Kane, Ben and Sun, Zhi-Wei (2010). ON ALMOST UNIVERSAL MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS. Trans. Am. Math. Soc., 362 (12). S. 6425 - 6456. PROVIDENCE: AMER MATHEMATICAL SOC. ISSN 0002-9947
Full text not available from this repository.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 (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 equal 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 equal 5, 7,..., where v(2)(a) is the 2-adic order of a.
Item Type: | Journal Article | ||||||||||||
Creators: |
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URN: | urn:nbn:de:hbz:38-490984 | ||||||||||||
DOI: | 10.1090/S0002-9947-2010-05290-0 | ||||||||||||
Journal or Publication Title: | Trans. Am. Math. Soc. | ||||||||||||
Volume: | 362 | ||||||||||||
Number: | 12 | ||||||||||||
Page Range: | S. 6425 - 6456 | ||||||||||||
Date: | 2010 | ||||||||||||
Publisher: | AMER MATHEMATICAL SOC | ||||||||||||
Place of Publication: | PROVIDENCE | ||||||||||||
ISSN: | 0002-9947 | ||||||||||||
Language: | English | ||||||||||||
Faculty: | Unspecified | ||||||||||||
Divisions: | Unspecified | ||||||||||||
Subjects: | no entry | ||||||||||||
Uncontrolled Keywords: |
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URI: | http://kups.ub.uni-koeln.de/id/eprint/49098 |
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