Why do statements which appear elementary have complicated proofs? The motivation for this question is : Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$ and some other problems in Mathematics which looks as if they are elementary but their proofs are very much sophisticated.
I would like to consider two famous questions: First the "Fermat's Last Theorem" and next the unproven "Goldbach conjecture". These questions appear elementary in nature, but require a lot of Mathematics for even comprehending the solution. Even the problem, which I posed in the link is so elementary but I don't see anyone even giving a proof without using the prime number theorem. 
Now the question is: Why is this happening? If I am able to understand the question, then I should be able to comprehend the solution as well. A Mathematician once quoted: Mathematics is the understanding of how nature works. Is nature so complicated that a common person can't understand as to how it works, or is it we are making it complicated. 
At the same time, I appreciate the beauty of Mathematics also: Paul Erdős' proof of Bertrand's postulate is something which I admire so much because, of its elementary nature. But at the same time i have my skepticism about FLT and other theorems.

I have stated 2 examples of questions which appear elementary, but the proofs are intricate. I know some other problems, in number theory which are of this type. Are there any other problems of this type, which are not Number Theoretical? If yes, I would like to see some of them.


There are elementary questions that have no elementary answers, other than questions in Number Theory and situations invoking Godel incompleteness. I have in mind finding antiderivatives of elementary functions such as $e^{-x^2}$, $(\sin x)/x$, $1/(\log x)$, and many others. It is known that the antiderivatives of these functions cannot be expressed in closed form in terms of powers, exponentials, logarithms, trig functions, etc. 
It's also known that the solutions of simple equations like $x+e^x=0$, $x=\cos x$, $x\log x=1$ and so on can't be expressed in closed form in terms of the standard elementary functions. 
