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Article: The triangular theorem of eight and representation by quadratic polynomials

TitleThe triangular theorem of eight and representation by quadratic polynomials
Authors
KeywordsQuadratic forms
Sums of odd square
Triangular numbers
Issue Date2013
Citation
Proceedings of the American Mathematical Society, 2013, v. 141 n. 5, p. 1473-1486 How to Cite?
AbstractWe 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 Identifierhttp://hdl.handle.net/10722/192201
ISSN
2015 Impact Factor: 0.7
2015 SCImago Journal Rankings: 1.082

 

DC FieldValueLanguage
dc.contributor.authorBosma, Wen_US
dc.contributor.authorKane, Ben_US
dc.date.accessioned2013-10-23T09:27:19Z-
dc.date.available2013-10-23T09:27:19Z-
dc.date.issued2013en_US
dc.identifier.citationProceedings of the American Mathematical Society, 2013, v. 141 n. 5, p. 1473-1486en_US
dc.identifier.issn0002-9939en_US
dc.identifier.urihttp://hdl.handle.net/10722/192201-
dc.description.abstractWe 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.languageengen_US
dc.relation.ispartofProceedings of the American Mathematical Societyen_US
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.rightsFirst published in [Proceedings of the American Mathematical Society] in [2013, v. 141 n. 5], published by the American Mathematical Society-
dc.subjectQuadratic forms-
dc.subjectSums of odd square-
dc.subjectTriangular numbers-
dc.titleThe triangular theorem of eight and representation by quadratic polynomialsen_US
dc.typeArticleen_US
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.1090/S0002-9939-2012-11419-4en_US
dc.identifier.scopuseid_2-s2.0-84874210917en_US
dc.identifier.volume141en_US
dc.identifier.issue5en_US
dc.identifier.spage1473en_US
dc.identifier.epage1486en_US

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