2-Adic zeros of diagonal forms and distance pebbling of graphs

The purpose of this article is threefold. Our first two goals are to prove two theorems, one in the field of 2-adic solubility of forms and another in the field of graph pebbling. Our third purpose is to show that while these two theorems may be in different areas, their proofs are essentially the same. Thus we will show that there is an interesting connection between some number theoretical statements and statements about graph pebbling. To make this analogy, we define what could reasonably be called the distance γ pebbling number of a graph G. It will transpire that an interesting statement about 2-adic zeros of an additive form of degree d is analogous to a statement about the γ-pebbling numbers of a directed cycle graph with d vertices.