Journal of the
Korean Mathematical Society JKMS

Let $P_8(x)=3x^2-2x$. A polynomial of the form $a_1P_8(x_1)+a_2P_8(x_2)+\cdots+a_kP_8(x_k)$ ($a_i\in \mathbb N$) is called a $k$-ary octagonal form. An octagonal form is called regular if it represents every positive integer which is locally represented. In this paper, we prove that there are at most 15 regular ternary octagonal forms and establish the regularity of 12 forms.

Keywords: Generalized $m$-gonal number, octagonal number, regular ternary octagonal form

MSC numbers: Primary 11D09, 11E20

Supported by: This work was supported by a 2-Year Research Grant of Pusan National University.