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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| Language: | English |
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Wiley
2015-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2015/306590 |
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| author | Lan Qi Xiaoxue Li |
| author_facet | Lan Qi Xiaoxue Li |
| author_sort | Lan Qi |
| collection | DOAJ |
| 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). |
| format | Article |
| id | doaj-art-070b4ef5934a428fb0f26490231a21d9 |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-070b4ef5934a428fb0f26490231a21d92025-02-03T07:23:55ZengWileyThe 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 |
| spellingShingle | Lan Qi Xiaoxue Li The Diophantine Equation 8x+py=z2 The Scientific World Journal |
| title | 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_short | The Diophantine Equation 8x+py=z2 |
| title_sort | diophantine equation 8x py z2 |
| url | http://dx.doi.org/10.1155/2015/306590 |
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