Currently displaying 1 – 17 of 17

Showing per page

Order by Relevance | Title | Year of publication

Diophantine m -tuples and elliptic curves

Andrej Dujella — 2001

Journal de théorie des nombres de Bordeaux

A Diophantine m -tuple is a set of m positive integers such that the product of any two of them is one less than a perfect square. In this paper we study some properties of elliptic curves of the form y 2 = ( a x + 1 ) ( b x + 1 ) ( c x + 1 ) , where { a , b , c } , is a Diophantine triple. In particular, we consider the elliptic curve E k defined by the equation y 2 = ( F 2 k x + 1 ) ( F 2 k + 2 x + 1 ) ( F 2 k + 4 x + 1 ) , where k ≥ 2 and F n , denotes the n -th Fibonacci number. We prove that if the rank of E k ( 𝐐 ) is equal to one, or k ≤ 50 , then all integer points on E k are given by ( x , y ) ∈ { ( 0 ± 1 ) , ( 4 F 2 k + 1 F 2 k + 2 F 2 k + 3 ± 2 F 2 k + 1 F 2 k + 2 - 1 × 2 F 2 k + 2 2 + 1 2 F 2 k + 2 F 2 k + 3 + 1 } .

Page 1

Download Results (CSV)