Representations of positive integers as sums of arithmetic progressions, II

As mentioned in the first part of this paper, our paper was motivated by two classical papers on the representations of integers as sums of arithmetic progressions. One of them is a paper by Sir Charles Wheatstone and the other is a paper by James Joseph Sylvester. Part I of the paper, though including some extensions of Wheatstone's work, was primarily devoted to extensions of Sylvester's Theorem. In this part of the paper, we will pay more attention on the problems initiated by of Wheatstone on the representations of powers of integers as sums of arithmetic progressions and the relationships among the representations for different powers of the integer. However, a large part in this portion of the paper will be devoted to the extension of a clever method recently introduced by S. B. Junaidu, A. Laradji, and A. Umar and the problems related to the extension. This is because that this extension, not only will be our main tool for study ing the relationships of the representations of different powers of an integer, but also seems to be interesting in its own right. In the process of doing this, we need to use a few results from the first part of the paper. On the other hand, some of our results in this part will also provide certain new information on the problems studied in the first part. However, for readers who are interested primarily in the results of this part, we have repeated some basic facts from Part I of the paper so that the reader can read this part independently from the first part.