I'm interested in finding a collection of basic results in mathematics that require rather advanced methods of proof. In this list we're not interested in basic results that have tedious simple proofs which can be shorted through more advanced methods but basic results that necessarily require advanced methods.
I appreciate the question asked here is very similar to another question asked on this website: It looks straightforward, but actually it isn't. Thank you for pointing this out. However, in my opinion, it does differ significantly (this is debatable). The main goal of the this discussion was to find examples that are easily digestible to non-advanced students of mathematics and related disciplines. This, I hope, will spur discussion of the dichotomy between what is considered trivial from a mathematics perspective and what may be considered intuitive. Quite often less experience students tend to gloss over fairly intuitive results under the assumption the proof follows easily. This I hope will be a good resource to show it is not the case.
In particular, I was hoping to find a list of problems that may seem intuitive on inspection, but are out of the reach of elementary methods. The statement of the theorem should be able to be understood by junior undergraduate students but the proof rather inaccessible. Can you also mention why elementary methods fail to shed any light on the problem.
Many thanks
Best Answer
The Kissing Number Problem asks, given a solid sphere, how many solid spheres of the same size can be placed tangent to it. For example, how many billiard balls can you place touching the surface of another billiard ball before you run out of space.
Trying it yourself, you'll pretty quickly come to the conclusion that 12 is the maximum. There just doesn't seem to be a way to fit a 13th in there, even though there is quite a bit of extra space. And mathematically, it's easy to construct a solution for 12 (inscribe icosahedron, place on the vertices) and prove that 14 is impossible (triangulate the tangent points--there's not enough surface area on the sphere for them).
But a full proof that 13 is impossible wasn't solved until 1953, even though the problem dated back to Isaac Newton.