In David Cox's book: Primes of the form $x^2+ny^2$, second edition, there is a theorem(Theorem 3.21, page 52) characterize whether two binary quadratic forms in the same genus. The contents of the theorem are:
Let $f(x,y)$ and $g(x,y)$ be primitive forms of discriminant $D\neq0$, positive definite if $D<0$. Then the following are equivalent:
(i) $f(x,y)$ and $g(x,y)$ are in the same genus, i.e., they represent the same values in $(\mathbb Z/D\mathbb Z)^\ast$.
(ii) $f(x,y)$ and $g(x,y)$ represent the same values in $(\mathbb Z/m\mathbb Z)^\ast$ for all nonzero integer $m$.
(iii) $f(x,y)$ and $g(x,y)$ are equivalent modulo $m$ for all nonzero integer $m$.
(iv) $f(x,y)$ and $g(x,y)$ are equivalent over the $p$-adic integer $\mathbb{Z}_p$ for all primes $p$.
(v) $f(x,y)$ and $g(x,y)$ are equivalent over $\mathbb Q$ via a matrix in $GL_2(\mathbb Q)$ whose entries have denominators prime to $2D$.
(vi) $f(x,y)$ and $g(x,y)$ are equivalent over $\mathbb Q$ without essential denominators, i.e., given any nonzero $m$, a matrix in $GL_2(\mathbb Q)$ can be found which takes one form to the other and whose entries have denominators prime to $m$.
The author does not give details of the proof, only some references.
Question: How to derive (ii) and (iii) from (i), and how to prove the equivalence of (iv) and (vi). Could someone please give me some hints？
Addition: $x^2+14y^2$ and $2x^2+7y^2$ are in the same genus, and for primes $p\neq 2,7$,
$$p=x^2+14y^2\,\text{or}\,2x^2+7y^2\Leftrightarrow p\equiv1,9,15,23,25,39\pmod {56}.$$
Is it right that for every congruent class of $1,9,15,23,25,39\pmod {56}$, both $x^2+14y^2$ and $2x^2+7y^2$ represent infinitely many prime numbers in the class?
