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 $3$.




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**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](https://mathoverflow.net/questions/149631/is-there-a-generalization-of-granville-langevin-conjecture-for-number-fields), 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 $f(|m|)=C |m|^{\frac{1}{(\deg(G)-2)}}$.



Diophantine equations with infinitely many large solutions
If the degree of the form is $d$, I think you get solutions with $f(|m|) > cm^{1/(d-2)}$ by applying Dirichlet's theorem in diophantine approximation to a root of $F(x,1)=0$. And you cannot do better than $cm^{1/(d-2)+\epsilon}$ by Thue-Siegel-Roth.


