S. S. Pillai proved that for a fixed positive integer , the exponential Diophantine equation , , has only finitely many solutions in integers and . We prove that when is of the form , the above equation has no solution in integers and with .
The famous problem of determining all perfect powers in the Fibonacci sequence and in the Lucas sequence has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations , with and , for all primes and indeed for all but primes . Here the strategy of [10] is not sufficient due to the sizes of...
We provide a lower bound for the number of distinct zeros of a sum for two rational functions , in term of the degree of , which is sharp whenever have few distinct zeros and poles compared to their degree. This sharpens the “-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface contains only finitely many rational or elliptic curves,...