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Article: The triangular theorem of eight and representation by quadratic polynomials
Title | The triangular theorem of eight and representation by quadratic polynomials |
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Authors | |
Keywords | Quadratic forms Sums of odd square Triangular numbers |
Issue Date | 2013 |
Citation | Proceedings of the American Mathematical Society, 2013, v. 141 n. 5, p. 1473-1486 How to Cite? |
Abstract | We investigate here the representability of integers as sums of triangular numbers, where the n-th triangular number is given by Tn = n(n+1)/2. In particular, we show that f(x1, x2, ...,xk) = b1Tx1+· · ·+bkTxk, for fixed positive integers b1, b2,. . ., bk, 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. © 2012 American Mathematical Society. |
Persistent Identifier | http://hdl.handle.net/10722/192201 |
ISSN | 2023 Impact Factor: 0.8 2023 SCImago Journal Rankings: 0.837 |
DC Field | Value | Language |
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dc.contributor.author | Bosma, W | en_US |
dc.contributor.author | Kane, B | en_US |
dc.date.accessioned | 2013-10-23T09:27:19Z | - |
dc.date.available | 2013-10-23T09:27:19Z | - |
dc.date.issued | 2013 | en_US |
dc.identifier.citation | Proceedings of the American Mathematical Society, 2013, v. 141 n. 5, p. 1473-1486 | en_US |
dc.identifier.issn | 0002-9939 | en_US |
dc.identifier.uri | http://hdl.handle.net/10722/192201 | - |
dc.description.abstract | We investigate here the representability of integers as sums of triangular numbers, where the n-th triangular number is given by Tn = n(n+1)/2. In particular, we show that f(x1, x2, ...,xk) = b1Tx1+· · ·+bkTxk, for fixed positive integers b1, b2,. . ., bk, 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. © 2012 American Mathematical Society. | - |
dc.language | eng | en_US |
dc.relation.ispartof | Proceedings of the American Mathematical Society | en_US |
dc.rights | First published in [Proceedings of the American Mathematical Society] in [2013, v. 141 n. 5], published by the American Mathematical Society | - |
dc.subject | Quadratic forms | - |
dc.subject | Sums of odd square | - |
dc.subject | Triangular numbers | - |
dc.title | The triangular theorem of eight and representation by quadratic polynomials | en_US |
dc.type | Article | en_US |
dc.description.nature | published_or_final_version | - |
dc.identifier.doi | 10.1090/S0002-9939-2012-11419-4 | en_US |
dc.identifier.scopus | eid_2-s2.0-84874210917 | en_US |
dc.identifier.volume | 141 | en_US |
dc.identifier.issue | 5 | en_US |
dc.identifier.spage | 1473 | en_US |
dc.identifier.epage | 1486 | en_US |
dc.identifier.issnl | 0002-9939 | - |