Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Sequences and Densities of Integers

abbott_1967_extremal_problem_number_theory/: Bounds the largest set of integers up to n with no k members having pairwise the same greatest common divisor when k grows with n, between powers of n for k about a power of log n and from below by about n over a power of log n for k a root of n, and shows that about n log log n over log n integers up to n can avoid three with pairwise the same least common multiple.

ailon_2004_torsion_points_curves_common_divisors/: Polynomial and matrix analogs of small gcd questions, with torsion-point, quadratic-norm and cyclotomic proofs and historical integer conjectures.

alexeev_2026_primitive_sets_von_mangoldt_chains_erdos/: Introduces a Markov chain method with von Mangoldt weights that bounds Erdős sums of primitive sets and settles several Erdős problems.

alford_1994_infinitely_many_carmichael_numbers/: Proves that there are infinitely many Carmichael numbers, with their count up to x exceeding x to the 2/7 for all large x, by building a modulus L with many primes p such that p minus 1 divides L and applying a zero-sum theorem for finite abelian groups.

alon_1988_sums_subsets_set_integers/: Proves that for n > n(eps) and 3n^{5/3+eps} < m < n^2/(20 log^2 n) the largest subset of {1,...,n} with no subset summing to m has floor(n/s) + s - 2 elements, s the least non-divisor of m, settling an Erdős-Graham conjecture, and bounds subsets of {1,...,n} with no r-th power among their subset sums.

alon_1996_sure_monochromatic_subset_sums/: Determines up to logarithmic factors the least number of colors needed to color 1..n-1 with no monochromatic subset summing to n.

anon_2026_primes_logarithmic_block_product/: A four-page anonymous note, linked from the site's Problem 457 thread, constructing infinitely many n for which every prime up to 2.1 log n divides the product of the next floor(log n) integers; the first written form of the construction the site accepted for Problem 457.

anon_2026_resolution_erdos_problem_38_sparse_dyadic_shift/: Digest of a six-page 2026 manuscript with no named author that constructs a sparse set, not an additive basis, giving every set of Schnirelmann density in (0,1) a uniform density increment under one shift, as Erdős Problem 38 asks.

araujo_2026_sarnaks_program_erdos_sieves_part_ii_measure_systems_applications/: Source record and research digest.

baier_2004_6/: Improves the counting bound for coprime P-sets, giving a count below N of at most (3+eps)N^(2/3)/log N infinitely often.

balandraud_2012_addition_theorem_maximal_zero_sum_free_sets/: Bounds the subsum set of a set disjoint from its negative in Z/pZ and thereby proves Selfridge's 1976 conjecture on maximal zero-sum free sets.

balasubramanian_1996_conjecture_r/: Proves Graham's conjecture unconditionally in its strong form for every set of N distinct integers with greatest common divisor 1 and N at least five.

balister_2019_structure_number_erdos_covering_systems/: Determines asymptotically the logarithm of the number of minimal covering systems of the integers of size n, via a structural theorem.

balister_2022_erdos_covering_systems/: An expository note presenting the distortion method and using it to reprove Hough's minimum modulus theorem for square-free moduli.

bambah_1947_numbers_which_can_be_expressed_as/: An elementary argument shows that for every eps > 0 and all large x some sum of two squares lies between x and x + 2 sqrt(2+eps) x^(1/4).

banks_2014_consecutive_primes_tuples/: Shows admissible tuples infinitely often contain consecutive primes, producing runs of prime gaps that are increasing, decreasing, or successively divisible.

bedert_2023_problem_erdos_sarkozy_about_sequences_no/: Proves that a subset of 1..n in which no term divides the sum of two larger terms has at most n/3 + O(1) elements, and at most the ceiling of n/3 for large n, matching the extremal example.

beker_2023_problem_erdos_graham_about_consecutive_sums/: Shows increasing sequences in 1..n exist whose consecutive-block sums take at least cn^2 distinct values, answering a question of Erdős and Graham.

belgikar_2024_new_applications_ergodic_theory_sets_differences/: Reproves and generalizes the Stewart-Tijdeman and Ruzsa theorems on difference sets by pointwise ergodic methods, including amenable-group versions.

bergelson_2017_density_coprime_tuples_form_where_are/: Proves that, under growth and separation conditions, the set of n making n and the floors of k Hardy-field functions coprime has natural density one over the zeta value at k+1.

