Who is to decide the main idea(s) of research in a PhD Proposal, the student or his/her supervisor? I'm a pure math phd student and the supervisor gave me two papers so as I find an idea to research but the subjects are unrelated and I can't concentrate on one of them and can't decide which one is likely  to be a fruitful subject, What should I do?
First, it depends on the country you're in, the type of funding you're applying for, and probably to a smaller extent on your field of research.

In case you already have an agreement with a supervisor, then you should probably look to him for advice on choosing a topic, but make the choice yourself. Given that he sent you papers on these two topics, you can probably assume (absent any evidence to the contrary) that both would make a good and fruitful subject. Thus, choose with your heart: what problem is most exciting to you?