<s>[INST] Find the value of the following Legendre Symbols and Use Gausslemma to compute each of the Legendre symbols below Find the value of the following Legendre Symbols:
a)  $(\frac{18}{43})$
b) $(\frac{19}{23})$
Use Gauss' Lemma to compute each of the Legendre symbols below:
a) $(\frac{8}{11})$
b) $(\frac{6}{31})$
 [/INST] $a)$ $$\left ( \frac{18}{43} \right )=\left ( \frac{2 \cdot 3^2}{43} \right )=\left ( \frac{2}{43} \right )\left ( \frac{3}{43} \right )^2=(-1)^({43^2-1}{8})=(-1)^{(\frac{42}{8} )\cdot (\frac{44}{8})}=[(-1)^6]^{\frac{11}{2}}=(-1)^{3 \cdot 11}=-1$$
$b)$ $$\displaystyle{\left ( \frac{19}{23} \right )=(-1)^{{9}\cdot{11}} \left ( \frac{23}{19} \right )=(-1)\left ( \frac{4}{19} \right )=(-1)\left ( \frac{2}{19} \right )^2=(-1) \cdot (-1)^{\frac{(19-1) \cdot (19+1)}{8}}= (-1) \cdot (-1)=(-1) \cdot (-1)^{45}=1}$$
The Gauss' Lemma is:
$$\text{Let prime } p>2 \text{ and } p \nmid a. \\If \ m \text{  from the numbers } \\
  a \cdot 1, a \cdot 2, \dots , a \cdot \frac{p-1}{2} \text{ have a remainder,when they are divided by } p, \text{ that is between } \frac{p+1}{2} \text{ and } p-1, \text{ then } \left ( \frac{a}{p} \right )=(-1)^m  $$
Can you apply it now?
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