Results 31 to 40 of about 2,597 (231)
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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Systems of Diophantine Equations [PDF]
Proceedings of the American Mathematical Society, 1951 where fi and gi are homogeneous polynomials with integral coefficients, fi being of degree n and gi being of degree m. If there are no integers s> 1, a k, 3' such that ak = sla , ij = s, where X, g are positive integers such that Xn =,m, then Xk= ak, yij=gi3 is defined to be a primitive solution of (1). If Xk=aQk, yij=fi3 is a primitive solution of (1),
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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 MathematicsWe 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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An Introduction to Refined Neutrosophic Number Theory [PDF]
Neutrosophic Sets and Systems, 2021 Number theory is concerned with properties of integers and Diophantine equations. The objective of this paper is dedicated to introduce the basic concepts in refined neutrosophic number theory such as division, divisors, congruencies, and Pell's equation
Mohammad Abobala, Muritala Ibrahim
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A CLASS OF DIOPHANTINE EQUATIONS [PDF]
Proceedings of the National Academy of Sciences, 1959 Nach Verf. hat \[ \alpha^x+\beta^x=\alpha^n+\beta^n,\quad \alpha,\beta=\tfrac 12 (1\pm\sqrt{-7}), \] für gegebene \(n\) höchstens zwei Lösungen und für \(n=2^m\) genau die triviale Lösung \(x=2^m\).
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Generating Pythagoras Quadruples in Symbolic 2-Plithogenic Commutative Rings [PDF]
Neutrosophic Sets and Systems, 2023 This paper is dedicated to find a general algorithm for generating different solutions for Pythagoras non-linear Diophantine equation .
Yaser Ahmad Alhasan +2 more
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On the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$ [PDF]
, 2013 summary:In this study, we determine when the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$ has an infinite number of positive integer solutions $x$ and $y$ for $0\leq n\leq 10.$ Moreover, we give all positive integer solutions of the same equation for ...
Keskin, Refik +2 more
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