Powerintegral points on elliptic curves
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE
Objective
Field of Science
/natural sciences/mathematics/pure mathematics
Topic(s)
FP7PEOPLEIEF2008  Marie Curie Action: "IntraEuropean Fellowships for Career Development"
Funding Scheme
MCIEF  IntraEuropean Fellowships (IEF)
Coordinator
UNIVERSITEIT UTRECHT
Address
Heidelberglaan 8
3584 Cs Utrecht
Netherlands
Activity type
Higher or Secondary Education Establishments
EU Contribution
€ 151 863,51
Administrative Contact
Martijn Dekker (Mr.)
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE
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 socalled 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 ‘Powerintegral 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 socalled 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.
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE
Discover other articles in the same domain of application
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE
Final Report Summary  PIP (Powerintegral points on elliptic curves)
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.
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE
Deliverables
Deliverables not available
Publications
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE
Project information
PIP
Grant agreement ID: 235210
Status
Closed project

Start date
1 July 2009

End date
30 June 2011
Funded under:
FP7PEOPLE