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!

  • $\begingroup$ Isn't your ABC example backwards? It has infinitely many exceptions if $\kappa=1$, but your question asks for a property that is satisfied for finitely many integers if $\kappa=1$. $\endgroup$ – TonyK Apr 7 at 23:01
  • $\begingroup$ @TonyK does not change the generality of the question, it is enough to reverse the inequality mark and substitute infinite by finite. $\endgroup$ – MasM Apr 7 at 23:44

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.

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    $\begingroup$ I'd bet that "finitely many solutions" was intended to exclude zero solutions, even though it wasn't explicitly stated. $\endgroup$ – JonathanZ supports MonicaC Apr 7 at 23:30
  • $\begingroup$ Well done, can you give another example which instead of null solution for $\kappa=1$ in your second example ,there be finitely many solutions? $\endgroup$ – MasM Apr 7 at 23:35
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    $\begingroup$ @MasM edited accordingly $\endgroup$ – Ethan MacBrough Apr 7 at 23:43
  • $\begingroup$ Great, that implicitly implies how vast and mighty the real realm is, compare to countable world ;) $\endgroup$ – MasM Apr 8 at 1:17

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