Good Integers: A Concise Completion of the Non-Coprime Case
Somphong Jitman
TL;DR
This paper extends the classical theory of good integers from the coprime setting to the non-coprime case by introducing a clean decomposition: write $A=ga$, $B=gb$ with $\gcd(a,b)=1$ and $g=\gcd(A,B)$. For a modulus $L$, decompose $L=\lambda_{\mathcal P(g)}(L)\ell$ and define $\gamma(L)=\max_{p|g}\lceil \nu_p(L)/\nu_p(g)\rceil$; a modulus $L$ is good with respect to $(A,B)$ iff the coprime core $\ell$ is good with respect to $(a,b)$ after a threshold $\kappa\ge\gamma(L)$. The set of admissible exponents for such $L$ forms a single arithmetic progression $\{K: K\equiv r \pmod{L_0}, K \ge \gamma(L)\}$, where $L_0$ and $r$ are determined by $\ell$ (with $L_0={\rm ord}_{\ell}(ab^{-1})$ for $\ell\ge3$, and $r=0$ or $L_0/2$ accordingly). The authors provide a complete, self-contained algorithm to decide $L\in G_{(A,B)}$, compute the minimal admissible exponent, and enumerate all admissible exponents, together with illustrative examples and several special cases. This unifies the non-coprime case with the classical coprime results and offers a practical tool for applications, such as coding theory, where divisors of $A^k+B^k$ are central.
Abstract
For coprime nonzero integers $a$ and $b$, a positive integer $\ell$ is said to be {\em good} with respect to $a$ and $b$ if there exists a positive integer $k$ such that $\ell |(a^{k}+b^{k})$. Since the early 1990s, such classical good integers have been studied intensively for their number theoretic structures and for applications, notably in coding theory. This work completes the study by relaxing the coprimality hypothesis and treating the non-coprime case $\gcd(a,b)\neq1$ in a concise and self-contained way. The results are presented in terms of the classical coprime criterion and $p$-adic valuations of $\ell$. As a consequence, whenever $\ell$ is good, all admissible exponents form a single arithmetic progression with an explicit starting point and period. Some special cases are discussed in the non-coprime setting. A practical decision procedure is developed that decides the goodness of a given integer and explicitly enumerates the full set of admissible exponents. Several illustrative examples are presented.