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Newton's formula and the continued fraction expansion of $\sqrt{d}$ .

Dujella, Andrej — 2001

Experimental Mathematics

Diophantine quadruples for squares of Fibonacci and Lucas numbers.

Dujella, Andrej — 1995

Portugaliae Mathematica

Square roots with many good approximants.

Dujella, Andrej; Petričević, Vinko — 2005

Integers

Diophantine quadruples in $ℤ [\sqrt{- 2}]$ .

Dujella, Andrej; Soldo, Ivan — 2010

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

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 $k \geq 2$ and $F_{n}$ , denotes the $n$ -th Fibonacci number. We prove that if the rank of $E_{k} (𝐐)$ is equal to one, or $k \leq 50$ , then all integer points on $E_{k}$ are given by $\begin{matrix} (x, y) \in {(0 \pm 1), (4 F_{2 k + 1} F_{2 k + 2} F_{2 k + 3} \pm (2 F_{2 k + 1} F_{2 k + 2} - 1) \\ \times (2 F_{2 k + 2}^{2} + 1) (2 F_{2 k + 2} F_{2 k + 3} + 1)} . \end{matrix}$