Assistance regarding a number theory project I am an IB student (HS equivalent), and I have to write a 12+ page long project for my mathematics class.
I am greatly interested in number theory and was thinking of starting by exploring and proving primitive Pythagorean triplets, and then moving on to prove Fermat's last theorem for the case of $n=4$. Then I was thinking of using my exploration of the Pythagorean triplets to prove for example that the radius of an incircle to a Pythagorean triangle is always an integer (or so I've heard), or that the area of a Pythagorean triangle can never take the form of $2a$ where $a$ is a perfect square.
However, I am concerned about the lack of complexity of my project. The proof for $n=4$ of formats theorem is unfortunately surprisingly easy. Would anyone please be willing to give me some ideas to improve the complexity of the project or perhaps take it a different route? Maybe explore different aspects of Fermat's last theorem, attempt a different proof, perhaps explore some more complex aspects of Pythagorean triples that I may not know about or take a different route altogether? Thanks a lot for your time.
 A: You might consider ways of finding "triples on demand" such as by side,  perimeter, area, area/perimeter ratio, product, and side difference. I've been working on such a paper for 10 years and have finally cut it down to 14 pagers.   Here is an example of something you might work with given Euclid's formula where
$$A=m^2-n^2\qquad B=2mn\qquad C=m^2+n^2$$
To find a triple, we solve for $n$ and test a range of $m$ values to se which yield integers, for example:
Finding side A using $F(m,n)$
$$A=m^2-n^2\implies n=\sqrt{m^2-A}\qquad\text{where}\qquad  \sqrt{A+1} \le m \le \frac{A+1}{2}$$
The lower limit ensures $n\in\mathbb{N}$ and the upper limit ensures $m> n$.
$$A=15\implies \sqrt{15+1}=4\le m \le \frac{15+1}{2} =8\quad\text{ and we find} \quad m\in\{4,8\}\implies n \in\{1,7\} $$
$$fF4,1)=(15,8,17)\qquad \qquad f(8,7)=(15,112,113) $$
This kind of work is easy.  but it gets harder when you get to area (a cubic equation) and product (a quintic equation) and side difference which is easy for $C-B\quad C-A\quad \text{and}\quad B-A=\pm1$ but not so for other $B-A$ differences. For primitives, if $X=B-A$,
$X$ can be any prime number $(p)$ where $p=\pm1\mod 8$, raised to any non-negative power.
Under $100$,  $X\in \{1,7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97\}$.
If you discover the patterns I did, it could be enough for the paper alone.
By the way, you might acquire and learn to use $LaTeX{} $ because it is much better than word in typesetting equations.
A: Some not so common elementary about Pythagorean triplets that you could possibly use:

*

*If two of the numbers in a Pythagorean triplet are primes $> 5000$ then the third number must have a prime factor $> 17$. Proof

*In a primitive Pythagorean triplet, the sum of the two perpendicular sides is never divisible by any of the following primes $2,3,5,11,13,19,29,37,43,53,59,61,67,83,101,107,109,131,\ldots$. Proof
