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

Updated Oct 8

Other Number Theory

alexeev_2026_short_proofs_combinatorics_number_theory/: Answers three Erdos questions: small prime factors of binomial coefficients, splitting additive bases, and equidistribution of alpha times primes.

applegate_lagarias_1995_density_bounds_1/: Proves by computer-assisted tree search that, for some c > 0 and all x >= 1, at least c x^0.65 of the positive integers up to x have a 3x+1 orbit reaching 1, a partial result toward the Collatz conjecture (problem 1135).

applegate_lagarias_1995_density_bounds_2/: Proves by a computer-solved linear program built from Krasikov's difference inequalities that for each a not divisible by 3 at least c_a x^0.81 integers n with |n| at most x reach a under the 3x+1 function, for all x at least a.

banks_2007_prime_numbers_beatty_sequences/: Gives asymptotic formulas, uniform in the modulus, for primes of the form qfloor(alphan + beta) + a and for Beatty-sequence primes in a residue class.

barina_2020_convergence_verification/: Reports and describes the computational verification that every starting value below 2^68 converges under the Collatz map (problem 1135).

barina_2025_improved_verification_limit_convergence_collatz/: Reports the distributed computation that verified the Collatz conjecture for every starting value below 2^71 (15 January 2025), with the algorithms and the project timeline from 2^68 in 2020; the current finite verification record for Problem 1135.

bell_lagarias_2014_genfun_natural_boundaries/: Shows unconditionally that the 3x+1 backward-orbit generating functions have the unit circle as natural boundary for every m >= 1 except possibly m = 1, 2, 4, 8, whose rationality is equivalent to the conjecture of problem 1135.

bernstein_1994_noniterative_2adic/: Gives an equivalent non-iterative 2-adic restatement of the 3N+1 conjecture, i.e. a reformulation of problem 1135.

bernstein_lagarias_1996_conjugacy_map/: Studies the conjugacy map Phi between the 2-adic shift and the 3x+1 map, proving that its reduction mod 2^n has order 2^(n-4) for n >= 6, and recalls from earlier work that the conjecture of problem 1135 is equivalent to Z^+ contained in Phi((1/3)Z).

bzdega_2010_bounds_ternary_cyclotomic_coefficients/: A theorem-indexed source review with a complete local Markdown reading copy.

canfield_1983_problem_oppenheim_factorisatio_numerorum/: Proves that highly factorable n have f(n) = n L(n)^{-1+o(1)} unordered factorizations, correcting Oppenheim, and gives a new lower bound for the smooth-number count Psi(x,y) with a uniform asymptotic.

chamberland_2003_update_survey/: Survey of the 3x+1 problem organized by attack surface, the same survey genre the site cites (La85/La10/La16) for problem 1135.

chamberland_2015_averaging_structure/: Its results are identities for the generating functions of the iterates of qx+r maps, whose polar coefficients stay fixed in n when q = 3, except the residue at x = 1, supporting only a heuristic for bounded orbits in problem 1135.

christie_et_al_2020_classifying_minimal_vanishing_sums_roots_unity/: A focused E0774 digest of the arXiv preprint.

chung_1981_irregularities_distribution_real_sequences/: Chung and Graham's one-page 1981 announcement of the sharp clustering bound for sequences in the unit interval: C at most 0.39441967..., the Fibonacci-digit extremal sequence and the permutation theorem, with the proofs deferred to their 1984 chapter.

chung_1984_irregularities_distribution/: Chung and Graham's full paper on the clustering measure C of a sequence in the unit interval: the sharp bound C at most 0.39441967..., the Fibonacci-digit sequence that attains it, and the permutation extremal problem behind both; the proof announced in their 1981 PNAS note.

cipollini_2026_optimality_van_doorn_upper_bound_mayer_erdos_farey/: A 2026 arXiv preprint, with declared AI assistance, claiming the matching lower bound f(n) >= (1/4 - o(1)) n for the minimum number of Farey fractions between a badly ordered pair, so that f(n) = (1/4 + o(1)) n; the source behind the site's SOLVED label for Problem 1005, accepted by the site and not refereed.

colliot_thelene_skorobogatov_2021_brauer_groups_schemes/: Surveys Brauer groups of schemes (Azumaya versus cohomological, residues, purity) and proves injectivity into the function-field Brauer group.

colliot_thelene_skorobogatov_2021_comparing_two_brauer_groups_ii/: Proves finite cokernel for the Brauer comparison map and that smooth complete intersections of dimension at least three in characteristic zero have only constant Brauer classes.

