A Note on squares in arithmetic progressions, II

Enrico Bombieri; Umberto Zannier

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2002)

  • Volume: 13, Issue: 2, page 69-75
  • ISSN: 1120-6330

Abstract

top
We show that the number of squares in an arithmetic progression of length N is at most c 1 ⁢ N 3 / 5 ⁢ log ⁡ N c 2 , for certain absolute positive constants c 1 , c 2 . This improves the previous result of Bombieri, Granville and Pintz [1], where one had the exponent 2 3 in place of our 3 5 . The proof uses the same ideas as in [1], but introduces a substantial simplification by working only with elliptic curves rather than curves of genus 5 as in [1].

How to cite

top

Bombieri, Enrico, and Zannier, Umberto. "A Note on squares in arithmetic progressions, II." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13.2 (2002): 69-75. <http://eudml.org/doc/252428>.

@article{Bombieri2002,
abstract = {We show that the number of squares in an arithmetic progression of length $N$ is at most $c_\{1\}N^\{3/5\}(\log N)^\{c_\{2\}\}$, for certain absolute positive constants $c_\{1\}$, $c_\{2\}$. This improves the previous result of Bombieri, Granville and Pintz [1], where one had the exponent $\frac\{2\}\{3\}$ in place of our $\frac\{3\}\{5\}$. The proof uses the same ideas as in [1], but introduces a substantial simplification by working only with elliptic curves rather than curves of genus $5$ as in [1].},
author = {Bombieri, Enrico, Zannier, Umberto},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Diophantine equations; Elliptic curves; Arithmetic progressions; elliptic curves; arithmetic progressions},
language = {eng},
month = {6},
number = {2},
pages = {69-75},
publisher = {Accademia Nazionale dei Lincei},
title = {A Note on squares in arithmetic progressions, II},
url = {http://eudml.org/doc/252428},
volume = {13},
year = {2002},
}

TY - JOUR
AU - Bombieri, Enrico
AU - Zannier, Umberto
TI - A Note on squares in arithmetic progressions, II
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2002/6//
PB - Accademia Nazionale dei Lincei
VL - 13
IS - 2
SP - 69
EP - 75
AB - We show that the number of squares in an arithmetic progression of length $N$ is at most $c_{1}N^{3/5}(\log N)^{c_{2}}$, for certain absolute positive constants $c_{1}$, $c_{2}$. This improves the previous result of Bombieri, Granville and Pintz [1], where one had the exponent $\frac{2}{3}$ in place of our $\frac{3}{5}$. The proof uses the same ideas as in [1], but introduces a substantial simplification by working only with elliptic curves rather than curves of genus $5$ as in [1].
LA - eng
KW - Diophantine equations; Elliptic curves; Arithmetic progressions; elliptic curves; arithmetic progressions
UR - http://eudml.org/doc/252428
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.