What seemingly innocuous results in mathematics require advanced proofs? 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

Perhaps the parallel postulate?

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point

It was only through trying to prove this "obvious" theorem that we discovered non-Euclidean geometries.
