Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic Structures
Let N,Z, and Q be the sets of natural, integers, and rational numbers, respectively. Our objective, involving a predetermined positive integer a, is to study a characterization of Diophantine equations of the form a+y2=z2. Building on this result, we aim to obtain a characterization for Pythagorean...
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Wiley
2025-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/ijmm/5516311 |
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| author | Roberto Amato |
| author_facet | Roberto Amato |
| author_sort | Roberto Amato |
| collection | DOAJ |
| description | Let N,Z, and Q be the sets of natural, integers, and rational numbers, respectively. Our objective, involving a predetermined positive integer a, is to study a characterization of Diophantine equations of the form a+y2=z2. Building on this result, we aim to obtain a characterization for Pythagorean n-tuples. Furthermore, we seek to prove the existence of a commutative infinite monoid in the set of Diophantine equations a+y2=z2 with elements in N. Additionally, we intend to establish a commutative infinite monoid with elements in N or Z on the set of Pythagorean quadruples. Moreover, in the set of Pythagorean quadruples, we aim to find a commutative infinite group with elements in Q or Z. To achieve these results, we prove the existence of some suitable binary operations. |
| format | Article |
| id | doaj-art-0332e05069fb449cb451697f35b97abc |
| institution | OA Journals |
| issn | 1687-0425 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-0332e05069fb449cb451697f35b97abc2025-08-20T02:01:00ZengWileyInternational Journal of Mathematics and Mathematical Sciences1687-04252025-01-01202510.1155/ijmm/5516311Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic StructuresRoberto Amato0Department of EngineeringLet N,Z, and Q be the sets of natural, integers, and rational numbers, respectively. Our objective, involving a predetermined positive integer a, is to study a characterization of Diophantine equations of the form a+y2=z2. Building on this result, we aim to obtain a characterization for Pythagorean n-tuples. Furthermore, we seek to prove the existence of a commutative infinite monoid in the set of Diophantine equations a+y2=z2 with elements in N. Additionally, we intend to establish a commutative infinite monoid with elements in N or Z on the set of Pythagorean quadruples. Moreover, in the set of Pythagorean quadruples, we aim to find a commutative infinite group with elements in Q or Z. To achieve these results, we prove the existence of some suitable binary operations.http://dx.doi.org/10.1155/ijmm/5516311 |
| spellingShingle | Roberto Amato Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic Structures International Journal of Mathematics and Mathematical Sciences |
| title | Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic Structures |
| title_full | Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic Structures |
| title_fullStr | Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic Structures |
| title_full_unstemmed | Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic Structures |
| title_short | Characterization of Diophantine Equations a+y2=z2, Pythagorean n-Tuples, and Algebraic Structures |
| title_sort | characterization of diophantine equations a y2 z2 pythagorean n tuples and algebraic structures |
| url | http://dx.doi.org/10.1155/ijmm/5516311 |
| work_keys_str_mv | AT robertoamato characterizationofdiophantineequationsay2z2pythagoreanntuplesandalgebraicstructures |