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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The triangular theorem of eight and representation by quadratic polynomials


Authors: Wieb Bosma and Ben Kane
Journal: Proc. Amer. Math. Soc. 141 (2013), 1473-1486
MSC (2010): Primary 11E25, 11E20, 11E45
Published electronically: September 21, 2012
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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.


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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
Email: bkane@mi.uni-koeln.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11419-4
PII: S 0002-9939(2012)11419-4
Keywords: Triangular numbers, quadratic forms, sums of odd squares
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.