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Article: On almost universal mixed sums of squares and triangular numbers

TitleOn almost universal mixed sums of squares and triangular numbers
Authors
Issue Date2010
Citation
Transactions of the American Mathematical Society, 2010, v. 362 n. 12, p. 6425-6455 How to Cite?
AbstractIn 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than $ 2719$ can be represented by the famous Ramanujan form $ x^2+y^2+10z^2$; equivalently the form $ 2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $ T_z$ denotes the triangular number $ z(z+1)/2$. Given positive integers $ a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ ax^2+by^2+cT_z$ represents sufficiently large integers and to establish similar results for the forms $ ax^2+bT_y+cT_z$ and $ aT_x+bT_y+cT_z$. Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form $ 2ax^2+y^2+z^2$ if and only if all prime divisors of $ a$ are congruent to 1 modulo 4. (ii) The form $ ax^2+y^2+T_z$ is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of $ a$ is congruent to 1 or 3 modulo 8. (iii) $ ax^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4. (iv) When $ v_2(a)\not=3$, the form $ aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4 and $ v_2(a)\not=5,7,\ldots$, where $ v_2(a)$ is the $ 2$-adic order of $ a$.
Persistent Identifierhttp://hdl.handle.net/10722/192194
ISSN
2015 Impact Factor: 1.196
2015 SCImago Journal Rankings: 2.168

 

DC FieldValueLanguage
dc.contributor.authorKane, Ben_US
dc.contributor.authorSun, ZWen_US
dc.date.accessioned2013-10-23T09:27:18Z-
dc.date.available2013-10-23T09:27:18Z-
dc.date.issued2010en_US
dc.identifier.citationTransactions of the American Mathematical Society, 2010, v. 362 n. 12, p. 6425-6455en_US
dc.identifier.issn0002-9947en_US
dc.identifier.urihttp://hdl.handle.net/10722/192194-
dc.description.abstractIn 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than $ 2719$ can be represented by the famous Ramanujan form $ x^2+y^2+10z^2$; equivalently the form $ 2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $ T_z$ denotes the triangular number $ z(z+1)/2$. Given positive integers $ a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ ax^2+by^2+cT_z$ represents sufficiently large integers and to establish similar results for the forms $ ax^2+bT_y+cT_z$ and $ aT_x+bT_y+cT_z$. Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form $ 2ax^2+y^2+z^2$ if and only if all prime divisors of $ a$ are congruent to 1 modulo 4. (ii) The form $ ax^2+y^2+T_z$ is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of $ a$ is congruent to 1 or 3 modulo 8. (iii) $ ax^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4. (iv) When $ v_2(a)\not=3$, the form $ aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4 and $ v_2(a)\not=5,7,\ldots$, where $ v_2(a)$ is the $ 2$-adic order of $ a$.-
dc.languageengen_US
dc.relation.ispartofTransactions of the American Mathematical Societyen_US
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.rightsFirst published in [Transactions of the American Mathematical Society] in [2010, v. 362 n. 12], published by the American Mathematical Society-
dc.titleOn almost universal mixed sums of squares and triangular numbersen_US
dc.typeArticleen_US
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.1090/S0002-9947-2010-05290-0en_US
dc.identifier.scopuseid_2-s2.0-78650279340en_US
dc.identifier.volume362en_US
dc.identifier.issue12en_US
dc.identifier.spage6425en_US
dc.identifier.epage6455en_US

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