$S$-integral solutions to a Weierstrass equation

Benjamin M. M. de Weger

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Displaying similar documents to “ $S$ -integral solutions to a Weierstrass equation”

The polynomial $x^{3} + x^{2} + x - 1$ and elliptic curves of conductor 11

Alfred J. Van der Poorten (1976-1977)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

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$S$ -integral points on elliptic curves - Notes on a paper of B. M. M. de Weger

Emanuel Herrmann, Attila Pethö (2001)

Journal de théorie des nombres de Bordeaux

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In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and $p$ -adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

On the diophantine equation $x^{2} = y^{p} + 2^{k} z^{p}$

Samir Siksek (2003)

Journal de théorie des nombres de Bordeaux

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We attack the equation of the title using a Frey curve, Ribet’s level-lowering theorem and a method due to Darmon and Merel. We are able to determine all the solutions in pairwise coprime integers $x, y, z$ if $p \geq 7$ is prime and $k \geq 2$ . From this we deduce some results about special cases of this equation that have been studied in the literature. In particular, we are able to combine our result with previous results of Arif and Abu Muriefah, and those of Cohn to obtain a complete solution for the equation...

The integral points on elliptic curves $y^{2} = x^{3} + (36 n^{2} - 9) x - 2 (36 n^{2} - 5)$

Hai Yang, Ruiqin Fu (2013)

Czechoslovak Mathematical Journal

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Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n > 1$ and both $6 n^{2} - 1$ and $12 n^{2} + 1$ are odd primes, then the general elliptic curve $y^{2} = x^{3} + (36 n^{2} - 9) x - 2 (36 n^{2} - 5)$ has only the integral point $(x, y) = (2, 0)$ . By this result we can get that the above elliptic curve has only the trivial integral point for $n = 3, 13, 17$ etc. Thus it can be seen that the elliptic curve $y^{2} = x^{3} + 27 x - 62$ really is an unusual elliptic curve which has large integral points. ...

Diophantine $m$ -tuples and elliptic curves

Andrej Dujella (2001)

Journal de théorie des nombres de Bordeaux

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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 $\begin{matrix} (x, y) \in {(0 \pm 1), (4 F_{2 k + 1} F_{2 k + 2} F_{2 k + 3} \pm (2 F_{2 k + 1} F_{2 k + 2} - 1) \\ \times (2 F_{2 k + 2}^{2} + 1) (2 F_{2 k + 2} F_{2 k + 3} + 1)} . \end{matrix}$

Solving elliptic diophantine equations: the general cubic case

Roelof J. Stroeker, Benjamin M. M. de Weger (1999)

Acta Arithmetica

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Division-ample sets and the Diophantine problem for rings of integers

Gunther Cornelissen, Thanases Pheidas, Karim Zahidi (2005)

Journal de Théorie des Nombres de Bordeaux

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We prove that Hilbert’s Tenth Problem for a ring of integers in a number field $K$ has a negative answer if $K$ satisfies two arithmetical conditions (existence of a so-called set of integers and of an elliptic curve of rank one over $K$ ). We relate division-ample sets to arithmetic of abelian varieties.

A new construction of $p$ -adic $L$ -functions attached to certain elliptic curves with complex multiplication

John L. Boxall (1986)

Annales de l'institut Fourier

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In this paper we apply the results of our previous article on the $p$ -adic interpolation of logarithmic derivatives of formal groups to the construction of $p$ -adic $L$ -functions attached to certain elliptic curves with complex multiplication. Our results are primarily concerned with curves with supersingular reduction.

Displaying similar documents to “ S -integral solutions to a Weierstrass equation”

Displaying similar documents to “ $S$ -integral solutions to a Weierstrass equation”