# Geometric generalization of Fermat Last Theorem

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?

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.