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.