besicovitch_1935_density_certain_sequences_integers/: Builds a primitive set whose set of multiples has no natural density, using sparse divisor windows and a gliding sequence of dyadic blocks.

bruedern_2017_local_oscillations_moderately_dense_sequences_primes/: Shows that p_{n+1}^2 - p_n p_{n+2} changes sign infinitely often for a set of primes whose counting function exceeds x/(log x)^(4/3) by an unbounded factor, and bounds the curvature of such sequences.

bugeaud_corvaja_zannier_2003_gcd_upper_bound/: Complete subexponential gcd proof for fixed independent bases, with sharp divisibility and lower-bound corollaries; coprimality remains separate.

cambie_2024_resolution_erdos_problem_least_common_multiples/: Shows the least common multiple of a short block of consecutive integers can exceed that of a longer block of larger integers by any factor.

cassels_1960_representation_integers_as_sums_distinct_summands/: Gives a circle-method criterion under which every large integer is a sum of distinct elements of a set, and shows growth and congruence do not suffice.

chan_2023_moments_gaps_between_consecutive_square_free/: Extends the range of exponents for which the gamma-th moment of gaps between consecutive squarefree numbers has an asymptotic to below 3.75.

chen_1998_sequences_bounded_l_c_m_each/: Proves the largest set of integers with all pairwise least common multiples at most x has size sqrt(9x/8) plus a smaller error term.

chen_2007_sequences_bounded_l_c_m_each/: Shows the error term in the largest size of an integer set with pairwise least common multiple at most x is unbounded, and bounds the excess over Erdős's construction by a double logarithm.

choi_1972_largest_subset_pairwise_l_c_m_not_exceeding_n/: Choi's 1972 upper bound for g(n), the largest number of integers in [1, n] whose pairwise least common multiples are all at most n: the Theorem g(n) < (1 + λ − λ*) n^(1/2) + o(n^(1/2)) with 1 + λ − λ* printed as at most 1.638, the source of the bound 1.638 sqrt(n) quoted on Problem 441, after the simpler estimate (6), g(n) < (1 + λ) n^(1/2) + o(n^(1/2)) with λ < 0.87.

chojecki_2026_distinct_consecutive_products/: Claims a density-one set of integers whose distinct consecutive blocks all have distinct products, answering a question of Erdos and Graham.

chojecki_2026_second_term_strongly_2_primitive_sets/: A five-page manuscript hosted at ulam.ai proving that the largest subset of one through n in which no member divides the product of two others has size pi(n) plus (27/2 + o(1)) n^(2/3)/(log n)^2, by tracking the constants in Erdős's 1938 factorization argument and packing linear prime triples; the author declares AI assistance and the site accepts it as the resolution of Problem 793.

chojecki_2026_truncated_congruence_sieves_erdos_problem_25/: Proves natural density for summable and pairwise-coprime truncated congruence sieves, and reduces Erdős problem 25 to uniform harmonic control of finite quotient sieves with globally sublogarithmic tower charges.

conlon_2021_subset_sums_completeness_colorings/: Solves several Erdos problems on Ramsey-complete and density-complete sequences, monochromatic subset sums and long homogeneous progressions in subset sums.

corvaja_zannier_2005_height_sunit_points/: Generalizes the gcd(a^n-1, b^n-1) upper bound to heights of rational functions at S-unit points, so it subsumes the bound bearing on problem 770.

dai_2006_sequences_bounded_l_c_m_each/: Gives an explicit error term for the maximum size of a set of integers whose pairwise least common multiples stay below x.

davenport_1936_sequences_positive_integers/: Shows the set of multiples of a sequence has logarithmic density and lower density both equal to the natural limit A, and deduces infinite divisibility chains.

davis_2026_forbidden_subgraphs_divisor_graphs/: Davis proves that the largest fork-free subset of one to n has size c_2 n+o(n) for an effectively computable constant; irrationality of c_2 remains open in this paper.

dietmann_2023_longer_gaps_between_values_binary_quadratic/: Improves Richards's lower bounds on large gaps between integers represented by a binary quadratic form, and between sums of two squares.

doorn_2025_growth_rates_sequences_governed_squarefree_properties/: Settles several Erdos questions on how fast a sequence must grow when its translates meet the squarefree numbers in prescribed ways.

doorn_2026_consecutive_integers_free_certain_prime_factors/: Proves an Erdős conjecture that the least n above 2k with no prime factor of the preceding k integers in (k,2k) grows superpolynomially in k.

