<s>[INST] Prove that there are no positive integers $a,b,c,d$ such that $a^2 + b^2 = c^2$ and $a^2 - b^2 = d^2$ Prove that there are no positive integers $a,b,c,d$ such that $a^2 + b^2 = c^2$ and $a^2 - b^2 = d^2$?
Could you show me the proof?
 [/INST] Summing and subtracting, you get $$\begin{cases} 2a^2=c^2+d^2\\2b^2=c^2-d^2\end{cases}$$ Thus $c,d$ have the same parity. If they are both even, we obtain $2a^2=4c'^2+4d'^2$ and $2b^2=4c'^2-4d'^2$, so $$\begin{cases} a^2=2(c'^2+d'^2)\\b^2=2(c'^2-d'^2)\end{cases}$$
Since $2\mid a^2,b^2$, we must have $a^2=4a'^2$, $b^2=4b'^2$, so we get that  $$\begin{cases} 2a'^2=c'^2+d'^2\\2b'^2=c'^2-d'^2\end{cases}$$
And now you're back to the same, but with smaller numbers. And thus we descend. 
It remains the case that both are odd, which I couldn't solve.
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