Final Report Summary - PIP (Power-integral points on elliptic curves)
The historical origins of this project lie in the theory of recurrent sequences, such as the famous Fibonacci sequence, in which each term is the sum of the previous two (the sequence starts 1,2,3,5,8,...). Such sequences have received much attention in mathematics, and have shown sometimes unexpected connections with population growth models, number theory, computer science and logic. In the early part of the 20th century, this theory was generalized to so-called 'elliptic divisibility sequences' (these are much more rapidly growing in size than Fibonacci). This generalisation was largely forgotten, but the past ten years have seen a revival, with a booming number of publications about the subject. One of the reasons is surely that elliptic divisibility sequences were found useful in cryptography (and directly relate to issues of implementation of fast security protocols), and useful in undecidability, a branch of logic that explores the boundaries of what is possible to compute using computers. Both of these theoretical issues underly many questions concerning the digital society.
It is important, also for the security issues, to understand any pattern or structure that might occur in the sequence. This project was about investigating the structure of such elliptic divisibility sequences, in particular, the question of pure powers in such sequences. Only recently, the corresponding problem was studied for the Fibonnacci sequence. In this project, we applied a novel 'modular method' to study this problem. This method originates with the deep and fundamental work of Andrew Wiles on Fermat's Last Theorem. During the project, these methods were enhanced and combined with primitive divisor results to find all of the perfect powers in some elliptic divisibility sequences. The results are effective, in the sense that there are finitely many such points and that there is a way to find them. Also, previous finiteness results for primes in elliptic divisibility sequences were improved upon and a more thorough examination of the criteria was given. The project output also consists of a further 'matrix' generalisation of the above concepts. As for number theory per se, the methods developed may lead to the solution of a whole new class of diophantine equations.
It is important, also for the security issues, to understand any pattern or structure that might occur in the sequence. This project was about investigating the structure of such elliptic divisibility sequences, in particular, the question of pure powers in such sequences. Only recently, the corresponding problem was studied for the Fibonnacci sequence. In this project, we applied a novel 'modular method' to study this problem. This method originates with the deep and fundamental work of Andrew Wiles on Fermat's Last Theorem. During the project, these methods were enhanced and combined with primitive divisor results to find all of the perfect powers in some elliptic divisibility sequences. The results are effective, in the sense that there are finitely many such points and that there is a way to find them. Also, previous finiteness results for primes in elliptic divisibility sequences were improved upon and a more thorough examination of the criteria was given. The project output also consists of a further 'matrix' generalisation of the above concepts. As for number theory per se, the methods developed may lead to the solution of a whole new class of diophantine equations.