Combinatorial Diophantine equations and a refinement of a theorem on separated variables equations

Abstract

Abstract. We look at Diophantine equations arising from equating classical counting functions such as perfect powers, binomial coefficients and Stirling numbers of the first and second kind. The proofs of the finiteness statements that we give use a variety of methods from modern number theory, such as effective and ineffective tools from Diophantine approximation. As a tool for one part of the statements we establish a theoretical result that gives a more precise description on the structure of the solution set in the theorem, due to Bilu and Tichy, on Diophantine equations with separate variables in the case when infinitely many solutions exist.

30 years of collaboration

“…Then there is an effectively computable constant C such 15 Instead of trying to review the huge related literature, we only refer to the books [159,106]. 16 In [178] the conditions were completely clarified by Zannier and a sharp bound on the number of solutions was given.…”

Composite values of polynomial power sums

“…Another example is that the decompositions of f exhibit arithmetical properties associated with f , which is used to solve equations of separated variable type (cf. [2,3]). The invertible elements in C[x] with respect to decomposition are the linear polynomials.…”

Decomposable polynomials in second order linear recurrence sequences

“…Another example is that the decompositions of f exhibit arithmetical properties associated with f , which is used to solve equations of separated variable type (cf. [2,3]). The invertible elements in C[x] with respect to decomposition are the linear polynomials.…”