Diophantine Equations with Linear Recurrences an Overview of Some Recent Progress

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called ”quotient problem” (resp. ”d-th root problem”), which in short asks whether the integrality of the values of the quotient (resp. d-th root) of two (resp. one) linear recurrences implies that this quotient (resp. d-th root) is itself a recurrence. We shall also relate such questions with certain natural diophantine equations, which in turn come from the simplest unknown cases of Vojta’s conjecture for integral points on algebraic varieties. Umberto Zannier Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa, ITALY E-mail : [email protected]