# Thread: A property of Pythagorean triples

1. ## A property of Pythagorean triples

Let $\displaystyle a,b \in\mathbb{N}^+$ such that $\displaystyle \sqrt{a^2+b^2} \in \mathbb{N}$.

Show that $\displaystyle \forall n\in\mathbb{N}^+\; (a+ib)^n \notin \mathbb{R}$

Where $\displaystyle i \in \mathbb{C}$ is the imaginary unit, and $\displaystyle \mathbb{N}^+,\mathbb{R, C}$ are the sets of all positive integers, reals and complex numbers respectively.

2. ## Re: A property of Pythagorean triples

Originally Posted by elim
Let $\displaystyle a,b \in\mathbb{N}^+$ such that $\displaystyle \sqrt{a^2+b^2} \in \mathbb{N}$.

Show that $\displaystyle \forall n\in\mathbb{N}^+\; (a+ib)^n \notin \mathbb{R}$

Where $\displaystyle i \in \mathbb{C}$ is the imaginary unit, and $\displaystyle \mathbb{N}^+,\mathbb{R, C}$ are the sets of all positive integers, reals and complex numbers respectively.
Hint:

Use mathematical induction.

3. ## Re: A property of Pythagorean triples

Originally Posted by elim
Let $\displaystyle a,b \in\mathbb{N}^+$ such that $\displaystyle \sqrt{a^2+b^2} \in \mathbb{N}$.

Show that $\displaystyle \forall n\in\mathbb{N}^+\; (a+ib)^n \notin \mathbb{R}$

Where $\displaystyle i \in \mathbb{C}$ is the imaginary unit, and $\displaystyle \mathbb{N}^+,\mathbb{R, C}$ are the sets of all positive integers, reals and complex numbers respectively.
Let $\displaystyle c = \sqrt{a^2+b^2}$ and let $\displaystyle \theta = \arccos(a/c)$. If $\displaystyle (a+ib)^n \in \mathbb{R}$ then $\displaystyle \theta$ will be a rational multiple of $\displaystyle \pi$ with a rational cosine. That can only happen if $\displaystyle \cos\theta\in\{0,\pm\tfrac12,\pm1\}$ (you can find a neat proof of that here). The result then follows quite easily.

4. ## Re: A property of Pythagorean triples

Thanks a lot Opalg!