Find all primes $x$ in a Euclidean domain $ Z[i]$ for which $|x|^2 \leq 16$.
Any help or hint is appreciated. From my understanding, is it true that $x$ can only be $2$ and $3$?
2026-03-26 04:36:47.1774499807
Find all primes $x$ in a Euclidean domain $ Z[i]$ for which $|x|^2 \leq 16$.
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If we write $x=a+ib$ where $a,b \in \mathbb Z$ : $$|x|^2\leq 16 \iff |x|\leq4 \iff a^2+b^2\leq 4\\\iff (a,b)=(0,\pm2)\text{ or }(a,b)=(\pm2,0)\text{ or }(a,b)=(0,\pm1)\text{ or }(a,b)=(\pm1,0)\text{ or }(a,b)=(\pm1,\pm1)\text{ or }(a,b)=(0,0)\\\iff x=\pm2i\text{ or }x=\pm2\text{ or }x=\pm i\text{ or }x=\pm1\text{ or }x=\pm1\pm i\text{ or }x=0$$ Now you can check which of these possible solutions are prime.