On Parametric and Matrix Solutions to the Diophantine Equation x2+dy2−z2=0 Where d Is a Positive Square‐Free Integer
The well‐known matrix‐generated tree structure for Pythagorean triplets is extended to the primitive solutions of the Diophantine equation x2+dy2−z2=0 where d is a positive square‐free integer. The proof is based on a parametrization of these solutions as well as on a dual version of the Fermat’s me...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2023-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2023/1505337 |
| _version_ | 1797668011666046976 |
|---|---|
| author | James D. Shaw James Guyker |
| author_facet | James D. Shaw James Guyker |
| author_sort | James D. Shaw |
| collection | DOAJ |
| description | The well‐known matrix‐generated tree structure for Pythagorean triplets is extended to the primitive solutions of the Diophantine equation x2+dy2−z2=0 where d is a positive square‐free integer. The proof is based on a parametrization of these solutions as well as on a dual version of the Fermat’s method of descent. |
| first_indexed | 2024-03-11T20:22:51Z |
| format | Article |
| id | doaj.art-f3d3dea40882409b919d04be6e998072 |
| institution | Directory Open Access Journal |
| issn | 1687-0425 |
| language | English |
| last_indexed | 2024-03-11T20:22:51Z |
| publishDate | 2023-01-01 |
| publisher | Hindawi Limited |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj.art-f3d3dea40882409b919d04be6e9980722023-10-03T00:00:04ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences1687-04252023-01-01202310.1155/2023/1505337On Parametric and Matrix Solutions to the Diophantine Equation x2+dy2−z2=0 Where d Is a Positive Square‐Free IntegerJames D. Shaw0James Guyker1Department of MathematicsDepartment of MathematicsThe well‐known matrix‐generated tree structure for Pythagorean triplets is extended to the primitive solutions of the Diophantine equation x2+dy2−z2=0 where d is a positive square‐free integer. The proof is based on a parametrization of these solutions as well as on a dual version of the Fermat’s method of descent.http://dx.doi.org/10.1155/2023/1505337 |
| spellingShingle | James D. Shaw James Guyker On Parametric and Matrix Solutions to the Diophantine Equation x2+dy2−z2=0 Where d Is a Positive Square‐Free Integer International Journal of Mathematics and Mathematical Sciences |
| title | On Parametric and Matrix Solutions to the Diophantine Equation x2+dy2−z2=0 Where d Is a Positive Square‐Free Integer |
| title_full | On Parametric and Matrix Solutions to the Diophantine Equation x2+dy2−z2=0 Where d Is a Positive Square‐Free Integer |
| title_fullStr | On Parametric and Matrix Solutions to the Diophantine Equation x2+dy2−z2=0 Where d Is a Positive Square‐Free Integer |
| title_full_unstemmed | On Parametric and Matrix Solutions to the Diophantine Equation x2+dy2−z2=0 Where d Is a Positive Square‐Free Integer |
| title_short | On Parametric and Matrix Solutions to the Diophantine Equation x2+dy2−z2=0 Where d Is a Positive Square‐Free Integer |
| title_sort | on parametric and matrix solutions to the diophantine equation x2 dy2 z2 0 where d is a positive square free integer |
| url | http://dx.doi.org/10.1155/2023/1505337 |
| work_keys_str_mv | AT jamesdshaw onparametricandmatrixsolutionstothediophantineequationx2dy2z20wheredisapositivesquarefreeinteger AT jamesguyker onparametricandmatrixsolutionstothediophantineequationx2dy2z20wheredisapositivesquarefreeinteger |