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. . We know ( is assumed to be greater than one here). Hence, . Moreover, we know . Hence, . Similarly, .
So we have the three inequalities: and .
satisfy the triangle inequalities! Hence, form a triangle.
Using the cosine rule, we get , where is the angle opposite side .
Raising both sides to the power , we get . Now if and , we get . This is the case of the right-angled triangle.
However, if , then the right hand side, which is , is unlikely to simplify to .
There are multiple flaws in this argument. Coming to terms with them was a huge learning experience.