Intersecting semi-disks and the synergy of three quadratic forms
In this paper, we study the Diophantine equation x2 = n2 + mn + np + 2mp with m, n, p, and x being natural numbers. This equation arises from a geometry problem and it leads to representations of primes by each of the three quadratic forms: a2 + b2, a2 + 2b2, and 2a2 − b2. We show that there are inf...
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| Format: | Article |
| Language: | English |
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Sciendo
2019-06-01
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| Series: | Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica |
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| Online Access: | https://doi.org/10.2478/auom-2019-0016 |
| _version_ | 1818307452028846080 |
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| author | Ionaşcu Andrew D. |
| author_facet | Ionaşcu Andrew D. |
| author_sort | Ionaşcu Andrew D. |
| collection | DOAJ |
| description | In this paper, we study the Diophantine equation x2 = n2 + mn + np + 2mp with m, n, p, and x being natural numbers. This equation arises from a geometry problem and it leads to representations of primes by each of the three quadratic forms: a2 + b2, a2 + 2b2, and 2a2 − b2. We show that there are infinitely many solutions and conjecture that there are always solutions if x ≥ 5 and x ≠ 7; and, we find a parametrization of the solutions in terms of four integer variables. |
| first_indexed | 2024-12-13T06:58:36Z |
| format | Article |
| id | doaj.art-ebcc78504c8d49bb9898135af12e4b26 |
| institution | Directory Open Access Journal |
| issn | 1844-0835 |
| language | English |
| last_indexed | 2024-12-13T06:58:36Z |
| publishDate | 2019-06-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica |
| spelling | doaj.art-ebcc78504c8d49bb9898135af12e4b262022-12-21T23:56:00ZengSciendoAnalele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica1844-08352019-06-0127251410.2478/auom-2019-0016Intersecting semi-disks and the synergy of three quadratic formsIonaşcu Andrew D.0High School Junior, Columbus High School, Columbus, GA 31907In this paper, we study the Diophantine equation x2 = n2 + mn + np + 2mp with m, n, p, and x being natural numbers. This equation arises from a geometry problem and it leads to representations of primes by each of the three quadratic forms: a2 + b2, a2 + 2b2, and 2a2 − b2. We show that there are infinitely many solutions and conjecture that there are always solutions if x ≥ 5 and x ≠ 7; and, we find a parametrization of the solutions in terms of four integer variables.https://doi.org/10.2478/auom-2019-0016diophantine equationprime numbersquadratic formsprimary 52c07secondary 05a15, 68r05, 51k05 |
| spellingShingle | Ionaşcu Andrew D. Intersecting semi-disks and the synergy of three quadratic forms Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica diophantine equation prime numbers quadratic forms primary 52c07 secondary 05a15, 68r05, 51k05 |
| title | Intersecting semi-disks and the synergy of three quadratic forms |
| title_full | Intersecting semi-disks and the synergy of three quadratic forms |
| title_fullStr | Intersecting semi-disks and the synergy of three quadratic forms |
| title_full_unstemmed | Intersecting semi-disks and the synergy of three quadratic forms |
| title_short | Intersecting semi-disks and the synergy of three quadratic forms |
| title_sort | intersecting semi disks and the synergy of three quadratic forms |
| topic | diophantine equation prime numbers quadratic forms primary 52c07 secondary 05a15, 68r05, 51k05 |
| url | https://doi.org/10.2478/auom-2019-0016 |
| work_keys_str_mv | AT ionascuandrewd intersectingsemidisksandthesynergyofthreequadraticforms |