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Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields

István Gaál, Gábor Petrányi (2014)

Czechoslovak Mathematical Journal

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $ξ$ of the polynomial $P_{t} (x) = x^{4} - t x^{3} - 6 x^{2} + t x + 1$ , assuming that $t > 0$ , $t \neq 3$ and $t^{2} + 16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal...

Characterizing the sum of two cubes.

Broughan, Kevin A. (2003)

Journal of Integer Sequences [electronic only]

Complete solution of a family of simultaneous Pellian equations

Andrej Dujella (1998)

Acta Mathematica et Informatica Universitatis Ostraviensis

Complete solutions of a family of cubic Thue equations

Alain Togbé (2006)

Journal de Théorie des Nombres de Bordeaux

In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation $\begin{matrix} Φ_{n} (x, y) & = x^{3} + (n^{8} + 2 n^{6} - 3 n^{5} + 3 n^{4} - 4 n^{3} + 5 n^{2} - 3 n + 3) x^{2} y \\ - (n^{3} - 2) n^{2} x y^{2} - y^{3} = \pm 1, \end{matrix}$ for $n \geq 0$ .

Computing integral points on elliptic curves

J. Gebel, A. Pethő, H. G. Zimmer (1994)

Acta Arithmetica

Computing integral points on Mordell's elliptic curves.

J. Gebel, A. Pethö, H. G. Zimmer (1997)

Collectanea Mathematica

Congruence properties of the number of solutions of some equations.

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

Journal für die reine und angewandte Mathematik

Congruent numbers over real number fields

Tomasz Jędrzejak (2012)

Colloquium Mathematicae

It is classical that a natural number n is congruent iff the rank of ℚ -points on Eₙ: y² = x³-n²x is positive. In this paper, following Tada (2001), we consider generalised congruent numbers. We extend the above classical criterion to several infinite families of real number fields.

Congruent numbers with higher exponents

Florian Luca, László Szalay (2006)

Acta Mathematica Universitatis Ostraviensis

This paper investigates the system of equations $x^{2} + a y^{m} = z_{1}^{2}, x^{2} - a y^{m} = z_{2}^{2}$ in positive integers $x$ , $y$ , $z_{1}$ , $z_{2}$ , where $a$ and $m$ are positive integers with $m \geq 3$ . In case of $m = 2$ we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient $a$ with the condition $gcd (x, z_{1}) = gcd (x, z_{2}) = gcd (z_{1}, z_{2}) = 1$ . Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero $a$ .

Corrigendum to the paper "The number of solutions of the Mordell equation" (Acta Arith. 88 (1999), 173–179)

Dimitrios Poulakis (2000)

Acta Arithmetica

Crossed ladders and Euler's quartic.

Bremner, A., Høibakk, R., Lukkassen, D. (2008)

Annales Mathematicae et Informaticae

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