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Generalizations of some conjectures of Sun on the relations between N(abcdn) and T(abcdn)

  • Olivia X. M. Yao
Article
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Abstract

Let T(abcdn) 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(abcdn) 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(abcdn) and T(abcdn) and posed 23 conjectures. In this paper, we not only confirm five conjectures due to Sun, but also generalize four of them.

Keywords

Theta function identities Sum of squares Sum of triangular numbers 

Mathematics Subject Classification

11D85 11E25 

Notes

Acknowledgements

The author is very grateful to the referee for his/her helpful comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsJiangsu UniversityZhenjiangPeople’s Republic of China

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