I apologized if the post is a bit long. I am not a mathematician, but I love playing with math. I was looking at Fermat last Theorem the other day, and consider this situation.
In the original formulation, it was positive integer solution to $ a^n + b^n = c^n $, for $n>2,n \epsilon \mathbb{N}$. I consider it to be equivalent to finding non zero rational points on this closed curve $$|x|^n + |y|^n = 1$$I then plot that in 3D just to make it easier ($z=n$), and got a weird tube (I called Lz tube).
I then realized that the closed curve $|x|^n + |y|^n = 1$, is just the intersection between the Lz tube with horizontal plane (perpendicular to z-axis).
Now, here is my question, what happened if you tilt the plane? When plane $ax+by+cz+d=0$ that intersect the tube $|x|^z + |y|^z = 1$, it will produce a curve. What $(a,b,c,d)$ will produced infinitely many rational points on the closed loop that curve?
- Exclude points that contain zero
- There are closed part and open parts of the resulting curve. I only want to know the solution on the closed loop part.