Writing $28913000$ as the sum of two squares A little number theory fun. I am given that $167^2 + 32^2 = 28913$, and I am asked to find integers $a$ and $b$, such that $a^2 + b^2 = 28913000$. Here's my thought proces...--prophetes.ai

Writing $28913000$ as the sum of two squares A little number theory fun. I am given that $167^2 + 32^2 = 28913$, and I am asked to find integers $a$ and $b$, such that $a^2 + b^2 = 28913000$. Here's my thought process so far: Knowing that $1000 = 10^2 + 30^2$, I rewrote $28913000$ as $28913\times 1000$, and proceeded to multiply the sums of squares: $$(167^2 + 32^2)(10^2 + 30^2).$$ However, after foiling, I ended up with the sum of $4$ squares, and cannot think of a way to just find two squares, $a$ and $b$. Any help would be greatly appreciated!

Very good!

Do you know _complex numbers_? Assume that $-1$ has a square root _somewhere_ (certainly not in $\mathbb R$), denote it $i$, and introduce $+$, $\cdot$ operations with reals and $i$. So, $i^2=-1$, thus $(a+bi)(a-bi) = a^2+b^2$. $$(a+bi)(a-bi)(c+di)(c-di) = (a+bi)(c+di)\cdot (a-bi)(c-di)$$ Can you calculate it?