doorn_2026_length_interval_distinct_multiples/: Confirms a conjecture of Erdos and Pomerance by exhibiting intervals of length about cn log n / log log n with no distinct multiples of 1 through n.

elliott_nd_problem_erdos_concerning_power_residue_sums/: Proves that sums over primes of the a-th powers of the least kth power non-residue, for a < 4e^(1-1/k), are asymptotic to a constant times x over log x, confirming an Erdos conjecture.

elsholtz_2017_erdos_sarkozy_s_sequences/: Constructs an infinite set with Property P whose counting function beats the Erdos-Sarkozy example by a power of log, which the authors believe is the first such improvement since 1970.

erdos_1938_sequences_integers_no_one_which_divides/: Bounds sequences in which no member divides the product of two others, or in which all pairwise products differ, near the count of primes.

erdos_1951_problems_results_elementary_number_theory/: Proves large gaps in sequences defined by prime divisibility conditions, including sums of two squares and squarefree numbers, with moment results.

erdos_1956_pseudoprimes_carmichael_numbers/: Proves upper bounds for the counting functions of pseudoprimes and of Carmichael numbers up to x, and conjectures that the Carmichael count exceeds x^{1-eps} for every eps > 0 and large x.

erdos_1959_megjegyzesek_egy_versenyfeladathoz_remarks_problem_bemerkungen/: Studies how many integers must be selected from a fixed consecutive interval for product divisibility, and bounds a separate interval-length function.

erdos_1961_satze_und_probleme_uber_german/: Shows the total variation of the sequence p_k/k up to x has order log squared x, so the sequence is nowhere eventually monotone.

erdos_1961_szamelmeleti_megjegyzesek/: Proves the average of the least quadratic non-residue over primes up to x tends to the sum of p_k over 2 to the k, answering a question of Mirsky.

erdos_1962_szamelmeleti_megjegyzesek_iv/: Fourth part of Erdős's Hungarian remarks on number theory: thirty-three extremal problems, numbered 1 to 34, about integers in a finite interval with literature notes, among them problem 14 on the largest set of integers up to n with no k members having pairwise the same greatest common divisor.

erdos_1964_addition_residue_classes_mod/: Shows every residue class mod p is a subset sum of any k distinct nonzero classes once k is at least 3(6p)^{1/2}, with an asymptotic count, and conjectures the zero-sum threshold 2p^{1/2} for every modulus.

erdos_1964_multiplicative_representation_integers/: Shows that if every large integer is a product of two members of a sequence then the number of such representations is unbounded.

erdos_1964_problem_elementary_number_theory_combinatorial_problem/: Bounds the largest set of integers up to n with no t members having pairwise the same gcd between 2 to the power c_t log n/log log n and n to the power 3/4 plus epsilon, and ties closing the gap to the sunflower conjecture.

erdos_1969_applications_graph_theory_number_theory/: Survey showing how extremal graph and hypergraph theorems bound sequences with prescribed divisibility and distinct-product conditions.

erdos_1970_applications_graph_theory_number_theory/: Survey showing how graph and combinatorial arguments bound sequences with distinct products or sums, including the prime-set function f(k,x).

erdos_1970_divisibility_properties_sequences_integers/: Proves that an infinite set in which no element divides the sum of two larger elements has density zero, and shows this cannot be much improved.

erdos_1972_extremal_problems_number_theory/: A survey note collecting seven number-theoretic extremal problems, reporting recent progress and posing refinements of each.

erdos_1973_number_solutions_additive_functions/: Proves upper and lower bounds on the density of integers on which a real additive function takes a single nonzero value.

erdos_1976_multiplicative_representations_integers/: Gives a simpler proof of Szemerédi's theorem that two subsets of one through x with all products distinct have size product below c x^2/log x, conjectures the constant is 1 + o(1), and bounds the size product when every integer has fewer than c such representations.

erdos_1976_problem_graham/: Proves Graham's conjecture for all sufficiently large primes: if every nonempty zero subset sum of p nonzero residues has the same number of terms, only two distinct residues occur.

erdos_1977_differences_sums_integers_ii/: Continues the study of difference and sum intersector sets, the sequences B meeting the difference set or the sum set of every sequence of positive density: the squares and the shifted primes intersect differences but not sums, the residue class 1 mod 3 of density 1/3 being the example with the guess that 1/3 is extremal; a finite difference intersector set can have bounded size while a sum intersector set cannot, and Tijdeman's conjecture on the growth of difference intersector sets is proved.

