Diophantine equations E(x) = P(x) with E exponential, P polynomial
Wolfgang M. Schmidt (2004)
Acta Arithmetica
Manisha Kulkarni, B. Sury (2005)
Acta Arithmetica
Bartosz Naskręcki (2016)
Banach Center Publications
We discuss the distribution of Mordell-Weil ranks of the family of elliptic curves y² = (x + αf²)(x + βbg²)(x + γh²) where f,g,h are coprime polynomials that parametrize the projective smooth conic a² + b² = c² and α,β,γ are elements from ℚ̅. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.
Andrej Dujella, Alan Filipin, Clemens Fuchs (2007)
Acta Arithmetica
Enrico Bombieri, Julia Mueller, Umberto Zannier (2001)
Acta Arithmetica
Filipin, Alan (2007)
International Journal of Mathematics and Mathematical Sciences
Aguiló-Gost, F., García-Sánchez, P.A. (2010)
The Electronic Journal of Combinatorics [electronic only]
Claude Levesque, Michel Waldschmidt (2012)
Acta Arithmetica
José María Grau, Antonio M. Oller-Marcén, Manuel Rodríguez, Daniel Sadornil (2015)
Czechoslovak Mathematical Journal
The structure of the group and Fermat’s little theorem are the basis for some of the best-known primality testing algorithms. Many related concepts arise: Euler’s totient function and Carmichael’s lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer’s totient problem, Giuga’s conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group . In particular, we characterize Gaussian Carmichael numbers...
Régis de la Bretèche (2009)
Journal de Théorie des Nombres de Bordeaux
Ce papier présente les récents progrès concernant les fonctions zêta des hauteurs associées à la conjecture de Manin. En particulier, des exemples où on peut prouver un prolongement méromorphe de ces fonctions sont détaillés.
Mihai Cipu (2015)
Acta Arithmetica
A set of m positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine m-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple a,b,c,d,e with a < b < c < d < ed < 1.55·1072b < 6.21·1035c = a + b + 2√(ab+1) and ...
Iskander Aliev, Lenny Fukshansky, Martin Henk (2012)
Acta Arithmetica
Nicolas Templier (2015)
Journal of the European Mathematical Society
Let be a Hecke–Maass cusp form of eigenvalue and square-free level . Normalize the hyperbolic measure such that and the form such that . It is shown that for all . This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.
Gaël Rémond (2000)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
Kevin Destagnol (2016)
Acta Arithmetica
Following the line of attack of La Bretèche, Browning and Peyre, we prove Manin's conjecture in its strong form conjectured by Peyre for a family of Châtelet surfaces which are defined as minimal proper smooth models of affine surfaces of the form Y² - aZ² = F(X,1), where a = -1, F ∈ ℤ[x₁,x₂] is a polynomial of degree 4 whose factorisation into irreducibles contains two non-proportional linear factors and a quadratic factor which is irreducible over ℚ [i]. This result...
Florian Luca, Pantelimon Stănică (2007)
Acta Arithmetica
Jan-Hendrik Evertse, Umberto Zannier (2008)
Acta Arithmetica
Kálmán Liptai (2006)
Acta Mathematica Universitatis Ostraviensis
A positive is called a balancing number if We prove that there is no balancing number which is a term of the Lucas sequence.
Alex Gorodnik, François Maucourant, Hee Oh (2008)
Annales scientifiques de l'École Normale Supérieure
Let be the wonderful compactification of a connected adjoint semisimple group defined over a number field . We prove Manin’s conjecture on the asymptotic (as ) of the number of -rational points of of height less than , and give an explicit construction of a measure on , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points on . Our approach is based on the mixing property of which we obtain with a rate of convergence.
Tim D. Browning, Ulrich Derenthal (2009)
Annales de l’institut Fourier
The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type .