The number of representations by a ternary sum of triangular numbers
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \in \mathbb Z^3$ of $a \frac{x(x-1)}{2} + b \frac{y(y-1)}{2} + c \frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some relations between $t(a,b,c;n)$ and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in \cite{s}.
Abstract
For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \in \mathbb Z^3$ of $a \frac{x(x-1)}{2} + b \frac{y(y-1)}{2} + c \frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some relations between $t(a,b,c;n)$ and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in \cite{s}.
참고문헌 (8)
- C. Adiga, "A general relation between sums of squares and sums of triangular numbers", Int. J. Number Theory, 1(2), pp.175~182 (2005)
- N. D. Baruah, "Sums of squares and sums of triangular numbers induced by partitions of 8", Int. J. Number Theory, 4(4), pp.525~538 (2008)
- Y. Kitaoka, "Arithmetic of Quadratic Forms" (1993), Cambridge University Press
- O. T. O'Meara, "Introduction to Quadratic Forms, Classics in Mathematics" (2000), Springer-Verlag
- Z. H. Sun, "Some relations between t(a, b, c, d; n) and N(a, b, c, d; n)", Acta Arith, 175, pp.269~289 (2016)
- Z. H. Sun, "Ramanujan's theta functions and sums of triangular numbers", preprint, ~
- T. Yang, "An explicit formula for local densities of quadratic forms", J. Number Theory, 72(2), pp.309~356 (1998)
- X. M. Yao, "The relation between N(a, b, c, d; n) and (p, k) and pp; kq-parametr-ization of theta functions", J. Math. Anal. Appl, 453(1), pp.125~143 (2017)