The Diophantine Equation 8x+py=z2

The Diophantine Equation 8x+py=z2

Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p≡±3(mod  8), then the equation 8x+py=z2 has no positive integer solutions (x,y,z); (ii) if p≡7(mod  8), then the equation has only the solutions (p,x,y,z)=(2q-1,(1/3)(q+2),2,2q+1), where q i...

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Main Authors: Lan Qi, Xiaoxue Li
Format: Article
Language:English
Published: Hindawi Limited 2015-01-01
Series:The Scientific World Journal
Online Access:http://dx.doi.org/10.1155/2015/306590
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spelling doaj-070b4ef5934a428fb0f26490231a21d92020-11-25T02:15:34ZengHindawi LimitedThe Scientific World Journal2356-61401537-744X2015-01-01201510.1155/2015/306590306590The Diophantine Equation 8x+py=z2Lan Qi0Xiaoxue Li1College of Mathematics and Statistics, Yulin University, Yulin, Shaanxi 719000, ChinaSchool of Mathematics, Northwest University, Xi’an, Shaanxi 710127, ChinaLet p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p≡±3(mod  8), then the equation 8x+py=z2 has no positive integer solutions (x,y,z); (ii) if p≡7(mod  8), then the equation has only the solutions (p,x,y,z)=(2q-1,(1/3)(q+2),2,2q+1), where q is an odd prime with q≡1(mod  3); (iii) if p≡1(mod  8) and p≠17, then the equation has at most two positive integer solutions (x,y,z).http://dx.doi.org/10.1155/2015/306590
collection DOAJ
language English
format Article
sources DOAJ
author Lan Qi
Xiaoxue Li
spellingShingle Lan Qi
Xiaoxue Li
The Diophantine Equation 8x+py=z2
The Scientific World Journal
author_facet Lan Qi
Xiaoxue Li
author_sort Lan Qi
title The Diophantine Equation 8x+py=z2
title_short The Diophantine Equation 8x+py=z2
title_full The Diophantine Equation 8x+py=z2
title_fullStr The Diophantine Equation 8x+py=z2
title_full_unstemmed The Diophantine Equation 8x+py=z2
title_sort diophantine equation 8x+py=z2
publisher Hindawi Limited
series The Scientific World Journal
issn 2356-6140
1537-744X
publishDate 2015-01-01
description Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p≡±3(mod  8), then the equation 8x+py=z2 has no positive integer solutions (x,y,z); (ii) if p≡7(mod  8), then the equation has only the solutions (p,x,y,z)=(2q-1,(1/3)(q+2),2,2q+1), where q is an odd prime with q≡1(mod  3); (iii) if p≡1(mod  8) and p≠17, then the equation has at most two positive integer solutions (x,y,z).
url http://dx.doi.org/10.1155/2015/306590
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