erdos_1977_problems_results_combinatorial_number_theory_iii/: Problem survey collecting open questions on arithmetic progressions, covering congruences, sum-distinct sequences and recursively defined integer sequences.

erdos_1980_megjegyzesek_az_american_mathematical_monthly_egy/: Shows the conjectured bound on how often the least common multiple of consecutive terms of a sequence is small holds for pairs but fails for longer blocks.

erdos_1981_many_old_some_new_problems_number_theory/: Collects problems on primes, consecutive integers, and more; III.3 poses E839's density questions, builds an avoiding sequence of upper density 1/2, and gives a defective proof of a reciprocal-sum bound.

erdos_1986_problems_number_theory/: A problem paper on primes and divisors that includes a full proof of the Erdos-Selfridge theorem on intervals with few distinct multiples of given primes.

erdos_1987_divisibility_properties_integers_form/: Bounds the largest subset of one to N whose pairwise sums are all squarefree between a constant times log N and three N to the three quarters times log N.

erdos_1992_my_forgotten_problems_number_theory/: Erdős collects neglected problems on divisibility in short intervals, Sidon sequences, sum-closed sets and sequences with no divisor of two larger terms.

erdos_1997_some_old_new_problems_various_branches_combinatorics/: Erdős's posthumous 1997 problem paper in twelve numbered items: the Erdős–Gyárfás Ramsey function whose conjectured C^{root n} bound is Problem 129, the Erdős–Sárközy divisibility problems of Problems 12, 13 and 131 (with the N^{1/5} bound credited to a Budapest student), the integer-distance graph of Problem 130, the triangle-free diameter-two question of Problem 133, the five-distances conjecture of Problem 135, and the odd-cycle and power-of-two cycle conjectures of Problems 57 and 63.

fan_2025_maximal_order_shifted_prime_divisor_function/: Proves new unconditional and GRH-conditional lower bounds for the maximal order of the shifted-prime divisor function.

filaseta_1992_gaps_between_squarefree_numbers_ii/: Proves that for large x the interval from x to x plus a constant times the fifth root of x times log x always contains a squarefree number.

filaseta_2007_sieving_large_integers_covering_systems_congruences/: Proves the Erdős-Selfridge and Erdős-Graham conjectures on covering systems with large moduli, bounding reciprocal sums and uncovered density.

ford_2010_prime_chains_pratt_trees/: Gives new upper and lower bounds for counts and heights of prime chains and Pratt trees, and settles a 1990 conjecture of Erdos, Granville, Pomerance and Spiro.

ford_2018_long_gaps_between_primes/: Proves the maximal prime gap below X is at least of order log X log log X log log log log X divided by log log log X.

ford_et_al_2018_long_gaps_sieved_sets/: Corrected source record: long gaps from general one-dimensional sieves, their residue-class covering construction, and the distinction from the admissible-survivor optimization in Problem 1204.

gao_2010_distinct_length_modular_zero_sum_subsequences_graham_conjecture/: Proves Graham's conjecture for every modulus n: a sequence of n integers between 0 and n minus 1 that takes at least three distinct values has two nonempty zero-sum subsequences modulo n of distinct lengths, extending the Erdős–Szemerédi theorem from large primes to all n.

gordon_rodemich_1998_dense_admissible_sets/: Source record and research digest.

granville_1999_set_differences_given_set/: Poses the problem of the least number of values of a over gcd(a, b) taken over a set of m distinct positive integers, proves that it lies between the square root of m and about (3/2)(2m) to the 2/3, with the lower bound (m/2) to the 2/3 when the members of the set involve only two primes, and shows that the values of ab over gcd(a, b) squared number at least |A|.

granville_2001_spectrum_multiplicative_functions/: Determines the spectrum of [-1,1], the set of limits of averages up to N of completely multiplicative functions with values in [-1,1] when the function may change with N, and bounds spectra of general value sets.

granville_2020_sieving_intervals_siegel_zeros/: Granville's conditional extremal interval-sieve results, their Jacobsthal context, and the consequence for the least diameter of admissible k-tuples.

granville_lumley_2020_primes_short_intervals_heuristics_calculations/: Relates maximal prime counts in very short intervals to the largest admissible subset of an interval, recording the factor-two sieve bounds relevant to Problem 1204 and the limits of the prime-tuple heuristic.

