The triangular theorem of eight and representation by quadratic polynomials
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- by Wieb Bosma and Ben Kane PDF
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Abstract:
We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n=$ $n(n+1)/2$. In particular, we show that $f(x_1, x_2, \ldots , x_k)=b_1T_{x_1}+\cdots +b_kT_{x_k}$, for fixed positive integers $b_1, b_2, \ldots , b_k$, represents every nonnegative integer if and only if it represents $1$, $2$, $4$, $5$, and $8$. Moreover, if ‘cross-terms’ are allowed in $f$, we show that no finite set of positive integers can play an analogous role, in turn showing that there is no overarching finiteness theorem which generalizes the statement from positive definite quadratic forms to totally positive quadratic polynomials.References
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Additional Information
- Wieb Bosma
- Affiliation: Radboud Universiteit, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
- Email: bosma@math.ru.nl
- Ben Kane
- Affiliation: Radboud Universiteit, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
- Address at time of publication: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
- MR Author ID: 789505
- Email: bkane@mi.uni-koeln.de
- Received by editor(s): December 6, 2010
- Received by editor(s) in revised form: August 4, 2011, and August 25, 2011
- Published electronically: September 21, 2012
- Communicated by: Kathrin Bringmann
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1473-1486
- MSC (2010): Primary 11E25, 11E20, 11E45
- DOI: https://doi.org/10.1090/S0002-9939-2012-11419-4
- MathSciNet review: 3020835