conway_jones_1976_trigonometric_diophantine_equations_vanishing_sums_roots_unity/: A focused E0774 digest with a complete local Markdown reading copy.

coppersmith_steinberg_2006_entry_sum_cyclotomic_arrays/: A focused E0774 digest with a complete local Markdown reading copy.

croot_2004_sums_reciprocal_powers_modulo_prime/: Proves from the Bourgain-Katz-Tao sum-product estimate that for every epsilon in (0, 1] and every k at least 1 there is an N(epsilon, k) such that every residue modulo every prime p is a sum of at most N inverses of k-th powers of integers at most p to the epsilon (exactly N as printed, which fails for small primes); with k equal to 1 the Erdős-Graham inverse question for every prime, extending Shparlinski.

davenport_1963_theorem_uniform_distribution/: Shows that for almost all a > 0 the multiples of a hit a union of intervals of positive density as often as its measure predicts when O(N^(2 - delta)) of the intervals start at or below N, and restates Khintchine's problem.

davenport_leveque_1963_uniform_distribution_relative_fixed_sequence/: Proves that when the gaps of a fixed increasing sequence z_n decrease, the multiples kx (and more generally a_k x with a_{k+1} - a_k at least C a_k/k) are uniformly distributed relative to the z_n for almost all x, removing the earlier restriction that the gaps be O(1/z_n).

de_mathan_1980_numbers_contravening_condition_density_modulo_1/: De Mathan's 1980 solution of Erdős's lacunary density question: for every sequence of positive reals with consecutive ratios at least a fixed lambda above 1 and every interval, the multipliers x in the interval for which (q_n x) is not everywhere dense mod 1 form a set of Hausdorff dimension 1, from a general theorem on sequences of monotonic functions whose derivative ratios are bounded between lambda and mu; independent of Pollington.

doorn_2025_improved_bounds_mayer_erdos_phenomenon_similarly/: Sharpens the bounds on f(n), the largest index distance within which every two Farey fractions of order n are similarly ordered.

dubickas_2006_fractional_parts_lacunary_sequences/: Proves every lacunary sequence has a multiplier whose fractional parts avoid an interval, giving a quadratic bound on a related chromatic number.

elliott_1969_conjecture_erdos_concerning_character_sums/: Elliott's 1969 proof that for each epsilon in (0,1] the two-sided eventual-time threshold g(epsilon,p), the least t with |sum_{n<=m} (n/p)| < epsilon m for every m >= t, has a mean value c(epsilon) over the primes p <= x, with error term O((log log x)^{-1/8}); the paper of Erdős's display (80), whose one-sided threshold it treats in a remark.

elman_karpenko_merkurjev_2008_algebraic_geometric_theory_quadratic_forms/: Proves the Cassels–Pfister, Representation, Quadratic Value and Springer theorems on values and isotropy of quadratic forms over field extensions.

erdos_1943_note_farey_series/: Proves an absolute constant c exists such that Farey fractions of order n at index distance k are similarly ordered once n exceeds ck.

erdos_1961_unsolved_problems/: A survey of unsolved problems in number theory, combinatorics, set theory, geometry, analysis and probability, with known partial results and references.

erdos_1965_recent_advances_current_problems_number_theory/: A wide survey of number theory stating many of Erdos problems on prime gaps, discrepancy of sign functions, greatest prime factors and additive sequences.

erdos_1966_szamelmeleti_megjegyzesek/: A Hungarian survey updating and extending Erdos extremal problems on integer sequences, sign functions and divisibility.

erdos_1974_remarks_problems_number_theory/: Original collective and pairwise coprimality questions, with five complete elementary or relative deductions and explicit corrections to the source.

erdos_1975_problems_results_diophantine_approximations_ii/: Erdős's 1975 chapter in the Marseille-Luminy volume Répartition modulo 1, reporting progress on the questions of his 1964 Compositio paper on discrepancy, well-distributed sequences, metric Diophantine approximation and Heilbronn's triangle problem, and closing with new "disconnected problems", among them the lacunary density question of Problem 464.

erdos_1979_unconventional_problems_number_theory_math_mag/: Erdős's Mathematics Magazine list of twelve problems, among them barriers for the number of prime factors and of divisors, gaps between squarefree numbers, least common multiples of blocks of consecutive integers, the equation x to the x times y to the y equals z to the z, integers of the form a p squared plus b, and two divisors with ratio close to one.

erdos_1980_old_new_problems_results_combinatorial_number_theory/: Collects problems throughout combinatorial number theory, including the historical plane-coloring question and its necessary step qualification.

