Good Integers: A Structural Review With Applications to Polynomial Factorization and Algebraic Coding Theory

For nonzero coprime integers a and b, a positive integer l is said to be good with respect to a and b if there exists a positive integer k such that l divides ak+bk. Since the early 1990s, the notion of good integers has attracted considerable attention from researchers. This continued interest stems from both their elegant number-theoretic structure and their noteworthy applications across several branches in mathematics, with coding theory being among the most prominent areas where they play a crucial role. This paper provides a comprehensive review of good integers, emphasizing both their theoretical foundations and their practical implications. We first revisit the fundamental number-theoretic properties of good integers and present their characterizations in a systematic manner. The exposition is enriched with well-structured algorithms and illustrative diagrams that facilitate their computation and classification. Subsequently, we explore applications of good integers in the study of algebraic coding theory. In particular, special emphasis is placed on their roles in the characterization, construction, and enumeration of self-dual cyclic codes as well as complementary dual cyclic codes. Several examples are provided to demonstrate the applicability of the theory. This review not only consolidates existing results but also highlights the unifying role of good integers in bridging number theory and coding theory.