Does there exist an infinite exact covering system of the integers by congruence classes with pairwise distinct moduli, minimum modulus at least 5, for which the sum of the reciprocals of the moduli equals exactly 1 (a saturated system)? Constructions are known only for minimum modulus 2, 3, and 4, and Lewis proved that any example with minimum modulus at least 5 must have infinitely many different primes dividing its moduli. Filaseta and Kalogirou, who settled the corresponding finite-version conjecture of Davenport and of Erdős–Selfridge, state that the infinite exact saturated case remains open and identify the missing tool as an infinite analogue of the distortion method, suggesting a probability space on the profinite completion of the integers.

Background

A covering system is a finite or infinite collection of congruences x≡ai(modmi)x\equiv a_i \pmod{m_i} with integer residues aia_i and positive integer moduli mi>1m_i>1 whose residue classes jointly cover every integer. The moduli are distinct (an incongruent system) when no two congruences share a modulus; the minimum modulus m0=min⁡imim_0=\min_i m_i measures how far the covering starts from trivial small moduli. A covering system is exact (also called disjoint) when every integer satisfies exactly one congruence of the system. For finite systems, the theorem of Mirsky–Newman and Davenport–Rado forces the two largest moduli of an exact covering to coincide, so no finite exact system with distinct moduli exists at all; the exact-distinct question is therefore genuinely infinite. A basic splitting construction — start from x≡0(mod2)x\equiv 0\pmod 2 and x≡1(mod2)x\equiv 1\pmod 2 and repeatedly split the class with the largest modulus into two classes of twice the modulus — produces infinite exact distinct systems whose moduli are powers of 2 and whose reciprocal sum ∑i1/mi\sum_i 1/m_i equals 1. Because for an infinite system the natural density of the covered set (the asymptotic fraction of integers covered) need not equal the sum of the reciprocals of the moduli, one singles out the saturated systems: an infinite covering system is saturated when ∑i1/mi=1\sum_i 1/m_i = 1 exactly. For exact saturated systems, Lewis showed the density of the covered set does equal the sum of reciprocals, so saturation is the natural consistency condition between an exact covering of the integers and its moduli. Stein asked in 1958 whether infinite exact distinct saturated systems exist whose moduli are not just the powers of 2 greater than 1; Krukenberg's 1971 dissertation and Beebee's 1988 examples answered this by constructing such systems with moduli built from the primes 2 and 3, achieving minimum modulus m0∈{2,3,4}m_0\in\{2,3,4\}, and Beebee posed the problem of deciding whether such systems can have m0≥5m_0\ge 5. The strongest structural constraints are due to Lewis: if only finitely many primes divide the moduli of such a system, then every modulus is of the form 2j3i2^j3^i, and any example with m0≥5m_0\ge 5 must involve infinitely many distinct prime divisors among its moduli. This infinite setting contrasts sharply with the finite regime, where Hough proved that every finite distinct covering system (exactness not required) has minimum modulus bounded by an absolute constant, later reduced to 616000 by Balister, Bollobás, Morris, Sahasrabudhe and Tiba, and where Filaseta and Kalogirou proved the 1952 conjecture of Davenport, anticipated by Erdős and Selfridge in 1973, that a finite distinct covering system with m0>4m_0>4 has its reciprocal sum bounded away from 1 by a positive constant. No minimum-modulus bound can hold without finiteness: a greedy construction with moduli 2t+1,2t+2,…2^{t+1},2^{t+2},\dots covers the integers exactly with reciprocal sum below any prescribed ε>0\varepsilon>0, which is why the saturation condition carries the content in the infinite case. Filaseta and Kalogirou, after proving their finite theorem via the distortion method of Balister, Bollobás, Morris, Sahasrabudhe and Tiba (a technique that assigns explicit finite-level probability measures and sums tail contributions), point out that settling the infinite saturated problem appears to require an infinite analogue of that method — for instance a rigorously controlled probability space on Z^\hat{\mathbb Z}, the profinite completion of the integers (the compact topological group obtained as the inverse limit of the finite residue rings Z/nZ\mathbb Z/n\mathbb Z, in which every covering by congruence classes forces the reciprocal sum of the moduli to be at least 1) — and they explicitly state the problem of deciding existence for m0≥5m_0\ge 5 as the main open problem of the area.

Problem Statement

Decide the existence of an infinite covering system of congruences x≡ai(modmi)x\equiv a_i \pmod{m_i}, i=1,2,…i=1,2,\dots, of the integers such that (i) the moduli mi>1m_i>1 are pairwise distinct, (ii) the minimum modulus m0=min⁡imim_0=\min_i m_i satisfies m0≥5m_0\ge 5, (iii) the system is exact, i.e. every integer satisfies exactly one congruence of the system, and (iv) the system is saturated, i.e. ∑i=1∞1/mi=1\sum_{i=1}^{\infty} 1/m_i = 1. Either an explicit construction of such a system, with proof of exactness and of the saturation identity, or a proof that no such system exists, resolves the problem. This is Beebee's 1988 formulation of Stein's 1958 question as stated in the closing section of Filaseta-Kalogirou; no minimum-modulus ceiling, model restriction, or bound on the primes dividing the moduli is imposed beyond conditions (i)-(iv).

References

  1. Michael Filaseta and Alexandros Kalogirou, Covering systems with the sum of the reciprocals of the moduli close to 1, Transactions of the American Mathematical Society, published online 22 January 2026, DOI: 10.1090/tran/9670; arXiv preprint arXiv:2407.15280 (2024), https://doi.org/10.1090/tran/9670
  2. Sherman K. Stein, Unions of arithmetic sequences, Mathematische Annalen 134 (1958), 289-294, DOI: 10.1007/bf01343822, https://doi.org/10.1007/bf01343822
  3. John Beebee, Examples of Infinite, Incongruent Exact Covers, The American Mathematical Monthly 95 (1988), No. 2, 121-123, DOI: 10.1080/00029890.1988.11971980, https://doi.org/10.1080/00029890.1988.11971980
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  6. Bob Hough, Solution of the minimum modulus problem for covering systems, Annals of Mathematics (2) 181 (2015), No. 1, 361-382, DOI: 10.4007/annals.2015.181.1.6, https://doi.org/10.4007/annals.2015.181.1.6
  7. János Barát and Péter P. Varjú, A contribution to infinite disjoint covering systems, Journal de Théorie des Nombres de Bordeaux 17 (2005), No. 1, 51-55, DOI: 10.5802/jtnb.476, https://doi.org/10.5802/jtnb.476
  8. Gerry Myerson, Jacky Poon, and Jamie Simpson, Incongruent restricted disjoint covering systems, Discrete Mathematics 309 (2009), No. 13, 4428-4434, DOI: 10.1016/j.disc.2009.02.002, https://doi.org/10.1016/j.disc.2009.02.002
  9. Yu Hashimoto, Finiteness of Disjoint Covering Systems with Precisely One Repeated Modulus, arXiv:2603.26043 (2026), https://arxiv.org/abs/2603.26043
  10. C. E. Krukenberg, Covering sets of integers, Ph.D. dissertation, University of Illinois at Urbana-Champaign, 1971 (published by ProQuest LLC, Ann Arbor, MI; no DOI exists for this 1971 thesis); bibliographic entry as given in the bibliography of Filaseta-Kalogirou, arXiv:2407.15280, https://arxiv.org/abs/2407.15280