Regular triangular forms of rank exceeding 3

Abstract

A triangular form is defined to be an integer-valued quadratic polynomial of the form $a_1{P}_3(x_1)+a_2{P}_3(x_2)+\cdots +a_k{P}_3(x_k)$ where $a_i^{\prime }s$ are positive integers and $P_3(x)=x(x+1)/2$. A triangular form is called regular if it represents all positive integers which are locally represented. In this article, we find all regular triangular forms of more than three variables.