What is the difference between elementary and advanced math? In any problem solving situation, if the solution doesn't come to mind, one wonders if it is even possible to use the methods in mind, and if it is something you haven't learned yet, or never have been invented yet.
So my question is: How can one systematically find out what kind of maths (elementary or advanced methods) can be used?
The famous example is the Collatz Conjecture. It has often been quoted as the easiest to state unsolved problem.
Which at first glance looks like an elementary problem.
 A: You can't tell.


*

*Pell's equation is quite easy to solve, Pythagorean triples were known to the Babylonians, Fermat's Last Marginalium lasted hundreds of years and requires cutting-edge work on general elliptic curves.

*Bisecting an angle is easy with straightedge and compass, trisecting an angle is impossible without neusis.

*The area of a cirle is $\pi r^2$. The area of an ellipse is $\pi a b$. The total arc-length of a circle is $2\pi r$, the total arc-length of an ellipse is $4 a E(e)$, where $E$ is an elliptic integral and $e$ the eccentricity.

*The general quadratic is easy to solve, the cubic and quartic harder, but about as difficult as each other, the quintic is impossible in general.

*$\int e^{-x} \, dx$ is trivial, $\int e^{-x^2} \, dx$ is not an elementary function. $\int \sqrt{\sin{x}} \, dx $ is elliptic, $\int \sqrt{\tan{x}} \, dx$ is elementary.


Finally, it was thought for a long time that the Prime Number Theorem was a deep result needing complex analysis. Hardy remarked in 1921:

No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one. Now we know that the theorem is roughly equivalent to a theorem about an analytic function, the theorem that Riemann’s zeta function has no roots on a certain line. A proof of such a theorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems quite clear. We have certain views about the logic of the theory; we think that some theorems, as we say ‘lie deep’ and others nearer to the surface. If anyone produces an elementary proof of the prime number theorem, he will show that these views are wrong, that the subject does not hang together in the way we have supposed, and that it is time for the books to be cast aside and for the theory to be rewritten.

Then in 1948, Selberg and Erdős produced elementary proofs.
