Abstract: | In 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 $$(3,0,1,2), (3,1,1,2), (3,1,2,2),
(3,1,2,3), (4,1,2,3).$$ |