Ternary universal sums of generalized polygonal numbers

Abstract

An integer of the form [Formula: see text], for some integer [Formula: see text], is called a generalized polygonal number of order [Formula: see text]. A ternary sum [Formula: see text] of generalized polygonal numbers, for some positive integers [Formula: see text] and some integers [Formula: see text], is said to be universal over [Formula: see text] if for any nonnegative integer [Formula: see text], the equation [Formula: see text] has an integer solution [Formula: see text]. In this paper, we prove the universalities of [Formula: see text] ternary sums of generalized polygonal numbers, which was conjectured by Sun.