Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Divisors and Multiples
ahlswede_1994_extremal_sets_without_coprimes/: Disproves an Erdos conjecture on largest sets without k+1 coprimes and proves Erdos's generalized case f(n,1,s), for every s, for squarefree numbers and for all large n.
ahlswede_1995_maximal_sets_numbers_not_containing_pairwise/: Shows the Erdos sets are extremal for all sufficiently large n for each k, so the disproved conjecture fails only finitely often.
ahlswede_1996_sets_integers_pairwise_common_divisor_factor/: Determines, for n at least the product of the given primes, the largest set of integers up to n that pairwise share a divisor and have a factor from that prime set, proving an Erdos conjecture.
alexander_nd_density_multiplicative_structure_sets_integers/: Extends Erdos-Davenport logarithmic density theory, sharpening results on division chains and on the irregularity of prime factors of almost all integers.
alexeev_2025_independence_clique_cover_numbers_squarefree_graph/: Confirms the Erdos-Sarkozy guess that the non-odd-squarefree numbers form a largest set with no squarefree pairwise product.
benkoski_1974_weird_pseudoperfect_numbers/: Shows the weird numbers have positive density, constructs infinite families of primitive pseudoperfect numbers, and poses several reciprocal-sum questions.
biro_2016_upper_bound_extremal_version_hajnal_s/: Proves that under optimal play the triangle-free saturation game on n vertices ends after at most (26/121)n^2 + o(n^2) edges.
cambie_2025_resolution_erdos_problems_about_unimodularity/: Disproves unimodality for the density of integers with one divisor in an interval and for the density of integers whose kth prime is p.
chan_2014_factors_perfect_square/: Proves that a large perfect square has at most five divisors in a short interval around its square root, settling the Erdos-Rosenfeld question for squares.
chan_2015_factors_almost_squares_lattice_points_circles/: Extends the bounded-divisor result near the square root from perfect squares to almost squares and bounds lattice points on circles near an axis.
davenport_1951_sequences_positive_integers/: Gives a direct elementary proof that a set of multiples has lower density and logarithmic density both equal to the inclusion-exclusion limit A.
doorn_2026_practical_numbers_egyptian_fractions/: Claims an explicit form of the Price bound, infinitely many practical n with h(n) at most (14/log 2)(log log n)^2, from a uniform construction that also gives claimed bounds for Problems 304 and 293; mostly AI-generated, with an author-side Lean formalization that is not held and not built here.
eberhard_2025_ratios_consecutive_values_divisor_function/: Proves that every positive rational occurs infinitely often as a ratio of consecutive divisor function values, confirming an Erdős prediction.
erdos_1935_note_sequences_integers_no_one_which/: Proves that for any primitive sequence the sum of 1/(a log a) converges, so every primitive set has lower density zero, and that the density of the integers with a divisor between a and 2a tends to zero.
erdos_1944_highly_composite_numbers/: Proves the count of highly composite numbers up to x exceeds (log x)^(1+c) for some positive constant c.
erdos_1952_distribution_values_divisor_function/: Determines the asymptotic order of the logarithm of the number of distinct values taken by the divisor function up to x, and bounds runs of distinct divisor counts.
erdos_1964_applications_probability_analysis_number_theory/: Survey of Erdos's probabilistic methods, announcing that almost all integers have two divisors within a factor of two of each other.
erdos_1966_divisibility_properties_sequences_integers/: Shows sequences of positive logarithmic density contain long divisibility chains, with a growth bound whose exponent 1/2 is shown to be sharp.
erdos_1967_theorem_behrend/: Proves the reciprocal sum of an infinite primitive sequence up to x is o(log x over the square root of log log x).
erdos_1968_solvability_certain_equations_sequences_positive_upper/: Shows every sequence of positive upper logarithmic density contains an infinite subsequence whose subset gcds and lcms all lie in it.
