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Are there diophantine equations with only 2 or more solutions known up till now and it is not known whether there are more solutions to the equation, and also it is not known whether number of solutions are finite or infinite?

I know some problems of such diophantine equations, such as Brocard's Problem, Wilson Primes etc.

What are such some simple problems of such equations compared to Brocard's, Wilson Primes?

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There are values of $n$ where only finitely many solutions in integers to $x^3+y^3+z^3=n$ are known. See this answer for some examples.

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Have a look at the Generalization heading in this Wikipedia article: https://en.wikipedia.org/wiki/Catalan%27s_conjecture

It's a generalization of Catalan conjecture which was proved by Preda Mihăilescu in $2002.$

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