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On x(ax + 1)+y(by + 1)+z(cz + 1) and x(ax + b)+y(ay + c)+z(az + d)
文摘
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 469" 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 6b613ec861a57b4a876bec1248663" title="Click to view the MathML source">a>2, we prove that any nonnegative integer can be written as x(ax+b)+y(ay+c)+z(az+d) with 469" title="Click to view the MathML source">x,y,z integers, if and only if the quadruple (a,b,c,d) is among
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