Competition problem — Number theory

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Number theory Difficulty 6.0 National olympiad Prove it

6. Let $\left(\frac{D}{n}\right)$ be the Kronecker symbol from the previous problem. Prove that,
(i) for a given $D$ , there always exists an $n$ such that $\left(\frac{D}{n}\right)=-1$ ;
(ii) $\left(\frac{D}{|D|-1}\right)=\frac{D}{|D|}$

Solution

6. Let $D=2^{l} k, 2 \nmid k$ . (i) Discuss in three cases: $l=0$ , $l$ is odd, and $l$ is even, and use the Chinese Remainder Theorem; (ii) Use (a) and (b) from Question 5.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.