Results 21 to 30 of about 2,595 (231)

On a Diophantine Equation [PDF]

Journal of the London Mathematical Society, 1951
Summary: Denote by \(N(a,b)\) the smallest integer \(n\) so that \[ \frac{a}{b}=\frac{1}{x_1}+\cdots+\frac{1}{x_n},\quad ...
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GEOMETRIC THEOREMS, DIOPHANTINE EQUATIONS, AND ARITHMETIC FUNCTIONS [PDF]

, 2002
This book contains short notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers ...
Sándor, József
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On an diophantine equation [PDF]

Bulletin of the Australian Mathematical Society, 2000
In this note, we find all solutions of the diophatine equation x2 + 3m = yn, where (x, y, m, n) are non-negative integers with x ≠ 0 and n ≥ 3.
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A diophantine equation [PDF]

Glasgow Mathematical Journal, 1985
I was recently challenged to find all the cases when the sum of three consecutive integral cubes is a square; that is to find all integral solutions x, y ofy2=(x−1)3+x3+(x+1)3=3x(x2+2)This is an example of a curve of genus 1. There is an effective procedure for finding all integral points on a given curve of genus 1 ([1, Theorem 4.2], [2]): that is, it
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A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

Open Computer Science, 2018
We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms xi· xj = xk and xi + 1 = xi has only finitely many solutions in non-negative integers x1, .
Tyszka Apoloniusz
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On simultaneous diophantine equations [PDF]

Acta Arithmetica, 2003
The authors investigate the number of solutions of the simultaneous Diophantine equations \[ x^2- (M^2+4)y^2= -4, \quad y^2-dz^2=1, \tag{1} \] where \(M\) is assumed to be an odd positive integer and where \(d\) is a squarefree integer. They show that for squarefree \(d\) with at most four distinct prime factors, system (1) can have at most one ...
Katayama, Shin-ichi, Levesque, Claude
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The Solution of a Diophantine Equation [PDF]

Proceedings of the American Mathematical Society, 1952
in which we suppose that f(y) =f(yi, * ya) is a homogeneous polynomial, with integral coefficients, of degree m, where m is of the form 2P(2q+1), q being a non-negative integer, p is one of the integers 0, 1, * * *, n -1, and thus m 0 0 (mod 2n). We suppose further that the rank of the matrix of the forms Enl aajx (i= 1, , 2n) is 2n -1 and thus we may ...
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ON LINEAR DIOPHANTINE EQUATION [PDF]

, 2017
In this paper, the concept of Diophantine equation are discussed and the theorem relating to the linear Diophantine equations of two variables x and y are ...
Dr. D. Ramprasad
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On the Relationship Between Matiyasevich's and Smorynski's Theorems

Scientific Annals of Computer Science, 2019
Let R be a non-zero subring of Q with or without 1. We assume that for every positive integer n there exists a computable surjection from N onto Rn. Every R \in {Z,Q} satisfies these conditions.
Agnieszka Peszek, Apoloniusz Tyszka
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A note on the Diophantine equation (xᵏ-1)(yᵏ-1)²=zᵏ-1 [PDF]

Notes on Number Theory and Discrete Mathematics
We prove that, for k≥10, the Diophantine equation (xᵏ-1)(yᵏ-1)²=zᵏ-1 in positive integers x, y, z, k with z>1, has no solutions satisfying ...
Yangcheng Li
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