On conjectures of Z.-H. Sun
Undergraduate project, Sun Yat-sen University, 2022
Y. Ding, D. Gu, T. Huang, Q. Jiang and D. Ye, On conjectures of Z.-H. Sun, Results in Mathematics, 77 (2022), no.3, Paper No. 105.
Project description
In this project, we gave a general formulation for certain conjectures of Z.-H. Sun that relate representations by sums of triangular numbers to representations by sums of squares, and provided a computationally feasible kit to validate specified cases. For example, if one defines
\[T(a_{1},\ldots,a_{k};n)=\left|\left\{(x_{1},\ldots,x_{k})\in\mathbb{Z}^{k}: a_{1}\frac{x_{1}(x_1+1)}{2}+\cdots+a_{k}\frac{x_{k}(x_k+1)}{2}=n \right\}\right|\]and
\[S(a_{1},\ldots,a_{k};n)=\left|\left\{(x_{1},\ldots,x_{k})\in\mathbb{Z}^{k}: a_{1}x_{1}^{2}+\cdots+a_{k}x_{k}^{2}=n \right\}\right|,\]then upon our computations, one can tell that
\[{T(1,2,3,10;n)=\frac{4}{3}S(1,2,3,10;2n+4)}\]for any \(n\equiv1\pmod{4}\).