TITLE: Finding the Sum of the square of two positive integers.
QUESTION [3 upvotes]: Write the following equation as the sum of the square of two integers, $a^2 +b^2$.
  $$(8^2+5^2)(13^2+7^2)$$

I remember that you are supposed to do something with complex numbers or at least that is what my teacher did in class.

REPLY [0 votes]: Hint $\quad\ \begin{eqnarray}\alpha = a+bi\\ \\ \beta = c+di\end{eqnarray} \quad \Rightarrow\quad  $ $\begin{eqnarray} \color{#c00} {(\alpha \bar\alpha)}&&\ \ \ \, \color{#c00}{(\beta\bar\beta)} &\,=\,& \quad\ \color{#c00}{\alpha\beta\  \overline{(\alpha\beta)}}\\
\color{#c00}{N(\alpha)}&&\color{#c00}{N(\beta)} &=& \color{#c00}{N(\alpha\beta)},\quad\ N := \rm norm\\
N(a\,+\,b\,i)&&\!N(c\,+\,d\,i) &\,=\,& N(ac\!-\!bd\ +\ (ad\!+\!bc)\,i) \\
(a^2+b^2)\ &&\ \ \, (c^2\!+d^2) &\,=\,&\ \ \ (ac\!-\!bd)^2\! + (ad\!+\!bc)^2 \end{eqnarray}$
{"set_name": "stack_exchange", "score": 3, "question_id": 609213}