Good Integers: Theory and Applications

• Good integers are positive numbers that, for coprime integers a and b, satisfy the condition l divides a^k+b^k for some k, forming the basis of their study.
• The theory utilizes multiplicative orders, 2-adic valuations, and prime-power factorizations to classify classical, oddly-good, evenly-good, and (T,k)-good integers.
• Applications in coding theory include controlling cyclotomic classes and self-dual structures in cyclic and negacyclic codes, leading to explicit enumeration formulas.

Searching arXiv for recent and foundational papers on “good integers” to ground the article in the literature. Good integers are positive integers characterized by a divisibility condition in exponential sums. Fix nonzero coprime integers $a$ and $b$. A positive integer $\ell$ is good with respect to $(a,b)$ if there exists $k\ge 1$ such that $\ell\mid a^{k}+b^{k}$. Since Moree’s 1997 introduction of the notion, the subject has developed into a number-theoretic theory organized by multiplicative orders, $2$-adic valuations, and prime-power factorization, and it now plays a substantial role in algebraic coding theory (Jitman, 27 Aug 2025). A later extension introduces $(T,k)$-good integers, defined by divisibility in the subsequence $a^{ks+T}+b^{ks+T}$, and recovers the classical, oddly-good, and evenly-good cases as specializations (Jitman et al., 27 May 2026).

1. Definition and foundational properties

For fixed coprime nonzero integers $a,b$, the classical set of good integers is

$b$0

The basic edge case is $b$1, which is always good. If $b$2, then $b$3; thus every prime divisor of a good integer is coprime to both $b$4 and $b$5 (Jitman, 27 Aug 2025).

Parity enters immediately. If $b$6 is even, then every good integer is odd; no even $b$7 is good. If $b$8 is odd, then $b$9 is good, and more generally

$\ell$0

The literature also distinguishes two subclasses. A positive integer $\ell$1 is oddly-good if $\ell$2 for some odd $\ell$3, and evenly-good if $\ell$4 for some even $\ell$5. One has

$\ell$6

and for $\ell$7, $\ell$8 cannot be both oddly-good and evenly-good (Jitman, 2016).

These definitions are stable under sign normalization. Replacing $\ell$9 by $(a,b)$0 does not change the set of divisors of $(a,b)$1, so membership questions may be treated with $(a,b)$2 without loss for the divisibility problem in the coprime setting (Jitman et al., 27 May 2026).

2. Order-theoretic characterization

The decisive local invariant is the multiplicative order. For an odd prime $(a,b)$3 and $(a,b)$4, the existence of $(a,b)$5 with $(a,b)$6 is equivalent to

$(a,b)$7

equivalently to the statement that $(a,b)$8 is even. Order lifting at odd prime powers has the form

$(a,b)$9

so the $k\ge 1$0-adic valuation of the order is already determined at the prime level (Jitman, 27 Aug 2025).

For odd integers, the global criterion is Moree’s characterization. Let $k\ge 1$1 be odd and define $k\ge 1$2 modulo the relevant odd primes. Then

$k\ge 1$3

Thus the exact power of $k\ge 1$4 dividing $k\ge 1$5 must be the same positive integer for all odd primes dividing $k\ge 1$6 (1804.01916).

Writing $k\ge 1$7 with $k\ge 1$8 odd yields the complete classification. If $k\ge 1$9 is odd, then $\ell\mid a^{k}+b^{k}$0 iff one of the following holds: $\ell\mid a^{k}+b^{k}$1 If $\ell\mid a^{k}+b^{k}$2 is even, then $\ell\mid a^{k}+b^{k}$3 iff $\ell\mid a^{k}+b^{k}$4 and either $\ell\mid a^{k}+b^{k}$5, or $\ell\mid a^{k}+b^{k}$6 and there exists $\ell\mid a^{k}+b^{k}$7 such that $\ell\mid a^{k}+b^{k}$8 for every prime $\ell\mid a^{k}+b^{k}$9 (1804.01916).

This formulation separates the theory into two independent components: the odd part is governed by a uniform $2$0-adic order condition, while the $2$1-power part is governed by the explicit divisibility $2$2.

3. Oddly-good, evenly-good, and generalized subclasses

The subclasses $2$3 and $2$4 refine the order criterion. For odd $2$5,

$2$6

whereas

$2$7

Thus oddly-good integers correspond exactly to the case of one factor of $2$8 in each relevant order, while evenly-good integers require at least two (Prugsapitak et al., 2018).

A further generalization fixes $2$9. A positive integer $(T,k)$0 is $(T,k)$1-good if $(T,k)$2; similarly one defines $(T,k)$3-oddly-good and $(T,k)$4-evenly-good. If $(T,k)$5 are odd and $(T,k)$6, then

$(T,k)$7

Let $(T,k)$8. Then $(T,k)$9 if $a^{ks+T}+b^{ks+T}$0, and

$a^{ks+T}+b^{ks+T}$1

For $a^{ks+T}+b^{ks+T}$2, one has

$a^{ks+T}+b^{ks+T}$3

so the higher $a^{ks+T}+b^{ks+T}$4-power levels are entirely oddly-good (Prugsapitak et al., 2018).

The $a^{ks+T}+b^{ks+T}$5-framework extends the theory again. For fixed integers $a^{ks+T}+b^{ks+T}$6 and $a^{ks+T}+b^{ks+T}$7, a positive integer $a^{ks+T}+b^{ks+T}$8 is $a^{ks+T}+b^{ks+T}$9-good with respect to $a,b$0 if

$a,b$1

The corresponding set is denoted $a,b$2. The classical families are recovered by

$a,b$3

At an odd prime power $a,b$4 with $a,b$5,

$a,b$6

For odd $a,b$7, the global criterion requires a common value $a,b$8 of $a,b$9 across all primes $b$00, together with explicit congruence and divisibility conditions on $b$01. In the even case, assuming $b$02 are odd, one has $b$03 for all $b$04, and for $b$05,

$b$06

(Jitman et al., 27 May 2026).

