On a problem of Diophantus for rationals

JOURNAL OF NUMBER THEORY

卷:132

Article

Dujella, Andrej²  Fuchs, Clemens¹ 

[1] Swiss Fed Inst Technol, Dept Math, Ramistr 101, CH-8092 Zurich, Switzerland

[2] Univ Zagreb, Dept Math, Zagreb 10000, Croatia

关键词: Diophantine m-tuples;    Linear polynomials;    Elliptic curves;    Twists;    Rank;    Parity conjecture;   

DOI  :  10.1016/j.jnt.2012.04.004

来源: Elsevier

【 摘 要 】

Let q be a nonzero rational number. We investigate for which q there are infinitely many sets consisting of five nonzero rational numbers such that the product of any two of them plus q is a square of a rational number. We show that there are infinitely many square-free such q and on assuming the Parity Conjecture for the twists of an explicitly given elliptic curve we derive that the density of such q is at least one half. For the proof we consider a related question for polynomials with integral coefficients. We prove that, up to certain admissible transformations, there is precisely one set of non-constant linear polynomials such that the product of any two of them except one combination, plus a given linear polynomial is a perfect square. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free