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I'm new to number theory. So now I'm starting my journey of 'number theory' by reading this book. I'm currently in chapter 2 which is Pythagorean Triples. I don't understand. It says there are numbers $x,y, z$ such that $$\begin{cases}a=2x+1\\ b=2y+1\\c=2z\end{cases}.$$ My problem is figuring out how these equations can be motivated/justified?

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I guess you mean that there are no numbers $x,y,z$ with this property. We can show, that it is impossible for $c$ to be even while the triple is primitive.

Since, you need that $a^2 + b^2 = c^2$, if $c$ is even, there are two cases:

  • Both $a$ and $b$ even: In this case, the triple is not primitive since $(a/2,b/2,c/2)$ is also a triple.
  • Both $a$ and $b$ are odd: In this case consider the whole equation mod $4$. Any odd number squared mod $4$ gives $1$ since both $1^2 \equiv 1 (\textrm{mod } 4)$ and $3^2 \equiv 1 (\textrm{mod } 4)$. On the other hand any even number squared gives $0$ mod $4$ since both $0^2 \equiv 1 (\textrm{mod } 4)$ and $2^2 \equiv 1 (\textrm{mod } 4)$. So it is impossible that $a^2 + b^2 = c^2$ since $1 + 1 \not \equiv 0 (\textrm{mod } 4)$.
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They are just the translation of the sentence

Next, suppose that $a$ and $b$ are both odd, which means that $c$ would have to be even.

An odd number $a$ can be writen as $2x+1$ with integer $x$, etc.

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