TITLE: A simple number theory confirmation
QUESTION [11 upvotes]: Suppose $a,b\in\Bbb N$ are odd coprime with $a,b>1$ then is it true that if all four of $$x_1a+x_2b,\mbox{ }x_2a-x_1b,\mbox{ }x_1\frac{(a+b)}2+x_2\frac{(a-b)}2,\mbox{ }x_2\frac{(a+b)}2-x_1\frac{(a-b)}2$$ are in $\Bbb Z$ for some $x_1,x_2\in\Bbb R$ then $x_1,x_2\in\Bbb Z$ should hold?

REPLY [5 votes]: This is a simplification of Cehrng-tiao Perng's proof. As observed in that proof, it suffices to show that the Gaussian integers
$$ \gamma:=a+bi\qquad\text{and}\qquad \delta:=\frac{1+i}{2}(a-bi) $$
are coprime. We know that some $\mathbb{Z}$-linear combination of $a$ and $b$ equals $1$, while also
$$\gamma+(1-i)\delta=2a\qquad\text{and}\qquad -i\gamma+(1+i)\delta=2b,$$
hence some $\mathbb{Z}[i]$-linear combination of $\gamma$ and $\delta$ equals $2$. This shows that $\gcd(\gamma,\delta)$ divides $2$, hence the only possible common prime divisor of $\gamma$ and $\delta$ is $1+i$. However,
$$ \frac{\delta}{1+i}=\frac{a-bi}{2}\not\in\mathbb{Z}[i],$$
because $a$ and $b$ are odd integers. Hence $\delta$ is not divisible by $1+i$, and we are done.