On some universal sums of generalized polygonal numbers

Fan Ge, and Zhi-Wei Sun. "On some universal sums of generalized polygonal numbers." Colloquium Mathematicae 145.1 (2016): 149-155. <http://eudml.org/doc/286657>.

@article{FanGe2016,
abstract = {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 $p₃(x) + p₅(y) + p_\{11\}(z)$ and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in contrast, we show that $p₃ + p₅ + p_\{11\}$ and 3p₃ + p₅ + p₇ are universal over ℤ. Our proofs are essentially elementary and hence suitable for general readers.},
author = {Fan Ge, Zhi-Wei Sun},
journal = {Colloquium Mathematicae},
keywords = {polygonal numbers; representations of integers; ternary quadratic forms},
language = {eng},
number = {1},
pages = {149-155},
title = {On some universal sums of generalized polygonal numbers},
url = {http://eudml.org/doc/286657},
volume = {145},
year = {2016},
}

TY - JOUR
AU - Fan Ge
AU - Zhi-Wei Sun
TI - On some universal sums of generalized polygonal numbers
JO - Colloquium Mathematicae
PY - 2016
VL - 145
IS - 1
SP - 149
EP - 155
AB - 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 $p₃(x) + p₅(y) + p_{11}(z)$ and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in contrast, we show that $p₃ + p₅ + p_{11}$ and 3p₃ + p₅ + p₇ are universal over ℤ. Our proofs are essentially elementary and hence suitable for general readers.
LA - eng
KW - polygonal numbers; representations of integers; ternary quadratic forms
UR - http://eudml.org/doc/286657
ER -