Fermat’s Last Theorem

by ayushkhaitan3437

When in high school, spurred by Mr. Scheelbeek’s end-of-term inspirational lecture on Fermat’s Last Theorem, I tried proving the same for…about one and a half long years!
For documentation purposes, I’m attaching my proof. Feel free to outline the flaws in the comments section.

Let us assume FLT is true. i.e. x^n + y^n =z^n. We know x^n + y^n<(x+y)^n (n is assumed to be greater than one here). Hence, z<x+y. Moreover, we know z^n-x^n<(z+x)^n. Hence, y<z+x. Similarly, y+z<x.

So we have the three inequalities: x+y<z, x+z<y, and y+z<x.

x,y,z satisfy the triangle inequalities! Hence, x,y,z form a triangle.

Using the cosine rule, we get z^2=x^2 +y^2 -2xy\cos C, where C is the angle opposite side z.

Raising both sides to the power \frac{n}{2}, we get z^n=(x^2 +y^2 -2xy\cos C)^{\frac{n}{2}}. Now if n=2 and c=\frac{\pi}{2}, we get z^2=x^2+y^2. This is the case of the right-angled triangle.

However, if n\geq 3, then the right hand side, which is (x^2 +y^2 -2xy\cos C)^{\frac{n}{2}}, is unlikely to simplify to x^n + y^n.

There are multiple flaws in this argument. Coming to terms with them was a huge learning experience.