Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to,
$$a m^2 + b m n + c n^2 = d\tag{1}$$
Identity 1:
$$a x^2 + b x y + c y^2 - d z^2 = (a m^2 + b m n + c n^2 - d)(a p^2+bpq+c q^2)^2\tag{2}$$
where,
$$x =(am+bn)p^2+2cnpq-cmq^2$$
$$y = -anp^2+2ampq+(bm+cn)q^2$$
$$z = z_1 = ap^2+bpq+cq^2$$
for arbitrary $p,q$.
Identity 2:
Given initial solutions $m,n$ to $(1)$. Then,
$$a x^2 + b x y + c y^2 - d z^2 = (a m^2 + b m n + c n^2 - d)(x/m)^2\tag{3}$$
where,
$$x = mu^2 - 2(b m + 2c n)u v + D m v^2$$
$$y = n u^2 + 2(2a m + b n)u v + D n v^2$$
$$z = z_2 = u^2 - D v^2$$
$$D = b^2-4ac$$
for arbitrary $u,v$.
Questions: Let,
$$z_1 = ap^2+bpq+cq^2 =\pm 1\tag{4}$$
$$z_2 = u^2 - D v^2 = \pm 1\tag{5}$$
Is it true:
- There are times we can solve $(4)$ in the integers, but not $(5)$?
- Likewise, there are times we can solve $(5)$, but not $(4)$?
- Or does being able to solve $(4)$ imply solvability of $(5)$?
See also this post.