Abstract
Following Bhargava and Hanke’s celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to 3. In particular, if a positive-definite quadratic form represents all positive integers coprime to 3 and \(\le \)290, then it represents all positive integers coprime to 3. We use similar methods to those used by Rouse to prove (assuming GRH) that a positive-definite quadratic form representing every odd integer between 1 and 451 represents all positive odd integers.
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Acknowledgements
The authors used the computer software package Magma [4] version 2.21-1 extensively for the computations. Magma scripts and log files from the computations performed are available at http://users.wfu.edu/rouseja/CCXC/. This work represents the master’s thesis of the first author completed at Wake Forest University in the spring of 2015.
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Appendix: Table of quadratic forms with given truants
Appendix: Table of quadratic forms with given truants
Form | Truant |
|---|---|
\(2x^{2}\) | 1 |
\(x^{2}\) | 2 |
\(x^{2} + 2y^{2}\) | 5 |
\(x^{2} + y^{2}\) | 7 |
\(x^{2} -xy + y^{2} + 2z^{2} + 2zw + 4w^{2}\) | 10 |
\(x^{2} + y^{2} + 5z^{2}\) | 11 |
\(x^{2} + 2y^{2} - xz + 2yz + 5z^{2} - xw - 2zw + 4w^{2}\) | 13 |
\(x^{2} + y^{2} + 2z^{2} - yw + 3w^{2}\) | 14 |
\(x^{2} + xy + 2y^{2} - xz - yz + 4z^{2} + zw + 5w^{2}\) | 17 |
\(x^{2} + xy + 2y^{2} - xz - yz + 3z^{2} - xw - 2zw + 6w^{2}\) | 19 |
\(x^{2} + y^{2} - xz + yz + 7z^{2} - 2zw + 10w^{2}\) | 22 |
\(x^{2} + 2y^{2} + yz + 5z^{2} - xw + 7w^{2}\) | 23 |
\(x^{2} + 2y^{2} + 3z^{2} - 2yw + 3zw + 9w^{2}\) | 26 |
\(x^{2} + 2y^{2} - xz - yz + 4z^{2}\) | 29 |
\(x^{2} + xy + 2y^{2} - xz + yz + 5z^{2} + 2zw + 10w^{2}\) | 31 |
\(x^{2} + xy + 2y^{2} - xz + yz + 3z^{2} + 17w^{2}\) | 34 |
\(x^{2} + 2y^{2} + 5z^{2} - xw - 2yw + zw + 10w^{2}\) | 35 |
\(x^{2} + 2y^{2} - xz + yz + 5z^{2} - xw - 2zw + 12w^{2}\) | 37 |
\(x^{2} + y^{2} - xz + 5z^{2} - xw - yw - 3zw + 11w^{2}\) | 38 |
\(x^{2} + y^{2} - xz + yz + 5z^{2} - xw + 5zw + 11w^{2}\) | 46 |
\(x^{2} + y^{2} + yz + 6z^{2} + 9w^{2}\) | 47 |
\(x^{2} + y^{2} + 7z^{2} - xw + 7zw + 8w^{2}\) | 55 |
\(x^{2} + 2y^{2} + 3z^{2} - xw - yw + 3zw + 8w^{2}\) | 58 |
\(x^{2} + xy + 2y^{2} + 5z^{2} - wz + 5zw + 6w^{2}\) | 62 |
\(x^{2} + y^{2} + yz + 2z^{2} - xw + 23w^{2}\) | 70 |
\(x^{2} + y^{2} - xz + yz + 5z^{2} - xw - yw + 12w^{2}\) | 94 |
\(x^{2} + y^{2} + yz + 3z^{2} + 22w^{2}\) | 110 |
\(x^{2} + y^{2} + 7z^{2} + 7zw + 14w^{2} + 35v^{2}\) | 119 |
\(x^{2} + 2y^{2} - xz - yz + 4z^{2} + 29w^{2}\) | 145 |
\(x^{2} + 2y^{2} - xz - yz + 4z^{2} + 29w^{2} + 29zv + 58wv + 145v^{2}\) | 203 |
\(x^{2} + 2y^{2} - xz - yz + 4z^{2} + 29w^{2} + 29zv + 87wv + 145v^{2}\) | 290 |
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DeBenedetto, J., Rouse, J. Quadratic forms representing all integers coprime to 3. Ramanujan J 46, 431–446 (2018). https://doi.org/10.1007/s11139-016-9883-0
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DOI: https://doi.org/10.1007/s11139-016-9883-0