I have two main conjectures relating to the Pythagorean Theorem that I desperately want to find out if they are true or not. Could somebody please help me?
- If $ \ a^2 + b^2 = c^2 \ $ for which $c$ is a prime number, then for some $n \in \mathbb{N}, \ c = 4n + 1$.
- $\forall \{x, y, z\} \subset \mathbb{N}, \ \big(x^2 + (x + 1)^2\big)^2 = y^2 + z^2 \ $ for which $x \neq y \neq z$.
Tested for $c, x \leqslant 10^6$.
I did also have another conjecture that if $ \ a^2 + b^2 = c^2 \ $ for which $a$ is odd (and greater than $1$), $b$ is even, and $a < b$ then there existed a solution such that $c = b + 1$, but I am $100$% certain this has been discovered before (assuming so, since it was not very difficult to notice). However I am not sure if the other two conjectures have been discovered before, although if it had to be one of them, I would pick the second one. So could somebody also please clarify that for me?
Thank you in advance.
Suppose that is $c=4k+3$. Then $c|a^2+b^2$ and since it is prime $c|a$ and $c|b$. Thus $a=mc$ and $b=nc$. But then we get $m^2+n^2=1$ which is impossible if $mn\ne 0$.