Let $F(x,y)$ be a squarefree binary form with integer coefficients, possibly reducible, $\deg(F) \ge 3$.
I am interested in ways of getting infinitely many integer solutions $(x,y,m), m \ne 0$ to $F(x,y)=m$, maximizing $\max(|x|,|y|)$ relative to $m$.
More formally, suppose $F$ is as above, $f$ is an increasing function, and for infinitely many $x,y,m$, $m \ne 0$ the following hold:
1) $F(x,y)=m$.
2) $\max(|x|,|y|) \ge f(|m|,\deg(F))$ or $\max(|x|,|y|) \ge f(|m|)$
How fast can $f$ grow?
Partial result: For $\deg(F)=3$, it is possible to have $f(m)=C m$ for $C$ an arbitrary large constant depending on $F$. This may be best possible for $F$ of degree $3$.
Added later The construction with $f(|m|)= C m$ as suggested in comments:
For $C \sim \varphi^{k+1}$, we can take $f(|m|)= C m$ and $$G(x,y) = (F_{k+1} x - F_k y) (x^2 + x y - y^2)$$ where $F_k$ is the $k^{th}$ Fibonacci number.
Then we can take $x=F_n$, $y=F_{n+1}$, $m=(-1)^{n+1} F_{n-k}$.
We have the identities
$$ F_{k+1} F_n - F_k F_{n+1} \ = \ F_{k+1} x - F_k y \ = \ F_{n-k} $$ $$ F_n^2 + F_n F_{n+1} - F_{n+1}^2 \ = \ x^2 + xy - y^2 \ = \ (-1)^{n+1}.$$
which yield $G(x,y) = m$.
This construction seems to give an unbounded number $G(x_i,y_i)$ of unbounded quality for the Granville-Langevin conjecture, which is equivalent to $abc$ without using the radical at all.
This is a lower bound for $f(|m|)$ and experimentally there are much better solutions with the same $G$, though I don't know if they are infinite (probably not).
Multiplying $G$ by similar linear factors, I believe one can get $f(|m|)=C |m|^{\frac{1}{(\deg(G)-2)}}$.
