Odd Covering Problem in Number Theory

• The odd covering problem is the inquiry into whether there exists a covering system of the integers with all moduli odd, distinct, and greater than 1.
• Sieve-theoretic methods and square-free constraints illustrate that a covering system with only odd, square-free moduli cannot exist, imposing strict arithmetic limitations.
• Recent advances using repeated-modulus variants and tree diagram constructions provide explicit bounds and transfer principles that progress understanding of the all-odd covering challenge.

The odd covering problem is the question, posed by Erdős, of whether there exists a covering system

$\{\,x\equiv a_i \pmod{n_i}\}_{i=1}^k$

in which all moduli $n_i$ are odd, distinct, and $>1$ (Harrington et al., 2021). A covering system is a finite collection of congruences whose union is $\mathbb Z$. The problem remains open in this classical form, but several neighboring regimes are now sharply delineated: the square-free case is ruled out, repeated-modulus variants admit explicit constructions and transfer principles, and sieve-theoretic obstructions constrain any hypothetical all-odd distinct covering (Balister et al., 2019).

1. Classical formulation and basic parameters

A covering system of the integers is a finite collection of congruences

$\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$

such that every integer satisfies at least one of them (Harrington et al., 2021). Erdős’s odd covering problem asks whether there exists a covering system in which all moduli are odd, distinct, and $>1$ (Harrington et al., 2021).

A standard variation fixes an odd prime $p$ and asks for the smallest nonnegative integer $t_p$ for which there is a covering system whose moduli are odd, $>1$, all distinct except that $p$ appears exactly $n_i$0 times (Harrington et al., 2021). The square-free analogue imposes the additional requirement that each modulus be square-free and denotes the corresponding repetition parameter by $n_i$1 (Harrington et al., 2021).

A later generalization replaces the repeated prime by an arbitrary odd integer $n_i$2. A $n_i$3-covering system is a covering system

$n_i$4

in which each $n_i$5 is an odd integer $n_i$6, distinct from one another and from $n_i$7, and the modulus $n_i$8 appears exactly $n_i$9 times; one then defines

$>1$0

(Bispels et al., 22 Jul 2025).

The repeated-modulus parameters are important because they convert the original existence question into a quantitative problem. In particular, the 2021 paper shows that if there is a square-free covering system in which all moduli are odd, square-free, distinct except that $>1$1 appears exactly twice, then in fact there is a full odd covering, meaning that all moduli are odd and distinct (Harrington et al., 2021).

2. Square-free impossibility and sieve-theoretic obstructions

The strongest unconditional negative result currently available for the arithmetic odd covering problem is the square-free theorem: in any finite covering system of $>1$2 by arithmetic progressions with distinct square-free moduli, at least one of the moduli is even (Balister et al., 2019). Equivalently, there is no covering system whose moduli are simultaneously distinct, square-free, and all odd (Balister et al., 2019).

The square-free analysis admits a geometric reformulation. Let $>1$3 be the first $>1$4 primes, and set

$>1$5

where $>1$6. A hyperplane in $>1$7 is a product set $>1$8 in which each $>1$9 is either a singleton or the full set $\mathbb Z$0. The square-free odd covering problem becomes a covering-by-hyperplanes problem with a non-parallelness condition coming from distinct square-free odd divisors (Balister et al., 2019).

The proof strategy in the square-free case builds on the probabilistic sieve of Hough and its refinements. One exposes coordinates one by one, maintains a probability measure $\mathbb Z$1 on the partial grid $\mathbb Z$2, and controls the mass $\mathbb Z$3 of newly removed points through first and second moments of a removal fraction $\mathbb Z$4 (Balister et al., 2019). Theorem 3.1 of that work gives a termination criterion in terms of $\mathbb Z$5 and $\mathbb Z$6, while Theorem 3.2 supplies explicit combinatorial bounds on those moments (Balister et al., 2019).

