Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Covering Systems
adenwalla_2025_question_erdos_graham_covering_systems/: Shows no integer has its divisors greater than one forming a distinct covering system in which any two overlapping congruences have coprime moduli.
baillie_et_al_1981_problem_sierpinski_concerning_k_2n_1/: Reports a computer search over odd k below 78557 for the least Sierpiński number, leaving 118 candidates whose terms k 2^n + 1 were composite for every tested n.
balister_2018_erdos_covering_problem_density_uncovered_set/: Gives weighted bounds for uncovered density, proves Schinzel's conjecture, and supplies counterexamples to bounds based only on reciprocal sums.
balister_2021_erdos_selfridge_problem_square_free_moduli/: Complete published square-free and small-prime-square-free obstructions, with full sieve proofs and exact finite certificates.
banks_et_al_2014_sierpinski_carmichael_numbers/: Proves that for almost every odd k no term 2^n k + 1 is a Carmichael number, and relates Sierpiński and Riesel numbers to Carmichael and Lehmer numbers.
berger_1986_necessary_condition_odd_covering_systems/: The first geometric odd-covering obstruction, its nilpotent-group extension, and complete comparisons with earlier necessary conditions.
berger_1987_necessary_condition_odd_covering_systems_ii/: A forest correction to the geometric covering bound, yielding a historical six-prime necessary condition with complete proofs.
bispels_2025_further_investigation_covering_systems_odd_moduli/: Gives repeated odd-modulus covering constructions for composite moduli and primes, and derives odd coverings for several arithmetic subsets of the integers.
cambie_2025_proving_it_is_impossible_erdos_problem/: Argues that the minimum uncovered density in Erdos problem 278 has no general closed formula, reducing it to hard knapsack instances.
chen_2000_integers_form_k2n_1/: Proves that the odd integers M for which every M 2^n + 1 has at least three distinct prime factors have positive lower density, using (2,1)-primitive covering systems.
chen_2003_integers_forms_kr_2n_kr2n_1/: Proves that for fixed odd r the odd k for which every k^r - 2^n, or every k^r 2^n + 1, has at least two distinct prime factors contain an infinite arithmetic progression.
chen_2005_disjoint_arithmetic_progressions/: Complete historical unrestricted upper bound for disjoint progressions, with bounded-exponent counts, prime selection and high-power reduction.
cochrane_1996_covering_congruences_higher_dimensions/: Constructs a primitive homogeneous cover of the integer plane from an explicit twenty-class composite covering system.
crootiii_2003_non_intersecting_arithmetic_progressions/: Croot's upper and constructive lower bounds for disjoint progressions, with the published source.
de_la_breteche_2013_non_intersecting_arithmetic_progressions/: The 2013 counting bounds, their complete original proof chain and conditional endpoint, with a separate unresolved sunflower corollary.
erdos_1968_problem_p_erdos_s_stein/: The complete original density-zero proof, its quantitative lower construction, and the distinct limitation of the pairwise-gcd method.
erdos_odlyzko_1979_density_odd_integers_form_p_1_2_n_related_questions/: Proves that the odd k for which k 2^n + 1 is prime for at least one n have positive lower density, with an effective constant, and extends this to several prime bases.
filaseta_2000_irreducibility_theorem/: Gives explicit effective versions of Schinzel's irreducibility theorem for lacunary polynomials, linking reducibility to a distribution problem on residues.
filaseta_2001_coverings_integers_schinzel_irreducibility/: Relates uniform reducibility of lacunary polynomials to odd coverings and constructs a covering with controlled repetition of odd moduli.
filaseta_2008_powers_associated_sierpinski_numbers_riesel/: Shows that for every r there are infinitely many odd k with k, k^2, ..., k^r all Sierpinski numbers, and gives a 24-digit number that is both Sierpinski and Riesel, smaller than earlier examples.
filaseta_2024_covering_systems_sum_reciprocals_moduli_close/: Proves that a finite distinct covering system with minimum modulus above four has reciprocal modulus sum bounded away from one.