green_2004_cameron_erdos_conjecture/: Proves the Cameron-Erdős conjecture that the number of sum-free subsets of {1,...,N} is O(2^{N/2}), in fact asymptotically c(N)2^{N/2}.

grynkiewicz_2011_note_conjecture_graham/: Proves Graham's conjecture for every finite abelian group of order n: a sequence of n elements all of whose nonempty zero-sum subsequences have the same length has at most two distinct terms, with a list of the forms such sequences take; for cyclic groups of prime order the proof needs only the Cauchy-Davenport theorem and the pigeonhole principle.

gyory_2020_additive_multiplicative_decompositions_sets_integers_restricted/: Extends and sharpens the Elsholtz-Harper theorem on additive indecomposability of smooth numbers using S-unit equation methods.

hall_1996_proof_conjecture_heath_brown_concerning_quadratic/: Proves that at least an absolute positive proportion of the integers up to n are quadratic residues mod p, uniformly in the prime p and in n.

hamidoune_1996_zero_free_subset_sums/: Shows a subset of an abelian group of size sqrt(2|G|) plus a small error term must contain a nonempty subset summing to zero.

hardy_2002_modified_problem_pillai_related_questions/: Proves there are infinitely many Pillai primes and infinitely many integers n admitting such a prime, and lists related open problems.

hildebrand_1987_quantitative_mean_value_theorems_nonnegative_multiplicative/: Gives a sharp lower bound for the mean value of a nonnegative multiplicative function, matching the known upper bound via Dickman's function.

hooley_1965_difference_between_consecutive_numbers_prime/: Shows that, as n tends to infinity with n/phi(n) also tending to infinity, the gaps between consecutive integers prime to n, normalized by n/phi(n), are asymptotically distributed like a gamma variable with parameter 1.

iwaniec_1978_problem_jacobsthal/: Iwaniec's 1978 upper bound for Jacobsthal's problem over arbitrary primes: every interval of length a constant times r^2 log r times the product of (1 - 1/q_i)^{-1} contains at least r^2 integers coprime to the r primes q_1, ..., q_r, hence the maximal run C(r) of consecutive integers each divisible by one of r arbitrary primes is O(r^2 log^2 r); the primorial case, Y(x) = O(x^2), follows at r = pi(x).

khalfalah_2002_tight_bound_density_sum_no_two_perfect_square/: Proves that a subset of the first N integers in which no two distinct elements sum to a perfect square has at most (11/32 plus o(1)) N elements, matching Massias's construction of density 11/32 and sharpening the 0.475 bound of Lagarias, Odlyzko and Shearer; read as the DIMACS technical report preprint of December 2000.

klarner_1974_arithmetic_properties_certain_recursively_defined_sets/: Studies sets of integers closed under given linear operations, showing broad classes are finite unions of arithmetic progressions.

kolpakov_2022_free_semigroups_affine_maps_real_line/: Gives geometric ping-pong criteria for semigroups of real affine maps to be free, generalizing Klarner's integer conditions, and limits when they apply.

konieczny_2015_consecutive_sums_permutations/: A random permutation of 1..n has about (1+e^-2)/4 times n^2 distinct sums of consecutive terms, answering a question of Erdos and Harzheim negatively.

konyagin_2004_problems_set_square_free_numbers/: Bounds the largest subset of the first N integers all of whose pairwise sums, doubles included, are squarefree between a constant times log squared N times log log N and N to the 11/15 times a subexponential factor, and shows that the longest arithmetic progression of squarefree numbers up to N has length of order log squared N.

konyagin_2022_construction_schinzel_many_numbers_short_interval_without_small_prime_factors/: Source record and research digest.

lagarias_1983_density_sequences_integers_sum_no_two/: Proves by the circle method that for all N >= N_0 a subset of [1,N] in which no sum of two distinct elements is a perfect square has at most .475N elements, so such an infinite sequence has upper density at most .475.

laishram_2006_grimm_s_conjecture_consecutive_integers/: Verifies Grimm's conjecture on distinct prime divisors of consecutive composite integers for all n up to 1.9 times 10 to the tenth.

lehmer_1963_pairs_consecutive_power_residues/: Gives the largest possible least pair of consecutive kth power residues modulo a prime for every k up to six, proving the cases k = 5 and 6.

