Kaplansky's ternary quadratic form
This paper proves that if N is a nonnegative eligible integer, coprime to 7, which is not of the form x2+y2+7z2, then N is square-free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integer np2 by two quadr...
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| Format: | Article |
| Language: | English |
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Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201005294 |
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| _version_ | 1832560968939339776 |
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| author | James Kelley |
| author_facet | James Kelley |
| author_sort | James Kelley |
| collection | DOAJ |
| description | This paper proves that if N is a nonnegative eligible integer,
coprime to 7, which is not of the form x2+y2+7z2, then N is square-free. The proof is modelled on that of a similar
theorem by Ono and Soundararajan, in which relations between the
number of representations of an integer np2 by two quadratic
forms in the same genus, the pth coefficient of an L-function
of a suitable elliptic curve, and the class number formula prove
the theorem for large primes, leaving 3 cases which are easily
numerically verified. |
| format | Article |
| id | doaj-art-c577393120514349908ff4b3d6410ebf |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-c577393120514349908ff4b3d6410ebf2025-02-03T01:26:18ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252001-01-0125528929210.1155/S0161171201005294Kaplansky's ternary quadratic formJames Kelley0Department of Mathematics, University of California at Berkeley, Berkeley 94709, CA, USAThis paper proves that if N is a nonnegative eligible integer, coprime to 7, which is not of the form x2+y2+7z2, then N is square-free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integer np2 by two quadratic forms in the same genus, the pth coefficient of an L-function of a suitable elliptic curve, and the class number formula prove the theorem for large primes, leaving 3 cases which are easily numerically verified.http://dx.doi.org/10.1155/S0161171201005294 |
| spellingShingle | James Kelley Kaplansky's ternary quadratic form International Journal of Mathematics and Mathematical Sciences |
| title | Kaplansky's ternary quadratic form |
| title_full | Kaplansky's ternary quadratic form |
| title_fullStr | Kaplansky's ternary quadratic form |
| title_full_unstemmed | Kaplansky's ternary quadratic form |
| title_short | Kaplansky's ternary quadratic form |
| title_sort | kaplansky s ternary quadratic form |
| url | http://dx.doi.org/10.1155/S0161171201005294 |
| work_keys_str_mv | AT jameskelley kaplanskysternaryquadraticform |