erdos_1980_survey_problems_combinatorial_number_theory/: A wide survey of Erdos's problems on progressions, primitive sequences, covering congruences and visible lattice points, mostly stating open questions.

erdos_1982_some_new_problems_results_number_theory/: Erdős's Mysore 1981 problem paper in three parts, additive number theory, prime numbers and miscellaneous problems, stating older problems only where they are hard to find, were misstated or have seen progress; p. 54 claims without proof that in any partition of the integers into k classes some class has subset sums of upper logarithmic density at least 1/2, with a block example said to cap the constant at 3/4 that in fact caps it at 14/15.

erdos_1990_characterization_unique_expansions_related_problems/: Characterizes, for a base 1 < q < 2, which expansions of one with digits 0 and 1 are the greedy and which the unique expansion, and shows that for almost every base in (1, 2) the greedy expansion of one has, for arbitrarily large m, more than log_2 m consecutive zeros in its first m digits.

erdos_1995_my_favourite_problems_number_theory_combinatorics/: Erdos's late problem collection in number theory, graph theory and geometry, with prize offers and the state of the art for each question.

erdos_1996_pisot_numbers/: Proves that for q below the golden ratio, q is Pisot exactly when the spacings of sums of powers of q with digits up to two stay bounded away from zero.

erdos_1998_sequence_numbers_form_sums_powers_q/: The 1998 Erdős-Joó-Komornik paper on the gaps of the ordered finite sums of distinct powers of q in (1, 2): a positive lower gap for every Pisot number, the bound L(q) <= (q^2 - 1)e tending to 0 as q tends to 1, the implication l(q^2) = 0 => L(q) = 0 below sqrt 2, and the explicit statement that whether L(q) = 0 for all q near 1 was not known.

erdos_komornik_1998_developments_non_integer_bases/: Erdős and Komornik's 1998 paper on the ordered finite sums of powers of q with digits 0, ..., m: Pisot numbers are the bases whose signed-digit sums do not accumulate (Theorem I), the gaps have lim inf and lim sup zero for non-Pisot q once m is large (Theorems II, III), the gaps tend to zero for every m when 1 < q <= 2^(1/4) and q is not the square root of the second Pisot number (Theorem IV, which first resolved Problem 1096), and gaps tending to zero give universal developments (Theorem V).

feng_2016_topology_polynomials_bounded_integer_coefficients/: Proves the set of values at q of polynomials with integer coefficients of absolute value at most m is dense in the reals exactly when q is under m plus one and not Pisot.

garcia_tal_1999_official/: Proves that for the Hasse maps with m > d >= 2, gcd(d,m) = 1 and m < d^{d/(d-1)} a set of representatives of eventual agreement has Banach density zero, so every orbit does; the (m,d) = (3,2) member with residues {0,-1} is the Collatz shortcut map of problem 1135.

glibichuk_2006_combinatorial_properties_sets_residues_modulo_prime/: Shows every residue mod a large prime is a sum of the inverses of 8([1/eps+1/2]+1)^2 distinct positive integers at most p^eps.

graham_pollak_1970_note_nonlinear_recurrence_related_sqrt2/: Graham and Pollak's 1970 note on the Hwang-Lin sequence a_1 = m, a_{n+1} = [sqrt(2 a_n (a_n + 1))]: the closed form a_n = [τ(2^{(n-1)/2} + 2^{(n-2)/2})] for n > 1 with τ the m-th smallest element of {1, 2, 3, ...} ∪ {√2, 2√2, 3√2, ...}, from which for m = 1 the difference a_{2n+1} − 2a_{2n−1} is the n-th binary digit of √2; the theorem behind Problem 482's first paragraph, closing with the question whether √3 and cube-root analogs behave alike.

guy_1991_western_number_theory_problems/: The problem set of the December 1991 Asilomar Western Number Theory meeting, edited by Richard K. Guy, with problems 91:01 to 91:25 and comments on earlier problems; nine of the new problems are Erdős's, among them the three-lcm question, the binomial coefficient function g(k), four Sidon questions, the unit-fraction coloring question with Graham and the gap question on sums of powers of q with Joó.

guy_2004_unsolved_problems_number_theory/: Guy's third edition of Unsolved Problems in Number Theory (2004), the 190-section problem book in six parts (primes, divisibility, additive number theory, Diophantine equations, sequences of integers, none of the above) that states results and conjectures, mostly with references and without proofs; cited on 95 problem pages, from the Erdős--Turán Sidon and basis questions of C9 and the nonaveraging sets of C16 to the barriers, totient iterates and weird numbers of part B and the 3x+1 problem of E16.