li_2016_lower_bound_least_prime_arithmetic_progression/: Shows that for almost every modulus k the largest least prime in a progression mod k is at least a constant times phi(k) log k log_2 k log_4 k / log_3 k, with log_j the j-fold iterated logarithm.

li_2026_resolution_erdos_problem_768_sylow_divisor/: Determines the exact decay constant for integers satisfying the Sylow divisor condition, with a Lean 4 formalization the author reports as verified.

linnik_1942_erdos_theorem_addition_numerical_sequences/: Archives Linnik's non-basic essential-component construction, with an explicit reconstruction of its power-sum proof and source corrections.

nesetril_2024_pisier_type_theorems/: Constructs counterexamples to fixed-order Pisier-type statements using partial Steiner systems and carry-free integer encodings, while isolating the missing uniformity needed for full dissociation.

nguyen_2010_squares_sumsets/: Shows the largest square-sum-free subset of {1,...,n} has at most n^{1/3}(log n)^C elements, so with Erdos's n^{1/3} construction its size is n^{1/3+o(1)}.

nguyendang_2026_sequence_gcd_n_1_b_n/: Shows gcd(a^n-1, b^n-1) is a linear recurrence only when a and b are multiplicatively dependent, and derives reductions toward the Ailon-Rudnick conjecture.

olson_1968_addition_theorem_modulo/: Olson's 1968 proof of the Erdős–Heilbronn conjecture for prime moduli: s distinct nonzero residues modulo a prime p with s > (4p - 3)^{1/2} represent every residue class, zero included, as a sum of a nonempty subfamily, so the zero-sum threshold for primes is at most 2 sqrt p; with Theorem 2, at least min{(p+3)/2, s(s+1)/2} distinct subset sums when no two residues are equal or opposite.

openai_2026_primitive_roots_admissible_integer_base/: A 92-page release manuscript claiming the infinitude part of Artin's primitive root conjecture for every integer base other than -1 and the squares, with at least $c_a x/(\log x)^2$ such primes in each large dyadic interval, by a sieve construction of primes $p=crQ+1$ and a claimed uniform zero-free strip for finite-order Hecke L-functions of cyclotomic fields; it names no Erdős problem and touches Problems 985 and 429 as background.

openai_2026_quadratic_bound_jacobsthal_function/: An 83-page manuscript of the OpenAI mathematics release claiming h(k) ≪ k^2/(log log 3k)^2 for Jacobsthal's function over integers with at most k distinct prime factors, by a lower-bound sieve at the critical parameter 2 with an inverse estimate and stopped tree; the displayed question of Problem 970, touching Problems 687 and 929 through Y(x) = j(P(x)) - 1.

polymath_2014_variants_selberg_sieve_bounded_intervals_containing/: Generalizes Maynard's multidimensional Selberg sieve to prove H_1 <= 246 unconditionally and H_1 <= 6 under the generalized Elliott-Halberstam conjecture; its admissible-tuple section bounds the diameter of the narrowest admissible k-tuple.

pomerance_1989_two_methods_elementary_analytic_number_theory/: Survey of a Rankin-type upper bound method and a combinatorial lower bound method, applied to smooth numbers, factorizations, Euler-function fibers and pseudoprimes, including an upper bound for Carmichael numbers.

prachar_1955_divisors_form_prime_minus_one/: Reconstructs the four original shifted-prime divisor theorems, with their precise external prime-distribution and sieve inputs.

price_2026_coprime_power_differences/: Public manuscript giving an eventual log-two upper bound for both coprimality thresholds in Problem 820; the full elementary chain is recorded.

raghavan_2025_sharp_bounds_sets_distinct_subset_products/: Proves the largest subset of 1..N with all subset products distinct has size pi(N) + pi(sqrt N) + O(N^(5/12)), answering an Erdos question.

richter_1976_uber_die_monotonie_von_differenzenfolgen/: Shows that any prime sequence with non-decreasing consecutive gaps satisfies liminf q_n over n squared at least 1 over 2.84010.

ruzsa_1977_general_multiplicative_functions/: Develops G-multiplicative arithmetic functions and proves local limit theorems giving asymptotic densities for their level sets.

saias_1998_applications_des_entiers_diviseurs_denses/: Determines the exact order x/log x for the Erdős-Ruzsa small sieve and for the longest paths in the divisor graph on integers up to x.

