Competition problem — Number theory

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Number theory Difficulty 6.2 National olympiad Prove it

8. Prove: There exist infinitely many $n$ such that $n=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$ has no solutions satisfying condition (i) $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=1$ , and no solutions satisfying condition (ii) $x_{1}>x_{2}>x_{3}>$ $x_{4} \geqslant 0$ .

Solution

8. Derived from the previous question.

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