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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). enter image description here

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?

enter image description here

enter image description here

Caveats:
  • 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.
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