Tight universal triangular forms

Abstract

For a subset S of nonnegative integers and a vector a " pa 1 , . . . , a k q of positive integers, let V 1 S paq " ta 1 s 1 `¨¨¨`a k s k : s i P Suzt0u. For a positive integer n, let T pnq be the set of integers greater than or equal to n. In this paper, we consider the problem of finding all vectors a satisfying V 1 S paq " T pnq, when S is the set of (generalized) m-gonal numbers and n is a positive integer. In particular, we completely resolve the case when S is the set of triangular numbers.