PIP

Power-integral points on elliptic curves

 Coordinatore UNIVERSITEIT UTRECHT 

 Organization address address: Heidelberglaan 8
city: UTRECHT
postcode: 3584 CS

contact info
Titolo: Mr.
Nome: Martijn
Cognome: Dekker
Email: send email
Telefono: -2531424
Fax: 31-30-2531645

 Nazionalità Coordinatore Netherlands [NL]
 Totale costo 0 €
 EC contributo 151˙863 €
 Programma FP7-PEOPLE
Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013)
 Code Call FP7-PEOPLE-IEF-2008
 Funding Scheme MC-IEF
 Anno di inizio 2009
 Periodo (anno-mese-giorno) 2009-07-01   -   2011-06-30

 Partecipanti

# participant  country  role  EC contrib. [€] 
1    UNIVERSITEIT UTRECHT

 Organization address address: Heidelberglaan 8
city: UTRECHT
postcode: 3584 CS

contact info
Titolo: Mr.
Nome: Martijn
Cognome: Dekker
Email: send email
Telefono: -2531424
Fax: 31-30-2531645

NL (UTRECHT) coordinator 151˙863.52

Mappa


 Word cloud

Esplora la "nuvola delle parole (Word Cloud) per avere un'idea di massima del progetto.

points    extensions    solving    century    mathematical    sequence    theorem    equations    curves    applicant    sequences    perfect    proof    fibonacci    divisibility    power    found    fermat    hilbert    powers    diophantine    elliptic    finite    integral    faltings    problem    curve    generalized   

 Obiettivo del progetto (Objective)

'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.'

Introduzione (Teaser)

EU-funded researchers developed a novel method to find mathematical solutions to an important family of recurrent sequences relevant to Internet security protocols. Advances in this area have potential use in solving a whole new class of mathematical equations.

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