Universal mixed sums of generalized 4- and 8-gonal numbers

2018/09/11 by Ju, Jangwon, Oh, Byeong-Kweon
#11E12 #11E20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1809.03673

Abstract

An integer of the form P_m(x)= ((m-2)x²-(m-4)x)/(2) for an integer x, is called a generalized m-gonal number. For positive integers α₁,…,α_u and β₁,…,β_v, a mixed sum Φ=α₁P₄(x₁)+⋯+α_uP₄(x_u)+β₁P₈(y₁)+⋯+β_vP₈(y_v) of generalized 4- and 8-gonal numbers is called universal if Φ=N has an integer solution for any nonnegative integer N. In this article, we prove that there are exactly 1271 proper universal mixed sums of generalized 4- and 8-gonal numbers. Furthermore, the "61-theorem" is proved, which states that an arbitrary mixed sum of generalized 4- and 8-gonal numbers is universal if and only if it represents the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 30, 60, and 61.

Related

• Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions
• Almost universal ternary sums of polygonal numbers
• Almost universal mixed sums of squares and polygonal numbers
• Ternary universal sums of generalized pentagonal numbers
• Almost Universal Weighted Ternary Sums of Polygonal Numbers