I tried to ask one of my tutors this, but didn't get a proper answer...
How do you actually research maths. Like how do you have breakthroughs and stuff. I've always been curious about this.
A lot of it involves proving conjectures. E.g. a mathematician may read some papers about prime numbers and conjecture that there are more prime numbers of the form

than of the form

(a conjecture is a statement that is believed to be true but is unproven). Then another mathematician will come along and apply some fancy techniques of complex analysis in order to prove this conjecture. At the end they might conjecture something else, and in ten years time another mathematician might come along and write a paper where they prove that result.
So yeah, a lot of it is proving results that people believe to be true but don't know how to prove yet. In order to prove these conjectures, you often have to combine lots of different areas of mathematics and difficult results about the properties of certain functions or subsets of integers, for example.
(This is all based on stuff I'm reading about right now - google "Prime Number Races" to read an interesting paper on prime numbers in arithmetic progressions.)