Decompositions of Natural Numbers: From a Case Study in Mathematical Theory Exploration

In the context of a scheme based exploration model proposed by Bruno Buchberger, we investigate the idea of decomposition, applied in the exploration of natural numbers. The free decomposition problem (i.e. whether an element can always be decomposed with respect to an operation) can be arbitrarily difficult, and we illustrate this in the theory of natural numbers. We consider a restriction, the decomposition in domains with a well-founded partial ordering: we introduce the notions of irreducible elements, reducible elements w.r.t. a composition operation, decomposition of domain elements into irreducible ones, and also the problem of irreducible decomposition which we then solve. Natural numbers can be classified as a decomposition domain, in which we know how to solve the decomposition problem. This leads to the prime decomposition theorem.