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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |