PIP · Power-integral points on elliptic curves
7РП — „Хора“ (Действия „Мария Кюри“)
- Период
- 2009-07-01 → 2011-06-30
- Финансиране от ЕС
- 151 864 €
- Участници
- 1
- Схема
- MC-IEF
Линиите свързват координатора с партньорите.
Накратко на български
Елиптичните делимостни последователности, които приличат на по-бързо растящи версии на числата на Фибоначи, се анализират за откриване на степени в тях. Това помага за разбирането на структури, използвани в криптографията и при определянето на границите на компютърните изчисления.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
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.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
The study of Diophantine equations is one of the oldest branches of pure mathematics. The 20th century saw Siegel’s theorem about the finitness of integral points on elliptic curves, the negative solution of Hilbert’s 10th problem, Faltings' Theorem and the proof of Fermat’s Last Theorem. In the 21st century much work has already been done to resolve extensions of Hilbert’s 10th problem and to make Faltings' theorem effective. Moreover, solving generalized Fermat equations has, for example, led to all of the perfect powers in the Fibonacci sequence being found, an unsolved problem for over 50 years. In 2006 the applicant proved that for each positive integer larger than 2, there corresponds a finite set of rational points on an elliptic curve which contains the integral points. The points in these finite sets have an important number theoretic structure and are called power-integral points. However, the proof given by the applicant uses Faltings' theorem and so gives no way to find them. Remarkably, in many cases the power-integral points can be found by solving generalized Fermat equations and by finding the perfect powers in an elliptic divisibility sequence. An elliptic divisibility sequence is in many ways an analogue of the Fibonacci sequence and its properties are receiving a lot of attention due to links with extensions of Hilbert’s 10th problem and Cryptography. It is believed that the study of these sequences combined with advances in solving Diophantine equations will achieve the objectives of this proposal. These are: to find all of the power-integral points on families of elliptic curves, and to give a quantitative bound for the number of power-integral points on an arbitrary elliptic curve.
Оригинален текст от CORDIS (на английски).
Участници
- UNIVERSITEIT UTRECHT · UtrechtКоординаторНидерландия
Връзки
Данни: CORDIS, © Европейски съюз