Define $S$ to be the set of integers $k$, such that the Mordell curve $$y^2=x^3+k$$ has at least one integral point. I would like to get an idea of the structure of $S$. A first step would be to find pairs $(a,b)$ of positive integers such that $an+b\in S$ for every $n\in \mathbb Z$. I am not sure whether such families exist.
My idea was to find a parametrisation $y=f(t)$, $x=g(t)$ leading to such a family, in other words :
Are there polynomials $f,g\in \mathbb Z[t]$, such that $\deg(f^2-g^3)=1$ ?
For a start, I checked the polynomials with coefficients in the range $[-7,7]$ and with $\deg(f)\le 3$ , $\deg(g)\le 2$ and found no example.
It seems that this arcticle
https://en.wikipedia.org/wiki/Hall%27s_conjecture
answers my question. If I interprete the (proven) inequality right, such polynomials cannot exist.
The article seems to contain a typo : Shouldn't it be "$g(t)^2\ne f(t)^3$" instead of "$g(t)^3\ne f(t)^2$" or alternatively "$\deg(f(t)^2-g(t)^3)$" instead of "$\deg(g(t)^2-f(t)^3)$" ? Perhaps someone knows the proof and can state the theorem in the correct way.