Including false attempts in resarch statement
It is probably fine to indicate that you have already studied the problems/questions that you are exploring and that you have "narrowed the field" by rejecting certain unfruitful approaches. If you already have "strong results" then the research is behind you, not ahead. Focus more on the importance of the questions you are exploring than the failed attempts to answer them. In mathematics, some problems are inherently difficult. Some are even impossible, but things can be learned and math can advance even for those problems. 

It is especially valuable, if true, to point out that some of the attempts to answer the main question have also raised new questions that are fruitful for exploration.