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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 $(\frac{-30}{137})$

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"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:

$$\left(\frac{3}{127}\right)= -\left(\frac{127}{3}\right)=-\left(\frac{1}{3}\right)=-1.$$

And so on.

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  • $\begingroup$ Remark on typo above: should be $(\frac{-30}{127})$. Unluckily edit requires more than one char change :-) $\endgroup$
    – Maestro13
    Nov 30, 2016 at 22:08
  • $\begingroup$ @Maestro13 Got it, thanks. $\endgroup$
    – B. Goddard
    Nov 30, 2016 at 22:20

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