TITLE: Legendre symbol question for infinitely many primes of form 4k + 3
QUESTION [4 upvotes]: Given a positive integer n, how would one show that there are infinitely many primes p of the form 4k + 3 that have Legendre symbol (n/p) = -1?  From the comments I have received thus far, it has been suggested to use either the Cebotorev Density Theorem or Quadratic Reciprocity to show there infinitely such primes with (n/p) = (-1/p) = -1, but I don't see how and welcome any suggestions about this.  I am asking the question for any positive integer n, though I am primarily interested in the case where n is the product of two prime numbers, where it does not happen that both of these prime numbers are congruent to 3 mod 4.

REPLY [1 votes]: If $n=-1$ then $(n/p)= -1$ for all $4k+3$ primes
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