# A polynomial as a quadratic residue mod a prime

I want to show if it's true that $$60m^2+6m-1$$ is a quadratic residue modulo $$6gm+1$$ for all $$m \in \mathbb{N}$$ and $$6gm+1$$ is prime, for infinitely many positive integers $$g$$. (I'm not 100% certain this is true, so a proof that it's wrong would be equally helpful).

I'm more looking for a solid method of attacking this sort of problem in general. How can this be shown?

For instance, $$(m=1):$$ is $$65$$ a qr mod a prime of the form $$6g+1$$ infinitely often? $$(m=2):$$ is $$251$$ a qr mod a prime of the form $$12g+1$$ infinitely often? $$(m=3):$$ is $$557$$ a rq mod a prime of the form $$18g+1$$ infinitely often?

Is it true for any positive integer $$m$$?

Here is what I think you are asking: for each natural number $$m$$, are there infinitely many primes $$p \equiv 1 \bmod 6m$$ such that $$60m^2 + 6m - 1 \bmod p$$ is a quadratic residue?
To avoid being distracted by the algebraic expressions, set $$a = 60m^2 + 6m-1$$ and $$b = 6m$$. I think you are asking if there are infinitely many primes $$p \equiv 1 \bmod b$$ such that $$a \bmod p$$ is a quadratic residue. I'll show this can be done for arbitrary nonzero integers $$a$$ and $$b$$.
For each nonzero integer $$n$$, quadratic reciprocity implies $$n \bmod p$$ is a quadratic residue if (not only if) $$p \equiv 1 \bmod 4|n|$$. So it suffices to find infinitely many primes $$p$$ such that $$p \equiv 1 \bmod b, \ \ \ p \equiv 1 \bmod 4|a|.$$ These congruences both hold for a prime $$p$$ such that $$p \equiv 1 \bmod 4|a|b.$$ Dirichlet's theorem tells us there are infinitely many primes $$p$$ satisfying that last congruence condition, and for all them you'll have $$p \equiv 1 \bmod b$$ and $$a \bmod p$$ is a quadratic residue.
Actually, it is overkill to appeal to Dirichlet's theorem here, because we are seeking primes satisfying a congruence condition of the form $$p \equiv 1 \bmod N$$, and in that case the infinitude of such primes follows purely algebraically using values of the $$N$$th cyclotomic polynomial by modifying Euclid's proof of the infinitude of the primes.