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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}.
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참고문헌 (8)
참고문헌 신청
C. Adiga / 2005 / A general relation between sums of squares and sums of triangular numbers / Int. J. Number Theory 1(2):175~182
N. D. Baruah / 2008 / Sums of squares and sums of triangular numbers induced by partitions of 8 / Int. J. Number Theory 4(4):525~538
Y. Kitaoka / 1993 / Arithmetic of Quadratic Forms / Cambridge University Press
O. T. O'Meara / 2000 / Introduction to Quadratic Forms, Classics in Mathematics / Springer-Verlag
Z. H. Sun / 2016 / Some relations between t(a, b, c, d; n) and N(a, b, c, d; n) / Acta Arith 175:269~289
N. D. Baruah / 2008 / Sums of squares and sums of triangular numbers induced by partitions of 8 / Int. J. Number Theory 4(4):525~538
Y. Kitaoka / 1993 / Arithmetic of Quadratic Forms / Cambridge University Press
O. T. O'Meara / 2000 / Introduction to Quadratic Forms, Classics in Mathematics / Springer-Verlag
Z. H. Sun / 2016 / Some relations between t(a, b, c, d; n) and N(a, b, c, d; n) / Acta Arith 175:269~289
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