Understanding the structure of recurrent sequences
The famous Fibonacci sequence, in which each successive term is the sum of the previous two (1, 2, 3, 5, 8 and so on), is one of a large class of so-called recurrent sequences. Elliptic divisibility sequences are another important group of such sequences that grow much more rapidly than the Fibonacci sequence. Elliptic divisibility sequences have been the subject of renewed interest in recent years due to their importance in cryptography (important in Internet fast security protocols) and undecidability (related to what is possible to calculate with computers). European researchers supported by funding of the ‘Power-integral points on elliptic curves’ (PIP) project sought to understand any pattern or structure that might occur in elliptic divisibility sequences with a focus on pure powers in such sequences, of fundamental importance to Internet security issues. In previous work, the scientist showed that for every positive integer larger than 2, there is a finite set of points on an elliptic curve (power integral points, PIPs) that contains the so-called integral points. It has since been shown that, in many cases, the PIPs can be found by finding the perfect powers in an elliptic divisibility sequence. Perfect powers are numbers of the form m^k, where m is a positive integer greater than 1 (2, 3, etc.) and k greater than or equal to 2 (2, 3, 4, etc.). Thus, the perfect powers are 2^2=4, 2^3=8 and so on. The goal of the PIP project was to determine all PIPs on families of elliptic curves. In fact, applying a novel modular method based on Andrew Wiles’ work regarding Fermat’s Last Theorem, investigators found all perfect powers in certain elliptic divisibility sequences, thus demonstrating the finite nature of the solution (a countable number of them) and the way to find them. Scientists improved previous results for primes in elliptic divisibility sequences and generalised the concepts to matrices. PIP results could have important impact on a variety of topics related to digital electronics and computing as well as help lead the way to solution of a whole new class of mathematical equations.