erdos_1970_extremal_problems_combinatorial_number_theory/: Proves that the density d_t of the integers n for which t is a sum of distinct divisors of n tends to zero as t grows, below a negative power of log t, and that F(4, x) > cx, plus related divisor-density results.
erdos_1978_unconventional_problems_divisors_integers/: Elementary bounds on unusual divisor statistics: coprime consecutive divisors, divisors that are products of consecutive integers, and separable numbers.
erdos_1979_propinquity_divisors/: Almost all integers n have no two divisors d < d' < d(1 + theta) with theta a slowly decaying function of n times the power (log d)^{1 - log 3}.
erdos_1979_unconventional_problems_number_theory_asterisque/: Erdős's Luminy problem paper: unproved claims and questions on close divisors, on the densities of integers by their k-th prime factor or by a divisor in an interval, on the largest prime factors of n and n plus one, on the count of totient values, and on divisors congruent to one modulo d.
erdos_1980_asymptotic_formulas_generalized_divisor_functions/: Shows that some integer up to x has more than any fixed multiple of a sequence's reciprocal sum up to x of its members as divisors, once that sum exceeds a constant and no member lies just below x.
erdos_1981_sur_la_structure_de_la_suite/: Disproves the conjecture that the count of dyadic ranges holding a divisor of n is a vanishing fraction of the divisor count for almost all n.
erdos_1997_factor_difference_set_integers/: Studies the set of differences of factor pairs of an integer, proving pairs of differences are shared by only finitely many integers.
erdos_1999_greedy_algorithm_arithmetic_progressions_subset_sums/: Erdős, Lev, Rauzy, Sándor and Sárközy's 1999 paper on greedy 3-free sequences and on subsets of the first n integers with divisibility restrictions on subset sums: the definition of non-dividing sets (Property Q), the bounds n^{1/5} << Q(n) < 3 n^{1/2} + 1 with the guess Q(n) > n^{1/2 - epsilon}, and Theorem 5, log n / log 2 - 1 < R(n) < log n / log 2 + log log n / (2 log 2) + c for sets whose distinct subset sums never divide one another.
fang_2022_searching_boundary_abundance_odd_weird_numbers/: An exhaustive computer search shows there is no odd weird number below 10^21, and none below 10^28 with abundance under 10^14.
ford_2008_distribution_integers_divisor_given_interval/: Determines the order of magnitude of the number of integers up to x with a divisor in an interval, and settles conjectures of Erdos and Tenenbaum.
gorodetsky_2024_erdos_sums_almost_primes/: Disproves the Banks-Martin monotonicity conjecture for Erdos sums of k-almost primes and gives an asymptotic with explicit secondary term.
hildebrand_1987_divisor_function_at_consecutive_integers/: Proves d(n) = d(n+1) for at least a constant times x(log log x)^-3 integers n <= x and that the limit points of log(d(n+1)/d(n)) have positive lower density and contain an interval around 0.
hughes_2026_sums_distinct_divisors_factorials/: Bounds the number of distinct divisors of n! needed to represent every integer up to n! by (2 log 2 + o(1)) n/log n, sharpening the Tenenbaum–Yokota and Yokota bounds through the Berend–Harmse factorial divisor-gap estimate.
jenw1n_2026_lean_proof_erdos_problem_18b/: Records the Conjectures.io-accepted Lean proof that for every positive epsilon h(n!) is below n^epsilon for all large n, the second question of Problem 18: site-kernel verified, review approved and certified in September 2026, not refereed and not built here.
koukoulopoulos_2025_erdos_s_integer_dilation_approximation_problem/: Resolves Erdos's 1948 integer dilation approximation problem for discrete sets of positive upper logarithmic density, using GCD graphs.
kovac_2025_number_divisors_mersenne_numbers/: Proves the doubling ratios of the summed divisor counts of Mersenne numbers are unbounded, so Erdos's limit cannot be finite.
letendre_2025_divisors_integer_short_interval/: Bounds the number of divisors of an integer in a short interval, proving a bounded count for windows of length n to the power theta squared minus epsilon.