fornal_2026_large_gcd_disjoint_residue_classes/: Fornal and Sun obtain a nearly linear pairwise gcd in any disjoint residue family and a structural consequence for extremal families.
graham_1964_fibonacci_like_sequence_composite_numbers/: Exhibits two relatively prime 34-digit integers whose Fibonacci-like sequence is claimed to have no prime term, by choosing them so that a covering set of eighteen primes divides every term.
granville_pappalardi_2026_two_dimensional_covering_systems/: Uses coverings of the integer lattice to classify finite-prime obstructions for prime values of differences of two powers.
guo_sun_2005_odd_covering_systems_distinct_moduli/: Proves that a distinct odd covering with square-free least common multiple would require at least twenty-two different prime divisors.
harrington_2015_two_questions_covering_systems/: Constructs a distinct-modulus three-cover, handles a repeated-modulus odd covering variant, and develops primitive multiple coverings with an application to b-Sierpiński numbers.
harrington_sun_wong_2022_covering_systems_odd_moduli/: Studies covering systems with odd moduli when one odd prime may repeat, proving square-free reduction and explicit multiplicity bounds for 7, 11, and large primes.
ho_2026_non_intersecting_arithmetic_progressions_spread_cores/: Ho proves the sharp logarithmic scales for the number of disjoint progressions with bounded moduli and their reciprocal sums above a cutoff.
hough_2015_solution_minimum_modulus_problem_covering_systems/: Compiles Hough's local-lemma proof of an absolute minimum-modulus bound and the certified published bound of ten to the sixteenth.
hough_2019_covering_systems_restricted_divisibility/: Proves every distinct covering system of congruences has a modulus divisible by 2 or by 3, progress on the Erdos-Selfridge odd covering problem.
ismailescu_2014_new_kind_fibonacci_like_sequence_composite/: Constructs a coprime Fibonacci-like sequence of composites using algebraic factoring on odd indices and a partial covering system on even ones.
izotov_1995_note_sierpinski_numbers/: Produces infinitely many Sierpinski numbers of a new kind, fourth powers whose compositeness rests partly on an algebraic factorization.
klein_2023_jth_smallest_modulus_covering_system/: Bounds the j-th modulus of every minimal distinct covering system through a bounded-multiplicity distortion argument.
margolis_2019_herzog_schonheim_conjecture_small_groups/: Verifies the Herzog-Schonheim conjecture for every group of order below 1440 and shows G-harmonic tuples of length at most four are Z-harmonic.
mcnew_2026_densities_covering_numbers_abundant_numbers/: McNew and Setty's covering-number theory and density-existence proof, with explicit counterexamples to two unrestricted bounds in arXiv v2.
mian_2026_kernel_checked_exclusions_odd_covering/: Mian and Siddique's finite exclusion through period 10000, with the complete capacity argument, an exact arithmetic replay and qualified public Lean evidence.
nielsen_2009_covering_system_smallest_modulus_40/: Nielsen's prime-tree construction claiming a distinct covering system with least modulus 40, with the exact local reconstruction boundary recorded.
obryant_2006_sun_disjoint_congruence_classes/: Proves Sun's large-common-divisor conjecture for at most twenty disjoint congruence classes and develops a finite search for minimal counterexamples.
owens_2014_covering_system_minimum_modulus_42/: Owens's 2014 master's thesis construction claiming a distinct covering system with least modulus 42, with the exact local reconstruction boundary.
park_2024_proof_kahn_kalai_conjecture/: Proves the Kahn--Kalai expectation-threshold conjecture by iterating minimum fragments and covering, at low expected cost, the edges whose minimum fragments stay large.
porubsky_1978_translated_geometric_progressions_covering_systems/: Connects translated geometric progressions with finite covering systems and gives prime-adic and size bounds for irredundant coverings.
schinzel_nd_reducibility_polynomials_covering_systems_congruences/: Shows the irreducibility of x^n + f(x) for all n in an arithmetic progression is equivalent to a structural condition on finite covering systems of congruences.
simpson_1985_regular_coverings_integers_arithmetic_progressions/: Proves Znam's conjecture that any minimal covering of the integers by arithmetic progressions has at least f(P)+1 progressions.
