On a few Diophantine equations, in particular, Fermat's last theorem
This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat's last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit ot...
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| Format: | Article |
| Language: | English |
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Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203210668 |
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| _version_ | 1850174784754679808 |
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| author | C. Levesque |
| author_facet | C. Levesque |
| author_sort | C. Levesque |
| collection | DOAJ |
| description | This is a survey on Diophantine equations, with the purpose being to
give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat's last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solved à la Wiles. We will exhibit many families of Thue equations, for which Baker's linear forms in logarithms and the knowledge of the unit groups of certain families of number fields prove useful for finding all the integral solutions. One of the most difficult conjecture in number theory, namely, the ABC conjecture, will also be described. We will conclude by explaining in elementary terms the notion of modularity of an elliptic curve. |
| format | Article |
| id | doaj-art-d76c2ebd8d564a3f8aa43303f8869b8a |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2003-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-d76c2ebd8d564a3f8aa43303f8869b8a2025-08-20T02:19:34ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252003-01-012003714473450010.1155/S0161171203210668On a few Diophantine equations, in particular, Fermat's last theoremC. Levesque0Département de Mathématiques et de Statistique, Université Laval, Québec G1K 7P4, CanadaThis is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat's last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solved à la Wiles. We will exhibit many families of Thue equations, for which Baker's linear forms in logarithms and the knowledge of the unit groups of certain families of number fields prove useful for finding all the integral solutions. One of the most difficult conjecture in number theory, namely, the ABC conjecture, will also be described. We will conclude by explaining in elementary terms the notion of modularity of an elliptic curve.http://dx.doi.org/10.1155/S0161171203210668 |
| spellingShingle | C. Levesque On a few Diophantine equations, in particular, Fermat's last theorem International Journal of Mathematics and Mathematical Sciences |
| title | On a few Diophantine equations, in particular, Fermat's last theorem |
| title_full | On a few Diophantine equations, in particular, Fermat's last theorem |
| title_fullStr | On a few Diophantine equations, in particular, Fermat's last theorem |
| title_full_unstemmed | On a few Diophantine equations, in particular, Fermat's last theorem |
| title_short | On a few Diophantine equations, in particular, Fermat's last theorem |
| title_sort | on a few diophantine equations in particular fermat s last theorem |
| url | http://dx.doi.org/10.1155/S0161171203210668 |
| work_keys_str_mv | AT clevesque onafewdiophantineequationsinparticularfermatslasttheorem |