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28 March 2024
 
  » arxiv » 1505.3679

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On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$
Zhi-Wei Sun ;
Date 14 May 2015
AbstractIn this paper we first investigate for what positive integers $a,b,c$ every nonnegative integer $n$ can be represented as $x(ax+1)+y(by+1)+z(cz+1)$ with $x,y,z$ integers. We show that $(a,b,c)$ can be any of the following six triples: $$(1,2,3), (1,2,4), (1,2,5), (2,2,4), (2,2,5), (2,3,3), (2,3,4),$$ and conjecture that any triple $(a,b,c)$ among $$(2,2,6), (2,3,5), (2,3,7), (2,3,8), (2,3,9), (2,3,10)$$ also has that property. For integers $0le ble cle dle a$ with $a>2$, we prove that any nonnegative integer can be represented as $x(ax+b)+y(ay+c)+z(az+d)$ with $x,y,z$ integers, if and only if the quadruple $(a,b,c,d)$ is among $$(3,0,1,2), (3,1,1,2), (3,1,2,2), (3,1,2,3), (4,1,2,3).$$
Source arXiv, 1505.3679
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