schinzel_1959_sur_un_probleme_de_paul_erdos/: Proves that integers up to n with all pairwise least common multiples above n have reciprocal sum at most 31/30, with equality only for 2, 3, 5 and n = 5.

schinzel_1961_remarks_paper_sur_certaines_hypotheses_concernant_les_nombres_premiers/: Source record and research digest.

schoen_2001_problem_erdos_sarkozy/: Schoen's 2001 note proving that a P-set of pairwise coprime integers, one in which no element divides the sum of two larger elements, has counting function below 2n^{2/3} for infinitely many n, by the analytic large sieve; with the p^2 example of Erdős and Sárközy showing that the exponent cannot go below 1/2.

sorenson_webster_2015_strong_pseudoprimes_twelve_bases/: Sorenson and Webster's exact values and algorithmic results for strong pseudoprimes to the first twelve prime bases.

stewart_1978_difference_sets_sets_integers/: Surveys the structure of ordinary, infinite and density difference sets of integer sets of positive upper density.

szemeredi_1970_conjecture_erdos_heilbronn/: Proves that for an absolute constant c, any c times the square root of n distinct elements of an abelian group of n elements have a nonempty subset summing to zero.

szemeredi_1976_problem_p_erdos/: Szemerédi's 1976 proof of Erdős's conjecture that two sets of positive integers up to n whose pairwise products across the sets are all distinct have size product below C n^2/log n, with the constant C made explicit in terms of the Brun and Mertens constants; the paper the site names as the proof of Problem 490, with Diviš's r-set extension and the announcement of the Erdős–Szemerédi bounded-representation theorem.

szemeredi_2005_long_arithmetic_progressions_sumsets_thresholds_bounds/: Locates the thresholds for the longest arithmetic progression in an l-fold sumset and settles conjectures of Folkman and Erdos on subcomplete and complete sequences.

tang_2025_harmonic_lcm_patterns_sunflower_free_capacity/: Gives explicit polylogarithmic lower bounds for the maximal harmonic sum of an LCM-k-free set and ties the problem to the sunflower conjecture.

tang_2026_hofstadter_consecutive_sum_sequence_omits_infinitely/: Proves the greedy consecutive-sum sequence omits infinitely many integers, with a_n at least n + log log n / log 20 - O(1) and at most C_eps n to the power 4175/2506 + eps.

tao_2024_dense_sets_natural_numbers_unusually_large/: Answers an Erdos-Graham question negatively by constructing sets far denser than the primes whose pairwise least common multiples stay large.

tao_2024_product_representations_squares/: Shows that for every k at least 4 the largest subset of 1..N with no k distinct elements multiplying to a square has size at most (1-c_k+o(1))N, with c_k > 0.

tenenbaum_1996_block_behrend_sequences/: Gives a sufficient condition for a block sequence of intervals to be Behrend, that is for its set of multiples to have density one, adjacent to the necessary condition of Hall and Tenenbaum, and confirms Erdős's conjecture of a critical exponent: blocks of relative length j to the minus alpha, with consecutive ratios between two constants greater than one, are Behrend when alpha is below log 2 and not when it is above.

wang_2026_proposed_complete_solution_erdos_problem_536/: Claims that sets avoiding three integers with equal pairwise least common multiples have size o(N), via the cap-set theorem.

wang_2026_proposed_solution_erdos_problem_486/: Constructs a delayed multi-residue sieve whose survivor set has no logarithmic density and isolates the finite-block transfer needed for the singleton-residue Problem 25.

weingartner_2025_schinzel_szekeres_function/: Gives asymptotics for counting functions of the Schinzel-Szekeres function and applies them to divisor paths, reciprocal sums, and a small sieve.

weisenberg_2024_sparse_admissible_sets_problem_erdos_graham/: Disproves the Erdős–Graham conjecture by building arbitrarily sparse infinite admissible sets that cannot be translated into the primes; the resolving paper of Problem 429.

zeng_2026_collective_coprimality_threshold/: A July 2026 public proof claim giving a square-root upper bound for the collective gcd threshold whenever it exceeds P(n), and an exact reduced-fraction counting criterion.

zeraoulia_2026_conditional_resolution_reciprocal_sums_primes/: Bibliographic record of a Zenodo preprint claiming the Problem 726 asymptotic under an unproved reciprocal-prime equidistribution hypothesis; its PDF was not retrieved and its argument has not been read here.

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