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Integral Solution to the Equation x2+2a.7b=yn

Diophantine equation is an equation in which solutions to it are from some predetermined classes and it is one of the oldest branches of number theory. There are many types of diophantine equations, for instance linear diophantine equation, exponential diophantine equation and others. In this resear...

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Main Author: Yow, Kai Siong
Format: Thesis
Language:English
English
Published: 2011
Online Access:http://psasir.upm.edu.my/id/eprint/20379/1/IPM_2011_15_ir.pdf
http://psasir.upm.edu.my/id/eprint/20379/
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spelling my.upm.eprints.203792014-01-13T09:31:34Z http://psasir.upm.edu.my/id/eprint/20379/ Integral Solution to the Equation x2+2a.7b=yn Yow, Kai Siong Diophantine equation is an equation in which solutions to it are from some predetermined classes and it is one of the oldest branches of number theory. There are many types of diophantine equations, for instance linear diophantine equation, exponential diophantine equation and others. In this research, we will investigate and find the integral solutions to the diophantine equation x² +2a.7b=yⁿ where a and b are positive integers and n is even. By fixing n = 2r , we determine the generators of and x and yr for 1 ≤ a ≤ 6 with any values of b. Then, we investigate the necessary conditions to obtain integral solutions of x and y under each value of af there is any. The approach is by looking at the possible combinations for the product 2a⋅7b and solving the equations simultaneously. Then, from the results obtained, we substitute the values of a followed by b to get integer values of and a and yr under each category. After that, the equations are grouped according to the pattern that emerged and a geometric progression formula is applied to create the general formulae for the generators of solutions to the equation examined. Besides that, we have to identify the range of i, the number of non-negative integral solutions associated with each b for different values of a. When b is even, we find some special cases of determining the generators of solutions for and x and yr with a certain condition. From our investigation, we find that there is no integral solution of and x and yr the diophantine equation x² +2a.7b=yⁿ, when n is even and a = 1. It is found that the number of generators to determine the integral solutions to the equation depend on the values of a. Values of y are determined by taking the r-th root of yr for certain values of r. 2011-09 Thesis NonPeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/20379/1/IPM_2011_15_ir.pdf Yow, Kai Siong (2011) Integral Solution to the Equation x2+2a.7b=yn. Masters thesis, Universiti Putra Malaysia. English
institution Universiti Putra Malaysia
building UPM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Putra Malaysia
content_source UPM Institutional Repository
url_provider http://psasir.upm.edu.my/
language English
English
description Diophantine equation is an equation in which solutions to it are from some predetermined classes and it is one of the oldest branches of number theory. There are many types of diophantine equations, for instance linear diophantine equation, exponential diophantine equation and others. In this research, we will investigate and find the integral solutions to the diophantine equation x² +2a.7b=yⁿ where a and b are positive integers and n is even. By fixing n = 2r , we determine the generators of and x and yr for 1 ≤ a ≤ 6 with any values of b. Then, we investigate the necessary conditions to obtain integral solutions of x and y under each value of af there is any. The approach is by looking at the possible combinations for the product 2a⋅7b and solving the equations simultaneously. Then, from the results obtained, we substitute the values of a followed by b to get integer values of and a and yr under each category. After that, the equations are grouped according to the pattern that emerged and a geometric progression formula is applied to create the general formulae for the generators of solutions to the equation examined. Besides that, we have to identify the range of i, the number of non-negative integral solutions associated with each b for different values of a. When b is even, we find some special cases of determining the generators of solutions for and x and yr with a certain condition. From our investigation, we find that there is no integral solution of and x and yr the diophantine equation x² +2a.7b=yⁿ, when n is even and a = 1. It is found that the number of generators to determine the integral solutions to the equation depend on the values of a. Values of y are determined by taking the r-th root of yr for certain values of r.
format Thesis
author Yow, Kai Siong
spellingShingle Yow, Kai Siong
Integral Solution to the Equation x2+2a.7b=yn
author_facet Yow, Kai Siong
author_sort Yow, Kai Siong
title Integral Solution to the Equation x2+2a.7b=yn
title_short Integral Solution to the Equation x2+2a.7b=yn
title_full Integral Solution to the Equation x2+2a.7b=yn
title_fullStr Integral Solution to the Equation x2+2a.7b=yn
title_full_unstemmed Integral Solution to the Equation x2+2a.7b=yn
title_sort integral solution to the equation x2+2a.7b=yn
publishDate 2011
url http://psasir.upm.edu.my/id/eprint/20379/1/IPM_2011_15_ir.pdf
http://psasir.upm.edu.my/id/eprint/20379/
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score 13.211869

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