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I'm now finishing my final year of high school and preparing for university. I've applied to England to Imperial College, UCL, King's College, Queen Mary and Edinburgh all for a physics w/ theoretical physics or simply theoretical physics Msci in four years and planning to do a PhD later on. I have only heard from King's so far (got an offer).

However one thing has been bothering me for a while: while I don't regret applying for theoretical physics at all because I love the subject, I've been thinking about going into mathematics instead.

I enjoy both physics and mathematics and had a really tough time deciding for which to apply. In the end, the deciding factor in choosing physics was that I come from the French school system (I go the the French school in Belgium) as did my sister before me and while she got a perfect score in mathematics (and physics) when she passed her baccalauréat she struggled quite a bit during her first year (and she isn't at the top of her class now) doing material science at Imperial. We both work in different styles: she is more focused on lots of practice and learning (while obviously understanding the concepts) while I rely on understanding the concepts and being able to rethink (with some previous knowledge of course) what we saw in class during the test without much revision (but I spend "revision time" reading about physics and maths so I guess I keep in touch with the subject to some extent?). It's understandable she had some trouble as we simply have less maths and physics classes (7.5hrs of maths counting maths speciality, we don't even do matrices and 3hrs of physics/chemistry and another 2hrs of physics/chemistry labs) compared to A levels. Given that she had some trouble in maths, I decided to go for theoretical physics.

Now my question is: all the university websites just label the maths you do in the theoretical physics programs as "maths", "mathematical methods for theoretical physics" and such so I'm not very clear on what I'll actually be studying on that side (and what I should prepare for). I imagine it'll have PDEs, real analysis, complex analysis and so on but I can't figure out if it includes say topology or other more advanced things..? Will it be as rigorous as if I were studying it in a maths program?

Also, I'm fairly sure it won't include number theory, my favourite part of mathematics I've studied so far (I love the logic and reasoning used) and I was very pleased to see it had some applications in physics, in string theory and quantum chaos notably. I'd like to work with string theory so combining it with number theory would be great. What would you suggest as the best method to learn number theory (or any math) on my own, just grab recommended books and drill trough them? =(

Finally, what made me enjoy mathematics was the whole process of mathematical proofs, I know theoretical physicists use mathematics to make predictions or explain a phenomena, how close are these to proofs? I know there are some for most theorems/theories yet we've never done any/seen any in class while we prove nearly everything we study in mathematics. As a physicist, would it be possible to come up with a new and relevant mathematical theorem/property (possibly one also relevant to physics but I feel like all mathematics is relevant to physics) to contribute to the field?

Thanks in advance for any answers that may come, sorry if this post seems a bit flooded with questions!