Jacobi symbol

DEFINITION 7.1   Let $m$ be a positive odd integer. Let us factor $m$ :

$$m=\prod_i p_i^{e_i}$$

where $p_i$ are primes. Then for any $n\in \mathbb{Z}$ , we define Jacobi symbols as follows

$${\left(\frac{n}{m}\right)}= \prod_i{\left(\frac{n}{p_i}\right)}^{e_i}$$

We further define

$${\left(\frac{a}{p}\right)}= 0$$$$\text { if } a \in p\mathbb{Z}.$$

THEOREM 7.2 (quadratic reciprocity theorem)   For any positive odd integers $n,m$ , we have

$$\left(\frac{m}{n}\right) \left(\frac{n}{m}\right) =(-1)^{(m-1)(n-1)/4}.$$

THEOREM 7.3   Let $n$ be a postive odd integer. Then:

1. ${\left(\frac{-1}{m}\right)}=(-1)^{(m-1)/2}$ .
2. ${\left(\frac{2}{m}\right)}=(-1)^{(m^2-1)/8}$ .

EXERCISE 7.1   $p=113357$ is a prime. (You may use the fact without proving it.) Is there any integer $n$ such that

$$n^2=11351$$    in $$\mathbb{Z}/p \mathbb{Z}$$$$\text { ?}$$

If so, can you find such $n$ ?