2005/12/01 by Arnold Knopfmacher, Michael Mays · 2 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Advanced Combinatorial Mathematics #Mathematics #Product (mathematics) #Factorization #Modulo #Prime (order theory) #Integer (computer science) #Discrete mathematics #Combinatorics #Field (mathematics) #Pure mathematics #Computer science #Algorithm
paper · doi:10.1142/s1793042105000315
openalex publication_date 2005/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
The general field of additive number theory considers questions concerning representations of a given positive integer n as a sum of other integers. In particular, partitions treat the sums as unordered combinatorial objects, and compositions treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distinct, or relatively prime, or all congruent to ±1 modulo 5. In this paper we review work on analogous problems concerning representations of n as a product of positive integers. We survey techniques for enumerating product representations both in the unrestricted case and in the case when the factors are required to be distinct, and both when the product representations are considered as ordered objects and when they are unordered. We offer some new identities and observations for these and related counting functions and derive some new recursive algorithms to generate lists of factorizations with restrictions of various types.