Without the square-free assumption, the original odd covering problem remains open, but several structural obstructions are known. Hough and Nielsen showed that every covering system must include at least one modulus divisible by $\mathbb Z$7 or by $\mathbb Z$8, and Hopper proved that every covering system of the integers has a modulus divisible by a prime number less than or equal to $\mathbb Z$9 (Hopper, 2017). A related sieve result states that if $\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$0 is a covering system with distinct moduli $\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$1 and $\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$2, then at least one of the following holds: $\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$3, $\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$4, or $\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$5 (Balister et al., 2018). These statements do not settle the odd covering problem, but they substantially narrow the arithmetic shape of any potential counterexample.

3. Repeated-modulus variants and quantitative bounds

The repeated-modulus program asks how many repetitions of one odd modulus are sufficient when all other moduli remain distinct, odd, and $\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$6. The bounds below collect the values recorded in 2021 together with the later prime-case improvement obtained in 2025 (Harrington et al., 2021, Bispels et al., 22 Jul 2025).

Parameter	Meaning	Bounds
$\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$7	prime $\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$8, odd moduli, all distinct except $\{\,x\equiv a_i\pmod{n_i}\}_{i=1}^k$9 repeats	$>1$0, $>1$1, $>1$2, $>1$3, $>1$4 for $>1$5; later, for every prime $>1$6, there exists a covering system in which $>1$7 appears exactly $>1$8 times
$>1$9	square-free version of $p$0	$p$1, $p$2, $p$3, and for all primes $p$4, $p$5
$p$6	odd integer $p$7, exactly one repeated odd modulus $p$8	$p$9, $t_p$0, $t_p$1, $t_p$2

For the square-free problem, a key transfer theorem states: fix an odd prime $t_p$3. If there is a covering system in which all moduli are odd, square-free, distinct except that $t_p$4 appears exactly twice, then in fact there is a full odd covering (Harrington et al., 2021). The stated consequence is that if $t_p$5 for any $t_p$6, then an odd covering exists (Harrington et al., 2021).

For the non-square-free odd-moduli problem, the 2021 paper proves that there is a covering system with moduli odd, distinct except that $t_p$7 appears exactly four times, hence $t_p$8; there is a covering system with all moduli odd, distinct except that $t_p$9 appears exactly seven times, hence $>1$0; and for every prime $>1$1 there is a covering system with moduli odd, distinct except that $>1$2 appears exactly $>1$3 times, hence $>1$4 (Harrington et al., 2021). The 2025 paper strengthens the last statement to every prime $>1$5, and further specifies that the construction avoids using modulus $>1$6 (Bispels et al., 22 Jul 2025).

These bounds do not produce a distinct-moduli all-odd covering, but they show that “almost odd” systems with one repeated modulus are abundant. A plausible implication is that progress on the exact values of $>1$7 and $>1$8 remains one of the most concrete routes toward the classical problem.

4. Tree diagrams, CRT notation, and transfer mechanisms

A major methodological feature of the modern literature is the use of tree diagrams. In the 2021 framework, a tree diagram is a rooted tree whose nodes alternate primes and sub-partitions; each prime-labeled node is called a $>1$9-node if it splits its parent subset into $p$0 child congruence classes, and leaves record final single congruences (Harrington et al., 2021). The same paper also uses compact CRT notation: $p$1 which denotes the single congruence $p$2 equivalent by CRT to the system

$p$3

when $p$4 are pairwise coprime (Harrington et al., 2021).

The core new method is a systematic use of condensed tree diagrams. One starts with a small seed covering system $p$5 whose tree diagram has a single $p$6-node at the root with $p$7 child branches, each branch leading to a subtree $p$8 that individually covers $p$9. By carefully replacing the root $n_i$00-node by a higher-power branch $n_i$01 and reattaching copies of $n_i$02 under those higher-power nodes, one builds a large covering with fewer repetitions of the base prime (Harrington et al., 2021).

This mechanism underlies the square-free “double modulus $n_i$03 odd covering” theorem. After arranging that both $n_i$04 and $n_i$05 occur, one partitions the rest into sub-coverings on residue classes $n_i$06, introduces a new odd prime $n_i$07, replaces the single $n_i$08-node at the root by the power branch $n_i$09, and then attaches $n_i$10-node branches covering large $n_i$11-adic valuations by congruences $n_i$12 (Harrington et al., 2021).

