I am not able to prove it with euclidean evaluation $$ d(m+n\sqrt{7}) = m^2  7 n^2 $$. Is there any other way?
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It is Euclidean with respect to this norm. – Angina Seng Oct 23 '18 at 16:48

1For the proof, compare with [this question](https://math.stackexchange.com/questions/2215087/mathbbzsqrt11isnormeuclidean?noredirect=1&lq=1), or [this one](https://math.stackexchange.com/questions/124484/showmathbbzsqrt6isaeuclideandomain?rq=1). – Dietrich Burde Oct 23 '18 at 16:58

There is an OEIS related list https://oeis.org/A048981 , and the ring of integers in $\Bbb Q(\sqrt 7)$ is Euclidean. References are inserted there. – dan_fulea Oct 23 '18 at 17:00