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

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## 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:
CreatorsEmailORCIDORCID Put Code
Kane, BenUNSPECIFIEDUNSPECIFIEDUNSPECIFIED
Sun, Zhi-WeiUNSPECIFIEDUNSPECIFIEDUNSPECIFIED
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:
KeywordsLanguage
HALF-INTEGRAL WEIGHT; TERNARY QUADRATIC-FORMS; FOURIER COEFFICIENTS; MODULAR-FORMS; REPRESENTATION; INTEGERS; GENERAMultiple languages
MathematicsMultiple languages
URI: http://kups.ub.uni-koeln.de/id/eprint/49098 View Item