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Combinatorial Diophantine equations and a refinement of a theorem on separated variables equations. (English) Zbl 1289.11035

The authors provide several effective and ineffective results concerning Diophantine equations involving combinatorial counting functions, like Stirling numbers of the first and second kind, binomial coefficients and perfect powers. In their proofs they combine several deep effective and ineffective methods of diophantine number theory, both classical and modern. They also give an extension of a theorem of Bilu and Tichy about the finiteness of solutions of separate variables Diophantine equations, by providing a full description of the degenerate situations. They use this new tool in some of their proofs, too.

MSC:

11D41 Higher degree equations; Fermat’s equation
11D61 Exponential Diophantine equations
05A19 Combinatorial identities, bijective combinatorics
11B65 Binomial coefficients; factorials; \(q\)-identities
11B73 Bell and Stirling numbers
14G05 Rational points
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