Abstract
Let T(a, b, c, d; n) denote the number of representations of n as \(a\frac{x(x+1)}{2}+ b\frac{y(y+1)}{2}+c\frac{z(z+1) }{2}+d\frac{w(w+1)}{2}\), where \(a,\ b,\ c,\ d\) are positive integers and \( n,\ x,\ y,\ z,\ w\) are arbitrary non-negative integers, and let N(a, b, c, d; n) denote the number of representations of n as \(ax^2+by^2+cz^2+dw^2\), where this time x, y, z and w are integers. Recently, Sun proved relations between N(a, b, c, d; n) and T(a, b, c, d; n) and posed 23 conjectures. In this paper, we not only confirm five conjectures due to Sun, but also generalize four of them.
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The author is very grateful to the referee for his/her helpful comments.
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This work was supported by the National Science Fund for Distinguished Young Scholars of Jiangsu Province (No. BK20180044) and Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers.
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Yao, O.X.M. Generalizations of some conjectures of Sun on the relations between N(a, b, c, d; n) and T(a, b, c, d; n). Ramanujan J 48, 639–654 (2019). https://doi.org/10.1007/s11139-018-0067-y
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DOI: https://doi.org/10.1007/s11139-018-0067-y