Is it possible for extremely ingenious but elementary proofs for famous problems to exist?

Question

As Erdős put it, "Mathematics is not ready for such problems." when faced with the great conjecture of Collatz.

So is it impossible altogether for simple but ingenious proofs for famous problems like Fermat's Last Theorem, Goldbach's conjecture...etc to exist?

In other words, is it possible to solve such problems without consulting the "higher" domains of mathematics or seeking new domains to reach a solution?

E.g; Fermat claimed a proof for his Last Theorem, and it is quite probable that it was within elementary bounds (if it even existed), atleast by our standard today.

Answer 1

The example which immediately comes to mind is the elementary proof of the prime number theorem by Erdős and Selberg.

Answer 2

In 2013, Yitang Zhang made major inroads on prime gaps.

Quite likely, there will eventually be very short proofs on the Sierpinski number conjecture and the Riesel number conjecture.

Pentagon tiling just got a fifteenth solution.

A Hadamard matrix of order 428 was found in 2005, and eventually the order 668 will be found.

Some perfect solutions in the Kobon Triangle Problem were found recently.

The optimal way to cover a triangle with 2 squares wasn't found until 2009.

For the Euler sum of powers conjecture, the proof it was wrong was simple.

$27^5 + 84^5 + 110^5 + 133^5 = 144^5$ (Lander & Parkin, 1966)

Euler also conjectured there was no Graeco-Latin square of order 10, which was proven wrong in 1959 by a counterexample.

It's inevitable that a highly elegant proof or new wonderful example will come out for some currently unsolved problem. For some of these, there might be thousands of computer-years involved in the attack, but that will boil down to a short list of facts that fit on a single page and are easily checked.

There likely won't be a simple proof for those problems, though. But other, less studied problems, particularly those vulnerable to computerized insight -- yes, some of them will be solved.