How to Get an A* in A-Level Maths

Just back from the cinema - went to see Spanish film 'La habitación de Fermat' (Fermat's Room)
Best described as Saw vs Maths, it's a thriller whereby the main characters have to use their maths skills to get themselves out of a sticky situation.

I should say that all the maths is correct and presented very well - it's easily accessible and should be interesting to an A-level student, say. The protaganists are occasionally a little bit slower at solving the puzzles than you might expect, but not very much so - I won't spoil the puzzles for you!

The key bit of maths in the plot is Goldbach's Conjecture (GBC)
Every even number (greater than 2) can be written as the sum of two prime numbers

e.g.
4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
100 = 53 + 47
1528 = 599 + 929
and so on

GBC illustrates an important idea of proof in mathematics:
It would be easy to disprove GBC. We'd just need to find a number which doesn't work - a counterexample. Easy - if there is one we can easily find, anyway.
Proving GBC is more difficult - we have to be able to show mathematically that GBC will be true for EVERY number, even ones that we will never test.

You can tell that checking every number is not good enough... even if we check all the numbers up to 10¹⁸, it might be that bigger numbers don't work but we haven't checked them yet. Especially in this field of maths (number theory), this unexpected disproof does occasionally happen to conjectures... and mathematicians aren't satisfied until there's a clear proof.
So, a Conjecture is some important mathematical idea that is believable but nobody's managed to prove yet - after a proof is found the conjecture is upgraded to a Theorem (or if we disprove it, it's downgraded to a failed conjecture).

Goldbach's Conjecture is at least 267 years old and still mathematicians are trying to find a proof.

Good luck!