Sums of S-units and Perfect Powers in Recurrence Sequences

Abstract

Diophantine equations are polynomial or exponential equations in two or more variables for which integer solutions are sought. These equations look very simple but some are very difficult to solve. The same is the situation for recurrence relations. The recurrence relations sometimes arise in connection with combinatorial problems and sometimes arise in connection with the solutions of certain Diophantine equations. The best examples are Pell equations which are mostly solvable and the solutions constitute one or more classes of recurrence sequences. An interesting problem in connection with recurrence sequences is to find the terms which are perfect powers and a lot of work is available in the literature to identify such terms for almost all well-known sequences. A generalization of this problem is the study of Diophantine equations involving terms from some recurrence sequences. So far as the balancing sequence is concerned, the perfect powers occur rarely in the sums, differences of squares and sums or differences of cubes of two balancing numbers. However, in the case of general binary recurrence sequences, with some restrictions, only in finitely many cases, the sum of two terms is a perfect power. Given a finite set of primes S, an S-unit is an integer generated by the elements of S. An interesting problem is the study the Diophantine equation obtained by equating a linear combination of terms of a recurrence sequence with the sum of S-units. Under certain conditions, such Diophantine equations can be seen to admit a finite number of solutions. This can be achieved by providing an effective bound to the largest solution. The general Pell equation is the most common Diophantine equation that arises in connection with a variety of problems. A problem related to recurrence sequences and general Pell equations consists of searching for sums of two terms of some recurrence sequences in the solution sets of generalized Pell equations. With some conditions, the finiteness of such occurrences can be established by providing an upper bound for the number of solutions.