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Displaying 261 – 280 of 314

The Diophantine equation X² - db²Y⁴ = 1

Gary Walsh (1998)

Acta Arithmetica

The Diophantine Equation X³ = u+v over Real Quadratic Fields

Takaaki Kagawa (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

Let k be a real quadratic field and let $_{k}$ and $_{k}^{\times}$ be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v ( $X \in_{k}$ , $u, v \in_{k}^{\times}$ ) is developed.

The diophantine equation X3 + Y3 = DZ3 and the conjectures of Birch and Swinnerton-Dyer.

N.M. Stephens (1968)

Journal für die reine und angewandte Mathematik

The Diophantine equation x⁴ - Dy² = 1, II

J. H. E. Cohn (1997)

Acta Arithmetica

The Diophantine equation X(X + 1) = Y(Y... + 1).

J.H.E. Cohn (1991)

Mathematica Scandinavica

The Diophantine equation $y^{2} = D x^{4} + 1$ , II

J. Cohn (1975)

Acta Arithmetica

The Diophantine equation y...=Dx...+1, III.

J.H.E. Cohn (1978)

Mathematica Scandinavica

The diophantine equations ${(x^{2} - c)}^{2} = (t^{2} \pm 2) y^{2} + 1$

J. Cohn (1976)

Acta Arithmetica

The diophantine equations $x^{2} - k = 2 \cdot T_{n} (\frac{b^{2} \pm 2}{2})$ .

Udrea, Gheorghe (1998)

Portugaliae Mathematica

The divisor problem for binary cubic forms

Tim Browning (2011)

Journal de Théorie des Nombres de Bordeaux

We investigate the average order of the divisor function at values of binary cubic forms that are reducible over $ℚ$ and discuss some applications.

The equation xyz = x + y + z = 1 in integers of a cubic field.

Andrew Bremner (1989)

Manuscripta mathematica

The Hasse problem for rational surfaces.

H.P.F. Swinnerton-Dyer, B.J. Birch (1975)

Journal für die reine und angewandte Mathematik

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

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.

The Ljunggren equation revisited

Konstantinos A. Draziotis (2007)

Colloquium Mathematicae

We study the Ljunggren equation Y² + 1 = 2X⁴ using the "multiplication by 2" method of Chabauty.

The method of infinite ascent applied on $A^{4} \pm n B^{3} = C^{2}$

Susil Kumar Jena (2013)

Czechoslovak Mathematical Journal

Each of the Diophantine equations $A^{4} \pm n B^{3} = C^{2}$ has an infinite number of integral solutions $(A, B, C)$ for any positive integer $n$ . In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when $A$ , $B$ and $C$ are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions $(A, B, C)$ of the Diophantine equation $a A^{3} + c B^{3} = C^{2}$ for any co-prime integer pair $(a, c)$ .

The method of Thue-Siegel for binary quartic forms

Shabnam Akhtari (2010)

Acta Arithmetica

The number of solutions of the Mordell equation

Dimitrios Poulakis (1999)

Acta Arithmetica

The number of solutions to cubic Thue inequalities

Jeffrey Lin Thunder (1994)

Acta Arithmetica

The number of zeroes of x3 + y3 + cz3 in certain finite fields.

S. Chowla, J. Cowles, M. Cowles (1978)

Journal für die reine und angewandte Mathematik

The $p$ -part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large

Remke Kloosterman (2005)

Journal de Théorie des Nombres de Bordeaux

In this paper we show that for every prime $p \geq 5$ the dimension of the $p$ -torsion in the Tate-Shafarevich group of $E / K$ can be arbitrarily large, where $E$ is an elliptic curve defined over a number field $K$ , with $[K : ℚ]$ bounded by a constant depending only on $p$ . From this we deduce that the dimension of the $p$ -torsion in the Tate-Shafarevich group of $A / ℚ$ can be arbitrarily large, where $A$ is an abelian variety, with $dim A$ bounded by a constant depending only on $p$ .

Currently displaying 261 – 280 of 314

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