halasz_1977_estimates_concentration_function_combinatorial_number_theory_probability/: Halász's 1977 bounds on how many of the 2^n signed sums of n vectors in d-space can fall into one unit ball: 2^n n^{-d/2} for d-dimensional configurations (Theorem 1), 2^n n^{-1-d/2} when the vectors are also 1-separated (Theorem 2) and 2^n n^{-3d/2} for scattered ones (Theorem 3), from his concentration inequality for sums of independent random vectors (Theorem 4); Theorem 2 confirms Erdős's conjecture that fixing the number of plus signs in the Sárközy–Szemerédi problem gives 2^n n^{-2}.

halbeisen_hungerbuhler_1997_optimal_rational_cycle_bounds/: Proves optimal length bounds for positive rational Collatz cycles and derives a lower bound of 102225496 on the length of any nontrivial integer cycle, conditional on the Collatz conjecture being verified up to 212366032807211; bears on cycle counterexamples to problem 1135.

hercher_2023_no_mcycles_91/: Excludes Collatz m-cycles for all m <= 91, the current cycle-exclusion frontier for problem 1135.

katznelson_2001_chromatic_numbers_cayley_graphs_z_recurrence/: Katznelson's 2001 answer to Erdős's 1987 question whether the Cayley graph on the integers defined by a lacunary sequence has finite chromatic number: yes, by coloring n through the position of n alpha on the circle for a multiplier alpha keeping every lambda alpha at distance more than epsilon(rho) from 0, with the printed bound epsilon(rho) > (rho - 1)^2 log^(-2)(rho - 1) for ratio rho near 1, and a characterization of finite chromatic number by non-recurrence in compact dynamical systems.

knight_2026_high_cycles/: Excludes Collatz high cycles, a class of potential counterexample cycles for problem 1135.

kontorovich_lagarias_2009_stochastic_models/: Surveys and develops the random-walk and branching-random-walk models of 3x+1 and 5x+1 orbits, states rigorous results about the models, proving or sketching the new 5x+1 ones, and derives conjectural predictions; a chapter of the AMS volume The Ultimate Challenge: The 3x+1 Problem.

konyagin_2001_distances_between_points_plane/: Shows that for each fixed delta the largest planar point set in a disc of radius X with every pairwise distance at least delta from the nearest integer has O of X to the one half points.

korec_1994_density_estimate/: Proves that for every c > log_4 3 the set of y whose 3x+1 trajectory drops below y^c has asymptotic density 1, an almost-all result on problem 1135 that does not decide it.

kovac_2024_set_points_represented_harmonic_subseries/: The set of triples of sums of 1/n, 1/(n+1), 1/(n+2) over infinite sets of positive integers with convergent reciprocal sum has non-empty interior.

krasikov_lagarias_2003_bounds_difference_inequalities/: Proves, by a computer-aided argument, that at least x^0.84 of the integers below x reach 1 under the 3x+1 map for all large x, improving the exponent 0.81 (problem 1135).

lagarias_1985_3x1_problem_generalizations/: Lagarias's 1985 Monthly survey of the 3x+1 (Collatz) problem: its history and names, the prizes offered for it (by Coxeter, Erdős and Thwaites), Erdős's dictum "Mathematics is not yet ready for such problems" (p. 3), Terras's stopping-time theory and the density bounds of Theorems A-F, the cycle results of Theorems H-J, the 2-adic connections, and Conway's undecidability theorems; the origin of the Erdős prize the site attaches to Problem 1135.

lagarias_2003_annotated_bibliography_1/: Lagarias's annotated bibliography of the 3x+1 literature 1963-1999, which poses the 3x+1 Conjecture of Problem 1135 and records Erdős's prize problem on the maps of Problem 1134, asked for positive density, as answered in the negative, unpublished.

lagarias_2006_annotated_bibliography_2/: Lagarias's annotated bibliography of the 3x+1 literature 2000-2009, the second installment, which poses the 3x+1 Conjecture of Problem 1135 and reports the ranges to which computation has verified it.

lagarias_2010_problem_overview/: Survey of the 3x+1 (Collatz) problem covering its history, generalizations, current records and why it appears out of reach of present methods.

