On admissible integers of cubic forms

Abstract

In this paper, we are mainly concerned with the form f(X,Y,Z)=X\(^{3}\)+Y\(^{3}\)+Z\(^{3}\) and more precisely, what integers this cubic can represent and which regions of R\(^{3}\) can cover at least one solution for almost all the potential integers that could be represented by it. We show first that for a family of regions in R\(^3\) to cover almost all the admissible integers of any diagonal cubic form, the number of solutions in each region must grow faster than any linear function, and next we choose a potential family for the cubic f(X, Y, Z).