Links for fulltext
(May Require Subscription)
- Publisher Website: 10.1090/S0002-9947-2010-05290-0
- Scopus: eid_2-s2.0-78650279340
- WOS: WOS:000282653100012
- Find via
Supplementary
- Citations:
- Appears in Collections:
Article: On almost universal mixed sums of squares and triangular numbers
Title | On almost universal mixed sums of squares and triangular numbers |
---|---|
Authors | |
Keywords | Half-integral weight modular forms Quadratic forms Representations of integers Sums of squares Triangular numbers |
Issue Date | 2010 |
Citation | Transactions of the American Mathematical Society, 2010, v. 362 n. 12, p. 6425-6455 How to Cite? |
Abstract | In 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 Identifier | http://hdl.handle.net/10722/192194 |
ISSN | 2023 Impact Factor: 1.2 2023 SCImago Journal Rankings: 1.581 |
ISI Accession Number ID |
DC Field | Value | Language |
---|---|---|
dc.contributor.author | Kane, B | en_US |
dc.contributor.author | Sun, ZW | en_US |
dc.date.accessioned | 2013-10-23T09:27:18Z | - |
dc.date.available | 2013-10-23T09:27:18Z | - |
dc.date.issued | 2010 | en_US |
dc.identifier.citation | Transactions of the American Mathematical Society, 2010, v. 362 n. 12, p. 6425-6455 | en_US |
dc.identifier.issn | 0002-9947 | en_US |
dc.identifier.uri | http://hdl.handle.net/10722/192194 | - |
dc.description.abstract | In 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.language | eng | en_US |
dc.relation.ispartof | Transactions of the American Mathematical Society | en_US |
dc.rights | First published in [Transactions of the American Mathematical Society] in [2010, v. 362 n. 12], published by the American Mathematical Society | - |
dc.subject | Half-integral weight modular forms | - |
dc.subject | Quadratic forms | - |
dc.subject | Representations of integers | - |
dc.subject | Sums of squares | - |
dc.subject | Triangular numbers | - |
dc.title | On almost universal mixed sums of squares and triangular numbers | en_US |
dc.type | Article | en_US |
dc.description.nature | published_or_final_version | - |
dc.identifier.doi | 10.1090/S0002-9947-2010-05290-0 | en_US |
dc.identifier.scopus | eid_2-s2.0-78650279340 | en_US |
dc.identifier.volume | 362 | en_US |
dc.identifier.issue | 12 | en_US |
dc.identifier.spage | 6425 | en_US |
dc.identifier.epage | 6455 | en_US |
dc.identifier.isi | WOS:000282653100012 | - |
dc.identifier.issnl | 0002-9947 | - |