For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = 0,1,2,... has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in contrast, we show that and 3p₃ + p₅ + p₇ are universal over ℤ. Our proofs are essentially elementary and hence suitable for general readers.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm6742-3-2016,
author = {Fan Ge and Zhi-Wei Sun},
title = {On some universal sums of generalized polygonal numbers},
journal = {Colloquium Mathematicae},
volume = {144},
year = {2016},
pages = {149-155},
zbl = {06602777},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6742-3-2016}
}
Fan Ge; Zhi-Wei Sun. On some universal sums of generalized polygonal numbers. Colloquium Mathematicae, Tome 144 (2016) pp. 149-155. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6742-3-2016/