In this paper we first investigate for what positive integers
a,b,c every nonnegative integer
n can be written as
x(ax+1)+y(by+1)+z(cz+1) with
fbc1d4cc089861c24b469" title="Click to view the MathML source">x,y,z integers. We show that
(a,b,c) can be either of the following seven triples
and conjecture that any triple
(a,b,c) among
also has the desired property. For integers
0⩽b⩽c⩽d⩽a with
a>2, we prove that any nonnegative integer can be written as
x(ax+b)+y(ay+c)+z(az+d) with
fbc1d4cc089861c24b469" title="Click to view the MathML source">x,y,z integers, if and only if the quadruple
(a,b,c,d) is among