4. Corrections, algorithms, and the non-coprime completion

The arithmetic theory underwent a significant correction in 2018. Two implications used in earlier papers were shown to be false. First,

$b$07

is false; a counterexample is $b$08 modulo $b$09, for which $b$10 but $b$11. Second,

$b$12

is false for odd composite $b$13 that are not prime powers; a counterexample is $b$14 modulo $b$15, since $b$16 but $b$17. The corrected criteria replace these false implications by explicit parity and order conditions prime-by-prime (1804.01916).

Algorithmic decision procedures follow directly from the corrected structure. In the classical coprime case, one writes $b$18, rejects immediately if $b$19 or $b$20, factors the odd part $b$21, computes $b$22 for each prime $b$23, and then applies the case distinction for $b$24 versus $b$25. In the $b$26-setting, an explicit algorithm decides whether a given integer $b$27 is $b$28-good with respect to $b$29 and, when it is, computes an exponent $b$30 such that

$b$31

the method combines prime-power local tests, CRT aggregation, and the parity check at $b$32 (Jitman et al., 27 May 2026).

A later completion treats the non-coprime case $b$33. Write

$b$34

and for $b$35 decompose

$b$36

Define

$b$37

Then, for $b$38, the following are equivalent: $b$39; there exists $b$40 with $b$41; and $b$42. Thus the non-coprime problem reduces to a coprime “core” plus a $b$43-adic threshold coming from $b$44 (Jitman, 17 Oct 2025).

This extension also identifies the full set of admissible exponents. If $b$45, then

$b$46

where $b$47 and $b$48 when $b$49. Hence all admissible exponents form a single arithmetic progression truncated below by the threshold $b$50 (Jitman, 17 Oct 2025).

5. Applications in coding theory

Coding-theoretic applications are a principal reason for the sustained study of good integers. In abelian coding theory, good integers control when cyclotomic classes are self-paired under reciprocal symmetries. For a finite abelian group $b$51, a $b$52-cyclotomic class $b$53 is type I iff $b$54, and in the Hermitian setting a $b$55-cyclotomic class is type I′ iff $b$56. These identifications lead to formulas for the average Euclidean and Hermitian hull dimensions of abelian codes, and to explicit bounds in terms of the counts of type I and type I′ classes (Jitman, 2016).

Generalized $b$57-good integers govern self-dual negacyclic codes. Let

$b$58

with $b$59, $b$60, $b$61 odd, and $b$62. Over $b$63, the reciprocity pattern of irreducible factors of $b$64 is controlled by $b$65. A Euclidean self-dual negacyclic code of length $b$66 over $b$67 exists iff $b$68 and

$b$69

Under this condition, the number of such codes is

$b$70

and otherwise $b$71. The Hermitian analogue over $b$72 satisfies the same existence condition and has enumeration

$b$73

with value $b$74 when $b$75 (Prugsapitak et al., 2018).

The $b$76-generalization extends these ideas from Euclidean and Hermitian duality to Galois duality. Specializing to $b$77, the divisors of

$b$78

control the reciprocal structure of $b$79 over $b$80. For a $b$81-cyclotomic class $b$82, the corresponding irreducible factor satisfies

$b$83

so $b$84 is $b$85-self-reciprocal iff $b$86. The arithmetic bridge is that a class is of type $b$87 iff the relevant order belongs to $b$88. This yields a description and enumeration of Galois LCD cyclic codes, with the number of $b$89-LCD cyclic codes equal to

$b$90

and it yields a characterization of Galois self-dual cyclic codes. Existence of a $b$91-self-dual cyclic code is equivalent to $b$92 and $b$93 being even (Jitman et al., 27 May 2026).

Across these applications, the same arithmetic data recur: multiplicative orders modulo prime powers, uniform $b$94-adic valuations, and the distinction between self-paired and paired cyclotomic factors.

6. Terminological scope and adjacent usages

The phrase “good integers” is not unique across current arXiv literature. In ergodic theory, a strictly increasing sequence $b$95 is called good if for every real $b$96 the limit

$b$97

exists. By the Riesz representation theorem, this is equivalent to the existence of an asymptotic distribution modulo $b$98, and by the spectral theorem it is equivalent to $b$99-convergence of ergodic averages along $\ell$00 in every probability measure preserving system (Lesigne et al., 2022). This notion concerns sequences of times, not divisors of $\ell$01.

A related dynamical paper studies collections of sequences that are “good for liminf-$\ell$02-recurrence,” meaning that for every measure-preserving system and every set $\ell$03 of positive measure,

$\ell$04

There, “good” refers to recurrence and characteristic factors, again rather than the arithmetic divisibility problem (Li et al., 2020).

In pseudorandom number generation, the phrase appears in a third sense: “good integers” can denote multipliers $\ell$05 that have good performance with respect to the spectral test for congruential generators. The criterion there is a lattice figure of merit, not the existence of $\ell$06 with $\ell$07 (Steele et al., 2020).

For number theory and coding theory, however, the standard meaning remains the classical one: positive integers divisible by some term of the sequence $\ell$08, together with the refined families $\ell$09, $\ell$10, $\ell$11-good, and $\ell$12-good. Within that framework, the subject now comprises a corrected coprime theory, an explicit non-coprime completion, algorithmic tests, and a wide array of applications to Euclidean, Hermitian, and Galois dualities in algebraic coding theory (Jitman, 17 Oct 2025).