Is it safe of me to assume that at some point 2x will be divisible by 3? Have I made any mistakes mathematically in this process?
If yes to the former and no to the later have I not just shown proof of concept that an angle can be trisected?
First of all, it has been conclusively proven that, while certain angles can be trisected (like a 90 degree angle), there is no algorithm that will trisect an arbitrary angle. Therefore, no matter how clever you hope to be, you will NEVER be able to come up with an angle trisection construction. It simply can't be done.
Secondly, you ask about 2^x being divisible by three. By the unique factorization theorem, any number that is a power of two is not equal to any number times 3. In particular, the only factors of 2^x are smaller powers of 2. So 2^x = 3y will NEVER be solvable in the integers.
Lastly, as for the actual proof that such a construction does not exist, I am at a loss. I do know it has something to do with looking at the coordinate plane and seeing which points can be constructed using only a straightedge and compass, and seeing that the points necessary to determine 1/3 of an arbitrary angle are simply not constructible.
For example, suppose we have the points (0,0) and (1,0). Now we can draw a line through those two points, and a circle centered at (0,0) and through (1,0), and we get an intersection at (-1,0). Using this method repeatedly it is straightforward to see how we can get every lattice point on the x-axis, and since we can construct perpendiculars it is quickly obvious that every lattice point on the plane is constructible (as in it is the intersection of 2 circles, 2 lines, or a circle and a line, that are constructible).
Then using the equations for lines and circles we see what types of real numbers can be constructed. At this point I am not familiar enough with galois theory to continue the proof.
Hey baby, I'm proving love at nth sight by induction and you're my base case.