# Properties of Legendre Symbols and Quadratic Reciprocity

## I'm still a little bit confused about solving these types of problems. I have the formulas but still not able to apply them properly:

Use properities of the legebdre symbol and quadratic reciprocity to evaluate the legendre symbols:

$(\frac{-30}{127})$, $(\frac{-30}{131})$, and $(\frac{-30}{137})$

"Factor and reduce". $$\left(\frac{-30}{127}\right) = \left(\frac{-1}{127}\right)\left(\frac{2}{127}\right)\left(\frac{3}{127}\right)\left(\frac{5}{127}\right).$$
There are special theorems for $-1$ and $2$. Then quadratic reciprocity works for the others:
$$\left(\frac{3}{127}\right)= -\left(\frac{127}{3}\right)=-\left(\frac{1}{3}\right)=-1.$$
• Remark on typo above: should be $(\frac{-30}{127})$. Unluckily edit requires more than one char change :-) – Maestro13 Nov 30 '16 at 22:08