Executive Summary : | In the present project, a systematic and in-depth study of zero sets of recurrence sequences and S-units in recurrence sequences is proposed. The zero set of a linear recurrence sequence is a union of a finite set and finitely many arithmetic progressions. This result is known as Skolem Mahler Lech (SML) theorem. There are many different proofs and extensions of the SML theorem in the literature though all proofs use p-adic methods in some way or other. However, these proofs are ineffective. This means that we do not know any algorithm that allows us to determine the zero set of a recurrence sequence. Furthermore, many quantitive results on a zero set of recurrence sequences have also been established. Recently, Derksen showed that the zero set of a linear recurrence sequence over a field of characteristic p is p-automatic. Next, a large number of interesting Diophantine equations arise when one studies the intersection of two sequences of positive integers. For example, the problem of determining all perfect powers in the Fibonacci sequence and in the Lucas sequence was a famous open problem for over 40 years and has been recently resolved using both classical and modular methods. In recent years, a combination of the development of general theory, computational tools, and computational techniques has greatly improved our ability to explicitly solve Diophantine equations concerning recurrence sequences. In recent years, we have proved few results related to S units and recurrence sequence. A sequence is holonomic if it satisfies a polynomial-linear recurrence. During the course of this project, we plan to explore in detail the zero set of holonomic sequences, S units in linear recurrence sequences, and prime divisors of some parametric linear recurrence sequences. We are hoping to accomplish these results, the concept of lower bounds for linear forms in logarithms of algebraic numbers, the Baker Davenport reduction method, continued fractions, elementary estimates from the theory of prime numbers, Schmidt subspace theorem, modular method and automatic sequences will be of use. |