CIRM - Videos & books Library - $D(n)$-sets with square elements

Virtualconference

Auteurs : Dujella, Andrej (Auteur de la Conférence)
CIRM (Editeur )

Loading the player...

Résumé : For an integer n, a set of distinct nonzero integers $\left \{ a_{1},a_{2},...a_{m} \right \}$ such that $a_{i}a_{j}+n$ is a perfect square for all 1 ≤ i < j ≤ m, is called a Diophantine m-tuple with the property $D(n)$ or simply a $D(n)$-set. $D(1)$-sets are known as Diophantine m-tuples. When considering $D(n)$-sets, usually an integer n is fixed in advance. However, we may ask if a set can have the property $D(n)$ for several different n's. For example, {8, 21, 55} is a $D(1)$-triple and $D(4321)$-triple. In a joint work with Adzaga, Kreso and Tadic, we presented several families of Diophantine triples which are $D(n)$-sets for two distinct n's with $n\neq 1$. In a joint work with Petricevic we proved that there are infinitely many (essentially different) quadruples which are simultaneously $D(n_{1})$-quadruples and $D(n_{2})$-quadruples with $n_{1}\neq n_{2}$. Morever, the elements in some of these quadruples are squares, so they are also $D(0)$-quadruples. E.g. $\left \{ 54^{2}, 100^{2}, 168^{2}, 364^{2}\right \} $ is a $D(8190^{2})$, $D(40320^{2})$ and $D(0)$-quadruple. In this talk, we will describe methods used in constructions of mentioned triples and quadruples. We will also mention a work in progress with Kazalicki and Petricevic on $D(n)$-quintuples with square elements (so they are also $D(0)$-quintuples). There are infinitely many such quintuples. One example is a $D(4804802)$-quintuple $\left \{ 225^{2}, 286^{2}, 819^{2}, 1408^{2}, 2548^{2}\right \}$.

Keywords : diophantine triples; diophantine quadruples; elliptic curves

Codes MSC :
11D09 - Quadratic and bilinear equations
11G05 - Elliptic curves over global fields
11Y50 - Computer solution of Diophantine equations

Informations sur la Vidéo

Réalisateur :

Hennenfent, Guillaume

Langue :

Date de publication :

Date de captation :

Sous collection :

arXiv category :

Domaine :

Format :

Durée :

Audience :

Download :

https://videos.cirm-math.fr/2020-11-26_Dujella.mp4

Informations sur la Rencontre

Nom de la rencontre : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire
Organisateurs de la rencontre : Rivat, Joël ; Tichy, Robert
Dates : 23/11/2020 - 27/11/2020
Année de la rencontre : 2020
URL Congrès : https://www.chairejeanmorlet.com/2256.html

Données de citation

DOI : 10.24350/CIRM.V.19686903
Citer cette vidéo: Dujella, Andrej (2020). $D(n)$-sets with square elements. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19686903
URI : http://dx.doi.org/10.24350/CIRM.V.19686903

Voir aussi

[Virtualconference] Improved cap constructions, and sets without arithmetic progressions / Auteur de la Conférence Elsholtz, Christian.
[Virtualconference] On S-Diophantine Tuples / Auteur de la Conférence Ziegler, Volker.
[Virtualconference] Poisson-generic points / Auteur de la Conférence Weiss, Benjamin.
[Virtualconference] Classification and statistics of cut-and-project sets / Auteur de la Conférence Weiss, Barak.
[Virtualconference] On binary quartic Thue equations and related topics / Auteur de la Conférence Walsh, Gary.
[Virtualconference] Multidimensional continued fractions and symbolic codings of toral translations / Auteur de la Conférence Thuswaldner, Jörg.
[Virtualconference] On generalised Rudin-Shapiro sequences / Auteur de la Conférence Stoll, Thomas.
[Virtualconference] Pseudorandomness at prime times and digits of Mersenne numbers / Auteur de la Conférence Shparlinski, Igor.
[Virtualconference] Zaremba's conjecture and growth in groups / Auteur de la Conférence Shkredov, Ilya.
[Virtualconference] Large values of the remainder term of the prime number theorem / Auteur de la Conférence Pintz, Janos.
[Virtualconference] Number of solutions to a special type of unit equations in two unknowns / Auteur de la Conférence Pink, István.
[Virtualconference] Bertini and Northcott / Auteur de la Conférence Pazuki, Fabien.
[Virtualconference] Dynamical irreducibility of polynomials modulo primes / Auteur de la Conférence Ostafe, Alina.
[Virtualconference] Diophantine exponents, best approximation and badly approximable numbers / Auteur de la Conférence Moshchevitin, Nikolay.
[Virtualconference] Some interactions between number theory and multifractal analysis / Auteur de la Conférence Martin, Bruno.
[Virtualconference] The sum-of-digits function in linearly recurrent number systems and almost primes / Auteur de la Conférence Madritsch, Manfred.
[Virtualconference] Fibonacci numbers and repdigits / Auteur de la Conférence Luca, Florian.
[Virtualconference] Skolem's conjecture and exponential Diophantine equations / Auteur de la Conférence Hajdu, Lajos.
[Virtualconference] Equidistribution of roots of unity and the Mahler measure / Auteur de la Conférence Habegger, Philipp.
[Virtualconference] Effective finiteness results for diophantine equations over finitely generated domains / Auteur de la Conférence Györy, Kalman.
[Virtualconference] Constructing abelian extensions with prescribed norms / Auteur de la Conférence Frei, Christopher.
[Virtualconference] (Logarithmic) densities for automatic sequences along primes and squares / Auteur de la Conférence Drmota, Michael.
[Virtualconference] Modularity of the q-Pochhammer symbol and application / Auteur de la Conférence Drappeau, Sary.
[Virtualconference] Higher moments of primes in intervals and in arithmetic progressions, II / Auteur de la Conférence De la Bretèche, Régis.
[Virtualconference] The Rudin-Shapiro function in finite fields / Auteur de la Conférence Dartyge, Cécile.
[Virtualconference] Independence of actions of (N,+) and (N,×) and Sarnak's Möbius disjointness conjecture / Auteur de la Conférence Bergelson, Vitaly.
[Virtualconference] On some diophantine equations in separated variables / Auteur de la Conférence Bérczes, Attila.

Bibliographie

ADŽAGA, Nikola, DUJELLA, Andrej, KRESO, Dijana, et al. Triples which are D (n)-sets for several n's. Journal of Number Theory, 2018, vol. 184, p. 330-341. - https://doi.org/10.1016/j.jnt.2017.08.024

DUJELLA, Andrej et PETRIČEVIĆ, Vinko. Diophantine quadruples with the properties $D (n_1)$ and $D (n_2)$ $D (n2)$. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2020, vol. 114, no 1, p. 21. - https://doi.org/10.1007/s13398-019-00747-9

DUJELLA, Andrej et PETRIČEVIĆ, Vinko. Doubly regular Diophantine quadruple. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2020, vol. 114, art 189 - https://doi.org/10.1007/s13398-020-00921-4

DUJELLA, Andrej, KAZALICKI, Matija, et PETRIČEVIĆ, Vinko. D (n)-quintuples with square elements. arXiv preprint arXiv:2011.01684, 2020. - https://arxiv.org/abs/2011.01684

Dujella.jpg

Sélection Signaler une erreur

TUTELLES

PARTENAIRES

Destination de la recherche

Raccourcis

Voir aussi