Practical Tips: Mathematical research and discoveries How to behave when you have the feeling of working on something innovative? What to do if there is a chance (even the 1%) that your
  work is leading you to something original?


For example what if I don't know mathematicians I trust to ask? Is a good idea to talk about your results (even if are not real results) to someone?


  Question: 
  
  What do you do if you have this feeling?
  
  I'm looking for a list (or links to related SE questions/guides/books or others)
  of practical tips to use while studying a certain subject you
  have the feeling of having found an innovative approach that can
  provide a new solution.


I am interested too in books about the mathematicians' way to research. (I think is related)


  I already asked this question on Math.SE but was closed two times. Here the link to MathematicsSE
0. To give original contribution to the understanding of mathematical objects is what mathematician do, and talk about all the time. There is a priori no need to fear for ideas being stolen; if you want to play secure, you can publish a preprint to ensure priority.

Given the wording of your question, I assume that you are an amateur mathematician, and that you feel that you may have found something about an important, well-known question. Sorry if this is not the case.

As most mathematicians, I receive demand for advice of this kind from time to time, so here is my usual answer.


Be prepared to have made a mistake,
be prepared to have found something known for a long time,
be prepared to have found something that will not attract interest.


This may sound very negative, but these are the worries that are much more likely to be relevant than seeing your contribution stolen. I have seen recently an amateur publishing on viXra after seeking advice from me, that was afraid of having her ideas stolen. It turned out that her contribution was a few hundred years behind current knowledge.

Researchers in mathematics only succeed in advancing knowledge because they spend much time learning their specialty and keeping up with what is being proven, and we do sometimes reinvent the wheel (I got scooped by 130 years once, realizing that Camille Jordan already solved a cute problem I was interested in), or make mistake, or do things that do not interest our colleagues. It is tremendously difficult to avoid this pitfalls when you don't have access to the literature, don't have colleagues to speak with about your research, don't have a regular seminar to listen to, don't have had a PhD advisor to guide you through your first problems.

So, for a positive piece of advice:

4. learn the field you are interested in (e.g. read books, from the point you are in your knowledge to the field you are interested in). Be prepared for this step to take much time.