Main Article Content
Abstract
Let denote the number of ways of expressing as a linear combination of square numbers and be the number of ways of writing as a linear combination of triangular numbers, where being a nonnegative integer. Zhi-Hong Sun [International Journal of Number Theory, 2018] found several relations between and and posed seven open conjectures. The five conjectures on the ternary case are proved by Kim and Oh [Journal of Korean Mathematical Society, 2019] by an elementary method. Xia and Yan [International Journal of Number Theory, 2019] proved conjectures 6.1 - 6.6. In this paper, we prove the last conjecture 6.7. We also find some new relations between and .