On primitive solutions of the Diophantine equation x2 + y2 = M
We provide explicit formulae for primitive, integral solutions to the Diophantine equation x2+y2=M{x}^{2}+{y}^{2}=M, where MM is a product of powers of Pythagorean primes, i.e., of primes of the form 4n+14n+1. It turns out that this is a nice application of the theory of Gaussian integers.
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De Gruyter
2021
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oai:doaj.org-article:9bf5dc1d11004d51977fd3000a10b3c92021-12-05T14:10:53ZOn primitive solutions of the Diophantine equation x2 + y2 = M2391-545510.1515/math-2021-0087https://doaj.org/article/9bf5dc1d11004d51977fd3000a10b3c92021-08-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0087https://doaj.org/toc/2391-5455We provide explicit formulae for primitive, integral solutions to the Diophantine equation x2+y2=M{x}^{2}+{y}^{2}=M, where MM is a product of powers of Pythagorean primes, i.e., of primes of the form 4n+14n+1. It turns out that this is a nice application of the theory of Gaussian integers.Busenhart ChrisHalbeisen LorenzHungerbühler NorbertRiesen OliverDe Gruyterarticlepythagorean primesdiophantine equation11d4511d0911a41MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 863-868 (2021) |
| institution |
DOAJ |
| collection |
DOAJ |
| language |
EN |
| topic |
pythagorean primes diophantine equation 11d45 11d09 11a41 Mathematics QA1-939 |
| spellingShingle |
pythagorean primes diophantine equation 11d45 11d09 11a41 Mathematics QA1-939 Busenhart Chris Halbeisen Lorenz Hungerbühler Norbert Riesen Oliver On primitive solutions of the Diophantine equation x2 + y2 = M |
| description |
We provide explicit formulae for primitive, integral solutions to the Diophantine equation x2+y2=M{x}^{2}+{y}^{2}=M, where MM is a product of powers of Pythagorean primes, i.e., of primes of the form 4n+14n+1. It turns out that this is a nice application of the theory of Gaussian integers. |
| format |
article |
| author |
Busenhart Chris Halbeisen Lorenz Hungerbühler Norbert Riesen Oliver |
| author_facet |
Busenhart Chris Halbeisen Lorenz Hungerbühler Norbert Riesen Oliver |
| author_sort |
Busenhart Chris |
| title |
On primitive solutions of the Diophantine equation x2 + y2 = M |
| title_short |
On primitive solutions of the Diophantine equation x2 + y2 = M |
| title_full |
On primitive solutions of the Diophantine equation x2 + y2 = M |
| title_fullStr |
On primitive solutions of the Diophantine equation x2 + y2 = M |
| title_full_unstemmed |
On primitive solutions of the Diophantine equation x2 + y2 = M |
| title_sort |
on primitive solutions of the diophantine equation x2 + y2 = m |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/9bf5dc1d11004d51977fd3000a10b3c9 |
| work_keys_str_mv |
AT busenhartchris onprimitivesolutionsofthediophantineequationx2y2m AT halbeisenlorenz onprimitivesolutionsofthediophantineequationx2y2m AT hungerbuhlernorbert onprimitivesolutionsofthediophantineequationx2y2m AT riesenoliver onprimitivesolutionsofthediophantineequationx2y2m |
| _version_ |
1718371619014967296 |