A second transfer principle is the lifting lemma. If for some prime $n_i$13 one has a covering system $n_i$14 in which $n_i$15 appears exactly $n_i$16 times and no modulus is divisible by $n_i$17, then for every larger prime $n_i$18 one can produce an analogous system in which $n_i$19 appears exactly $n_i$20 times and no modulus is divisible by $n_i$21; moreover, oddness or square-freeness is preserved (Harrington et al., 2021).

The 2025 paper adds a complementary splitting lemma. If $n_i$22 is a $n_i$23-covering system and $n_i$24, then there exists a covering system in which the modulus $n_i$25 is used at most

$n_i$26

times; if $n_i$27, this improves to

$n_i$28

If $n_i$29 never used modulus $n_i$30, each bound may be reduced by $n_i$31 (Bispels et al., 22 Jul 2025). The key identity is

$n_i$32

combined with the option of retaining one copy of $n_i$33 (Bispels et al., 22 Jul 2025).

5. Explicit constructions and arithmetic consequences

The papers supply explicit constructions rather than solely existential arguments. In the square-free setting, one theorem gives a square-free covering system in which $n_i$34 appears exactly six times and no other modulus repeats, hence $n_i$35 (Harrington et al., 2021). In condensed notation, the root is

$n_i$36

with six branches; beneath these, the construction wedges progressively by $n_i$37, then $n_i$38, then $n_i$39, and continues up to the prime $n_i$40 (Harrington et al., 2021).

For composite repeated moduli, the 2025 paper gives direct tree-diagram constructions proving $n_i$41, $n_i$42, $n_i$43, and $n_i$44 (Bispels et al., 22 Jul 2025). In the case $n_i$45, the notation

$n_i$46

denotes the three congruences $n_i$47, $n_i$48, and $n_i$49, and the construction extends two of these branches through higher odd prime moduli in successive layers until all residue classes are exhausted (Bispels et al., 22 Jul 2025).

The later paper also records an application to special integer sets. Let

$n_i$50

It states that every sum of two squares, every sum of two cubes, every powerful number, every prime power, every derangement number, every Fermat number, and every perfect number lies in $n_i$51. By Corollary 3.2 of that paper, any subset of $n_i$52 having only finitely many points outside $n_i$53 admits an odd covering; hence the union of these seven classical families has an odd covering (Bispels et al., 22 Jul 2025).

These constructions illustrate a recurring pattern: repeated-modulus coverings are often built by finite tree expansions in increasing primes, with the depth chosen so that the leaves exhaust the required residue classes. This suggests that explicit finite diagrams, rather than purely asymptotic arguments, remain central to the subject.

6. Open problems and current research directions

The central open question remains Erdős’s original one: does an odd covering with all moduli odd, distinct, and $n_i$54 exist? (Harrington et al., 2021). The square-free case is settled negatively, but the unrestricted case is still unresolved (Balister et al., 2019).

A second family of open problems concerns exact repetition counts. For fixed odd $n_i$55, one asks for the exact value of $n_i$56; the 2021 paper records

$n_i$57

and the 2025 paper extends the $n_i$58 construction to all primes $n_i$59 (Harrington et al., 2021, Bispels et al., 22 Jul 2025). In the square-free setting, the exact values of $n_i$60 are likewise unknown.

The 2021 paper also formulates an asymptotic question: does there exist $n_i$61 so that for all sufficiently large primes $n_i$62 one has

$n_i$63

It adds that an odd covering would imply $n_i$64, and that any positive $n_i$65 would represent progress (Harrington et al., 2021).

The 2025 paper identifies several concrete next goals: show $n_i$66; find systems in which two moduli may repeat but all others remain distinct; and refine the splitting method so as to reduce counts such as $n_i$67 toward something closer to $n_i$68 (Bispels et al., 22 Jul 2025). It also notes that showing $n_i$69 would imply an odd covering of sums of three cubes or of the Bell numbers (Bispels et al., 22 Jul 2025).

Taken together, these results place the odd covering problem in a distinctive position. The square-free obstruction is definitive, the repeated-modulus theory is increasingly explicit, and the remaining gap lies precisely at the interface between exact distinctness and the constructional flexibility provided by prime powers and repeated odd moduli.