simpson_1997_crittenden_vanden_eynden_coverings/: Gives necessary conditions and an explicit size bound for counterexamples to the Crittenden–Vanden Eynden conjecture with a lower modulus cutoff.
simpson_zeilberger_1991_squarefree_distinct_covering_systems/: Gives prime replacement, a sieve lower bound, and a Bell-number formula that rule out odd square-free distinct covering systems using at most 17 primes.
sun_1995_covering_integers_arithmetic_sequences/: Characterizes real arithmetic-sequence m-covers through finite consecutive blocks and root-of-unity identities, with exact-cover consequences.
sun_1996_covering_integers_arithmetic_sequences_ii/: Develops Egyptian-fraction and fractional-part consequences for exact and nonexact m-covers of the integers.
sun_1999_covering_multiplicity/: Relates the multiplicity of a congruence covering to repeated fractional subset sums, and gives constraints on the moduli of minimal multiple covers.
sun_2004_herzog_schonheim_conjecture_uniform_covers/: Proves the Herzog-Schonheim conjecture for uniform coset covers when the subgroups are subnormal, with bounds on repeated indices.
sun_2004_range_covering_function/: Shows that divisibility-maximal moduli constrain the residue classes that can contain the range of a covering function.
sun_2005_introduction_papers_covers/: An author-written 2005 survey that states a generating theorem for weighted covers and routes readers to Sun's earlier covering papers.
sun_2007_covering_numbers/: Gives sufficient conditions for an integer to be a covering number and answers affirmatively a 1980 question of Erdos on forced divisor moduli.
sun_fang_2008_density_integers_form_p_1_2_n_arithmetic_progressions/: Proves that in an arithmetic progression of odd integers the members k with k 2^n + 1 prime for some n have positive lower density, unless a coprimality test modulo the odd part of the modulus fails, when they have density zero and the progression comes from a covering system.
zribi_2026_conditional_sharp_estimate_erdos_problem_1190/: Shows, assuming the sharp asymptotic for the maximum number of disjoint residue classes with distinct moduli at most N, that the supremum of reciprocal moduli sums for disjoint classes with distinct moduli above m is exp of minus (1+o(1)) times the square root of log m log log m.
This folder holds sources whose primary subject is Covering Systems.
Sources with other primary subjects
Explicit links to this subject's problems support these cross-references.
- erdos_1957_unsolved_problems
- akman_sissokho_2025_steiner_coset_partitions_groups
- akman_sissokho_2025_transversal_coset_partitions_groups
- akman_sissokho_2026_steiner_coset_partitions_five_mutually_commuting_subgroups
- akman_sissokho_2026_steiner_coset_partitions_groups_erratum
- berger_et_al_1986_herzog_schonheim_conjecture_finite_nilpotent_groups
- berger_et_al_1987_remark_multiplicity_partition_group_into_cosets
- garonzi_margolis_2025_herzog_schonheim_conjecture_simple_symmetric_groups
- ginosar_schnabel_2011_prime_factorization_conditions_multiplicities_coset_partitions_groups
- itabe_2026_herzog_schonheim_conjecture_coset_partitions_at_most_seventeen_cells
- korec_znam_1977_disjoint_covering_groups_cosets
- lettl_sun_2008_covers_abelian_groups_cosets
- menon_2026_two_questions_g_harmonic_tuples
- sun_1990_finite_coverings_groups
- balister_2019_structure_number_erdos_covering_systems
- davenport_1936_sequences_positive_integers
- filaseta_2007_sieving_large_integers_covering_systems_congruences
- canfield_1983_problem_oppenheim_factorisatio_numerorum
- erdos_1980_old_new_problems_results_combinatorial_number_theory
- erdos_1980_survey_problems_combinatorial_number_theory
- guy_2004_unsolved_problems_number_theory
- norton_1994_frequencies_large_values_divisor_functions
- dusart_1999_kth_prime_lower_bound
- erdos_1950_integers_form_related_problems
- ellis_2010_irredundant_families_subcubes