Affiliation:
1. Department of Mathematics, University of Ulsan, Ulsan 44610, Republic of Korea
2. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
Abstract
An integer of the form [Formula: see text] for an integer [Formula: see text] is called a generalized [Formula: see text]-gonal number. For positive integers [Formula: see text] and [Formula: see text], a mixed sum [Formula: see text] of generalized [Formula: see text]- and [Formula: see text]-gonal numbers is called universal if [Formula: see text] has an integer solution for every nonnegative integer [Formula: see text]. In this paper, we prove that there are exactly 1271 proper universal mixed sums of generalized [Formula: see text]- and [Formula: see text]-gonal numbers. Furthermore, the “[Formula: see text]-theorem” is proved, which states that an arbitrary mixed sum of generalized [Formula: see text]- and [Formula: see text]-gonal numbers is universal if and only if it represents the integers [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text].
Funder
the National Research Foundation of Korea
Publisher
World Scientific Pub Co Pte Lt
Subject
Algebra and Number Theory
Cited by
3 articles.
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