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Instytut Matematyczny, Colloquium Mathematicum, p. 1-7
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On some universal sums of generalized polygonal numbers
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Abstract
For m=3,4,... those $p_m(x)=(m-2)x(x-1)/2+x$ with $x𝟄\Z$ are called generalized $m$-gonal numbers. Recently the second author studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\Z$ (i.e., any $n𝟄\N=\{0,1,2,...\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z𝟄\Z$). In this paper we proved that $p_5+bp_5+3p_5 (b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\Z$; this partially confirms Sun's conjecture on $ap_5+bp_5+cp_5$. Sun also conjectured that any $n𝟄\N$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z𝟄\N$; in contrast we show that $p_3+p_5+p_{11}$ and $3p_3+p_5+p_7$ are universal over $\Z$. Comment: 7 pages