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Solutions of the Diophantine Equation $7 X^{2} + Y^{7} = Z^{2}$ from Recurrence Sequences

Hayder R. Hashim (2020)

Communications in Mathematics

Consider the system $x^{2} - a y^{2} = b$ , $P (x, y) = z^{2}$ , where $P$ is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation $7 X^{2} + Y^{7} = Z^{2}$ if $(X, Y) = (L_{n}, F_{n})$ (or $(X, Y) = (F_{n}, L_{n})$ ) where ${F_{n}}$ and ${L_{n}}$ represent the sequences of Fibonacci numbers and Lucas numbers respectively....

Solutions of x³+y³+z³=nxyz

Erik Dofs (1995)

Acta Arithmetica

The diophantine equation (1) x³ + y³ + z³ = nxyz has only trivial solutions for three (probably) infinite sets of n-values and some other n-values ([7], Chs. 10, 15, [3], [2]). The main set is characterized by: n²+3n+9 is a prime number, n-3 contains no prime factor ≡ 1 (mod 3) and n ≠ - 1,5. Conversely, equation (1) is known to have non-trivial solutions for infinitely many n-values. These solutions were given either as "1 chains" ([7], Ch. 30, [4], [6]), as recursive...

Solutions to infinite families of complex cubic Thue equations.

Emery Thomas (1993)

Journal für die reine und angewandte Mathematik

Solutions to xyz = 1 and x + y + z = k in algebraic integers of small degree, II

H. G. Grundman, L. L. Hall-Seelig (2015)

Acta Arithmetica

Let k ∈ ℤ be such that $|_{k} (ℚ) |$ is finite, where $_{k} : y ² = 1 - 2 k x + k ² x ² - 4 x ³$ . We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.

Solutions to xyz = 1 and x+y+z = k in algebraic integers of small degree, I

H. G. Grundman, L. L. Hall-Seelig (2014)

Acta Arithmetica

Let k ∈ ℤ be such that $|_{k} (ℚ) | = 3$ , where $_{k} : y ² = 1 - 2 k x + k ² x ² - 4 x ³$ . We determine all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.

Solvability of 𝔭-adic diagonal equations

Christopher M. Skinner (1996)

Acta Arithmetica

Solving a family of Thue equations with an application to the equation x²-Dy⁴=1

A. Togbe, P. M. Voutier, P. G. Walsh (2005)

Acta Arithmetica

Solving a linear equation in a set of integers I

Imre Z. Ruzsa (1993)

Acta Arithmetica

Solving a linear equation in a set of integers II

Imre Z. Ruzsa (1995)

Acta Arithmetica

Solving an exponential Diophantine equation

Mario Huicochea (2010)

Acta Arithmetica

Solving diophantine equations $x^{4} + y^{4} = q z^{p}$

Luis V. Dieulefait (2005)

Acta Arithmetica

Solving elliptic diophantine equations avoiding Thue equations and elliptic logarithms.

de Weger, Benjamin M.M. (1998)

Experimental Mathematics

Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms

R. J. Stroeker, N. Tzanakis (1994)

Acta Arithmetica

Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations

N. Tzanakis (1996)

Acta Arithmetica

Solving elliptic diophantine equations: the general cubic case

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

Acta Arithmetica

Solving linear systems of equations over integers with Gröbner bases

Amir Hashemi (2014)

Acta Arithmetica

We introduce a novel application of Gröbner bases to solve (non-homogeneous) systems of integer linear equations over integers. For this purpose, we present a new algorithm which ascertains whether a linear system of equations has an integer solution or not; in the affirmative case, the general integer solution of the system is determined.

Solving parametrized systems of Pell equations

Paulo Ribenboim (2005)

Acta Arithmetica

Solving the equation x3 - y2 = r in number fields.

Michael Laska (1982)

Journal für die reine und angewandte Mathematik

Some applications of Baker's sharpened bounds to diophantine equations

Robert Tijdeman (1974/1975)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Some applications of decomposable form equations to resultant equations

K. Győry (1993)

Colloquium Mathematicae

1. Introduction. The purpose of this paper is to establish some general finiteness results (cf. Theorems 1 and 2) for resultant equations over an arbitrary finitely generated integral domain R over ℤ. Our Theorems 1 and 2 improve and generalize some results of Wirsing [25], Fujiwara [6], Schmidt [21] and Schlickewei [17] concerning resultant equations over ℤ. Theorems 1 and 2 are consequences of a finiteness result (cf. Theorem 3) on decomposable form equations over R. Some applications of Theorems...

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