Ternary universal sums of generalized polygonal numbers

2016/12/04 by Ju, Jangwon, Oh, Byeong-Kweon, Seo, Bangnam
#11E12 #11E20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1612.01157

Abstract

An integer of the form p_m(x)= ((m-2)x²-(m-4)x)/(2) (m≥ 3), for some integer x is called a generalized polygonal number of order m. A ternary sum Φ_i,j,k^a,b,c(x,y,z)=ap_i+2(x)+bp_j+2(y)+cp_k+2(z) of generalized polygonal numbers, for some positive integers a,b,c and some integers 1≤ i≤ j ≤ k, is said to be universal over ℤ if the equation Φ_i,j,k^a,b,c(x,y,z)=n has an integer solution x,y,z for any nonnegative integer n. In this article, we prove the universalities of 17 ternary sums of generalized polygonal numbers, which was conjectured by Sun.

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