<s>[INST] Is there a deep reason why $(3, 4, 5)$ is pythagorean? The triple $(3, 4, 5)$ is a pythagorean triple - it satisfies $a^2 + b^2 = c^2$ and, equivalently, its components are the lengths of the sides of a right triangle in the Euclidean plane.
But of course, the first thing anybody notices is that the triple $(3, 4, 5)$ also happens to be an arithmetical succession of small numbers.
Is there a deep reason why choosing these three successive numbers just so happens to yield a pythagorean triple?
To anyone who feels the question is silly: consider $3^3+4^3+5^3$.
 [/INST] $(3,4,5)$ is Phythagorean because $5$ is a prime of the form $4k+1$. Some known facts:


*

*Every prime $p$ of the from $4k+1$ can be rewritten as a sum of squares of two distinct positive integers:


$$\forall k \in \mathbb{Z}_{+}, p = 4k+1\text{ prime} \implies \exists \alpha, \beta \in \mathbb{Z}_{+} \text{ s.t. } \alpha \neq \beta \wedge p = \alpha^2 + \beta^2$$


*

*Every number $n$ that can be written as a sum of squares of two distinct positive integers
is part of a Pyhthagorean triplet because of an algebraic identity:


$$n = (\alpha^2+\beta^2) \implies n^2 = (\alpha^2+\beta^2)^2 = (\alpha^2-\beta^2)^2 + (2\alpha\beta)^2$$


*

*Every Phythagorean triplet $(a,b,c)$ has a parametrization of the form:


$$a^2 + b^2 = c^2 \implies \begin{cases}a = (\alpha^2-\beta^2)\mu\\b = 2\alpha\beta\mu\\c = (\alpha^2 + \beta^2)\mu\end{cases}\quad\quad\text{up to order of }a, b$$ 


*

*When $a, b$ is relative prime to each other, we can set $\mu$ above to 1.


Take $5 = 2^2+1^2$ as an example, we get:
$$\begin{cases}a = 2^2-1^2 = 3\\b = 2\cdot 2 \cdot 1 = 4\\c = 2^2 + 1^2 = 5\end{cases}
\quad\quad\text{is a Pythagorean triplet}$$
$c = 5$ is the smallest example of such Pythagorean triplet. Since there are only 4 numbers smaller than 5, it is just a coincidence that $(3,4,5)$ are successive integers.
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