Elementary logarithmic lower bound for primes in the quadratic-residue classes of two
= Elementary logarithmic lower bound for primes in the quadratic-residue classes of two
Let $\pi_k(x)$ count primes at most $x$ congruent to $k$ modulo eight. Bounding the prime-power divisibility of
$$
\prod_{\sqrt x/2\leq n\leq\sqrt x}(n^2-2)
$$
above and its size below gives, for all sufficiently large $x$,
$$
\pi_1(x)+\pi_7(x)+1\geq\frac{\log x}{6\log3}.
$$