lichtman_2020_almost_primes_banks_martin_conjecture/: Disproves the Banks-Martin monotonicity conjecture for sums over k-almost primes, locating the global minimum at k = 6 and the limit at 1.
lichtman_2022_proof_erdos_primitive_set_conjecture/: Proves that the sum of one over a times log a over any primitive set is at most its value for the primes, settling the Erdos primitive set conjecture.
maier_1984_set_divisors_integer/: Proves Erdős's conjecture that almost all integers have two divisors with ratio below two, in the sharp form that the least logarithmic ratio of two divisors is at most log n to the power one minus log three, up to a factor exp(xi(n) sqrt(log log n)) with xi any function tending to infinity, for almost all n.
melfi_2015_conditional_infiniteness_primitive_weird_numbers/: Proves that 2 to the k times p times q is a primitive weird number when p and q are primes just below and just above 2 to the power k+2 at suitable odd distances, and deduces infinitely many primitive weird numbers from a prime-gap bound of size one tenth of the square root.
nicolas_1971_repartition_des_nombres_hautement_composes/: Proves that the number of highly composite numbers up to X is O((log X)^{1+c}) and improves Erdos's lower bound exponent.
price_2026_sparse_divisor_sums/: A proof claim posted on the erdosproblems.com proof-claims tab that infinitely many practical n have h(n) at most a constant times (log log n)^2, with an AI-generated write-up behind an Overleaf read link that is not held and unread here.
tenenbaum_1984_sur_la_probabilite_qu_un/: Gives matching upper and lower bounds, up to slowly varying factors, for the number of integers below x having a divisor in a given interval.
tenenbaum_1986_sur_un_probleme_de_crible_et/: Bounds the distribution of integers whose divisors are closely spaced, and applies it to practical numbers and the Erdos-Ruzsa small sieve.
tenenbaum_1995_sur_un_probleme_de_crible_et/: Corrects and slightly improves the author's earlier lower bound for closely spaced divisors, then bounds the longest path in the divisor graph.
tenenbaum_2013_erdos_unconventional_problems_number_theory/: Survey revisiting Erdos's 1979 unconventional number theory problems and reporting the progress made on them since.
weingartner_2015_practical_numbers_distribution_divisors/: Proves Margenstern's conjecture that the count of practical numbers up to x is asymptotic to cx/log x, with error term.
weingartner_2019_constant_factor_asymptotic_practical_numbers/: Narrows the constant in the asymptotic count of practical numbers to 1.336073 < c < 1.336077.
This folder holds sources whose primary subject is Divisors and Multiples.
Sources with other primary subjects
Explicit links to this subject's problems support these cross-references.
- erdos_1981_problems_results_additive_multiplicative_number_theory
- erdos_sarkozy_1992_arithmetic_progressions_subset_sums
- erdos_1985_problems_results_number_theory
- erdos_1987_locally_repeated_values_certain_arithmetic_functions
- tao_2025_quantitative_correlations_problems_prime_factors_consecutive
- erdos_1982_my_favourite_problems_which_recently_have
- erdos_1992_my_favourite_problems_various_branches_combinatorics
- alexeev_2026_primitive_sets_von_mangoldt_chains_erdos
- besicovitch_1935_density_certain_sequences_integers
- davenport_1936_sequences_positive_integers
- erdos_1992_my_forgotten_problems_number_theory
- wang_2026_proposed_solution_erdos_problem_486
- erdos_1979_unconventional_problems_number_theory_math_mag
- erdos_1980_old_new_problems_results_combinatorial_number_theory
- erdos_1980_survey_problems_combinatorial_number_theory
- guy_2004_unsolved_problems_number_theory
- baker_2001_difference_between_consecutive_primes
- erdos_1997_some_my_favorite_problems_results
- chvatal_1974_intersecting_families_edges_hypergraphs_hereditary_property