Some Generalized Formula For Sums of Cube

Abstract

The study of integer representations as a sum of powers is still a very long standing problem. In this work, the study of integer representation as a sum of cube is introduced and investigated for non-zero distinct integer solution. Let \(a_1\), \(a_2\), \(a_3\), ... , \(a_n\) and d be any positive integers such that \(a_n\) - \(a_n\)-1= \(a_n\)-1 - \(a_n\)-2 = ... = \(a_2\) - \(a_1\) = d. This study formulates some general results for sums of n cube. In particular, this research introduces and develops the diophantine equation I =(\(a_1\)+\(a_2\)+\(a_3\)+...+\(a_n\)) L = \(a_1^3\)+\(a_2^3\)+\(a_3^3\)+...+\(a_n^3\) for some integer L. The method involves decomposing integer I into sums of n cube and determination of general representation of integer L using case by case basis.