Given $k\in\Bbb N$ is there coprime $1<a,1<b$ with $(k,a^2-b^2)=1$ and coprime $1<c,1<d$ such that $k|(ac-bd)$ and $k|(ad-bc)$?
What is the smallest $\max(|a|,|b|,|c|,|d|)=\max(a,b,c,d)$ among such $a,b,c,d$?
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From your assumptions follows that $$k\mid (ac-bd\pm(ad-bc))=(a\mp b)(c\pm d).$$ So $k\mid(c\pm d)$, i.e. $k\mid(c+d,c-d)\in\{1,2\}.$
answered Jul 17, 2017 at 12:53
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$\begingroup$ @AJ. You may take $a=c=2$, $b=d=3$ for $k=1$, and $a=c=3$, $b=d=5$ for $k=2$. $\endgroup$ Jul 18, 2017 at 0:59
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$\begingroup$ that is what I meant and no other $k$ is possible. $\endgroup$– TurboJul 18, 2017 at 3:55