Can anybody give an example of a system of equivalence and/or inequality relations dependent to just a real value number like $\kappa$, which satisfies for finitely many integers if $\kappa =1$ and holds for infinitely many integers if we change $\kappa$ from $1$ by infinitely small real value like $\epsilon \in \mathbb{R}$.
an example like the statement of ABC conjecture below, but surely be much simpler to be resolved in an easy way.
If :
i) $\mathrm{rad}\,(n)$ is the product of the distinct primes in $n$,
ii) $A,B,C$ are three positive coprime integers,
iii) $A+B=C\ $,
iv) $\kappa \in \mathbb{R}$,
then for infinitely many exceptions we have: $$C<\mathrm{rad}\,(ABC)^{\kappa }\\ for\\ \kappa=1 .\tag{1}$$ and if we change $\kappa$ infinitely small $\epsilon >0$ from $1$ upward then for finitely many exceptions we have: $$C<\mathrm{rad}\,(ABC)^{\kappa }\\ for\\ \kappa=1+ \epsilon .\tag{2}$$
I asked this question because it is somehow odd to me that a slight change in a real value parameter may cause a sudden change to the state of a system dependent to that value from $\aleph _0$ to finite!
Let $x$ be the set of numbers $x\in\mathbb{R}$ satisfying $1<x<\kappa$. Then for $\kappa=1$ there are no solutions, but for any $\kappa>1$ there are continuously infinite solutions!
If you want a set of solutions that are restricted to integers, consider the set of tuples $(x,n)\in\mathbb{Z}^2$ with $1<x<\kappa^n$.
Edit: for a system where the number of solutions is finite but nonzero at $\kappa = 1$, take the following:
$x\in\mathbb{Z}$ such that $1\leqslant x\leqslant\kappa^x$.
Then when $\kappa = 1$, there is exactly one solution, namely $x=1$. But if $\kappa > 1$, then for large enough $x$ it will always be true that $x<\kappa^x$, giving infinitely many solutions.