idea. I write quotation marks because there is a grey zone between really taking it and getting own similar ideas based on pieces of information.

One way to establish "ownership" of the idea is to provide something tangible that provides documentation of the idea with your name on it such as an abstract from a conference, poster contribution or something similar. I cannot see if this would be possible in your case, given that you say you are lacking some pieces.

Approaching another scientist to establish a collaboration is a good idea. I would not use the suspicion that the idea will be stolen as a basic assumption, the large majority of scientists do not steal ideas in my opinion. So the problem is to increase your knowledge about the person(s) you intend to contact. Does someone in your neighbourhod know the person(s)? Try to hear if they have a reputation (good or bad, no partuclar reputation is also good in my opinion). Then write and ask about the interest in collaborating, specifying what expertise you seek in the collaoration. How much you wish to reveal of your complete idea is difficult to say but you will have to provide enough detail to clearly explain why the sought expertise is required. If you then have some way to indicate the your idea is known as your idea by others, it may work as a deterrent but I would avoid being to "paranoid" in the first approach.

I do not know if this will help you in your quest but getting to know your possible contact will help to remove the unknowns which are instrumental in building your uncertainty.

I don't have a big academic experience yet, but would like to share what I learnt.

First of all if you have an idea you can write a technical report where you describe what you doing, what you found up to that point. If you place the technical report on the University website, you can always say that you started with the idea and you wanted to develop it to a research paper but you have a trace that it is yours. (If you don't have data and results yet, you can write technical note where you describe what you going to do)

Second thing I learned is that if you have a good name on your paper just after your one, the paper will be somehow "stronger". Probably researchers will look first for the name of a well known professor and his publications instead of your one if you don't have many. Your name should be first anyway as it is your idea. In future your paper will be cited more often and will get better score. As far as I know it is worth to have co-authors.