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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 x,y,z integers. We show that (a,b,c) can be either of the following seven 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 the desired property...
Generalized octagonal numbers are those p8(x)=x(3x−2) with x∈Z. In this paper we show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's theorem on sums of four squares. Moreover, for 35 triples (b,c,d) with 1⩽b⩽c⩽d (including (2,3,4) and (2,4,8)), we prove that any nonnegative integer can be expressed...
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