On Conjectures of Z.-H. Sun

In recent work, Z.-H. Sun proposes a large number of conjectural formulas on interrelations between representations by a sum of triangular numbers and representations by ternary or quaternary quadratic forms. In this note, we establish a general statement to illustrate that all these conjectural formulas can be rigorously verified with the aid of modular forms, and as a consequence, we provide a computationally straightforward simple procedure for validating these formulas.