lagarias_2016_erdos_klarner_3x1_problem/: Lagarias's 2016 Monthly history of Erdős, Klarner and Rado on semigroups of integer affine maps: Erdős's 1972 orbit-size bound (Theorem 3), his prize problem on whether the smallest set containing 1 and closed under 2x+1, 3x+1 and 6x+1 has positive lower density, with a reconstructed proof of Crampin and Hilton's negative answer (Theorem 6), Klarner's six open free variants including Guy's E36, and the closing remark that the orbit bound is the closest Erdős came to the 3x+1 problem.

leveque_1953_uniform_distribution_modulo_subdivision/: Gives criteria for uniform distribution of a sequence relative to a general subdivision of the half-line, including almost-all results.

neklyudov_2021_functional_analysis_collatz/: Recasts cycles and divergent trajectories of the Collatz map as fixed points of a linear operator and bounds the number of cycles by an operator index; it proves no case of the Collatz conjecture.

norton_1994_frequencies_large_values_divisor_functions/: Norton's upper tail estimate supplies a precise external input for counting primitive covering numbers.

openai_2026_bounded_step_walks_gaussian_primes/: Claims the Gaussian moat conjecture (Problem 952) in a uniform form: for each step bound D the components of the graph joining Gaussian primes at distance at most D have at most B_D vertices, B_D nonexplicit; derived from a finite periodic sieve obstruction via entropy estimates on a sampled walk.

openai_2026_deterministic_polynomial_factorization_over_prime_fields/: A 48-page release manuscript claiming a deterministic algorithm that factors any polynomial over a prime field in time polynomial in the input length, by Berlekamp reduction, an even-degree pair-orientation splitter and an odd-degree splitter on cyclic covers, with the auxiliary primes it needs bounded through a cited uniform Hecke zero-free strip from a companion manuscript; it names no Erdős problem, and only its auxiliary-prime bound touches Problems 980 and 981, as background.

pollington_1979_density_sequence_n_k_xi/: Proves that for every sequence of positive numbers with consecutive ratios at least a fixed alpha above 1 and every s_0 below 1 there are a beta > 0 and an uncountable set of multipliers xi, of Hausdorff dimension at least s_0, whose fractional parts along the sequence all lie in [beta, 1 - beta], so that the xi with non-dense fractional parts form a set of dimension 1; a solution of Erdős's 1975 lacunary density question independent of de Mathan's.

sarkozi_1965_uber_ein_problem_von_erdos_und/: Proves the Erdős-Moser conjecture that the number of subset sums of n distinct positive reals hitting one value is at most about 2^n/n^{3/2}.

sarkozy_1976_distances_near_integers_i/: Proves Erdős's conjecture that the largest set of points in a disc of radius X with all mutual distances at least delta from the integers has o(X) points: at most (4 times 10^4 over delta cubed) X over log log X for large X.

sarkozy_1976_distances_near_integers_ii/: Reports Graham's construction showing N(X, 1/10) > (log X)/10 and improves it to N(X, delta) > X^(1/2 - delta^(1/7)) for small delta, so N(X, delta) tends to infinity with X; also builds infinite point sets with all mutual distances near an integer plus one half.

schmidt_1969_disproof_conjectures_diophantine_approximations/: Disproves the Davenport-Erdos uniform distribution conjecture and Croft's conjecture on infinite-measure sets containing multiples of almost every real.

shparlinski_2002_question_erdos_graham/: Shparlinski's 2002 answer to the Erdős–Graham question of Problem 1180: for every epsilon > 0, every sufficiently large prime p and every integer c there are k = 4 epsilon^{-3} + O(epsilon^{-2}) pairwise distinct integers x_i in [1, p^epsilon] whose inverses modulo p sum to c (Theorem 3), by Karatsuba's exponential-sum bounds in the form of Friedlander and Iwaniec, applied to products of two primes.

simons_de_weger_2010_mcycles_bounds/: Rules out nontrivial m-cycles of the 3n+1 map for m <= 75, direct cycle exclusion for the Collatz conjecture (problem 1135).

stanley_1980_weyl_groups_hard_lefschetz_theorem_sperner/: Uses the hard Lefschetz theorem to prove posets from algebraic varieties have the k-Sperner property, settling an Erdos-Moser conjecture.

steinberger_2012_lowest_degree_polynomial_nonnegative_coefficients_divisible_by_n_th_cyclotomic_polynomial/: A focused E0774 digest of the journal article.

stoll_2005_families_nonlinear_recurrences_related_digits/: Two infinite families of two-step floor recurrences whose differences u_{2n+1} - 2u_{2n-1} are the binary digits of an arbitrary positive real w, and one family for every integer base g >= 2 whose differences u_{2n+1} - g u_{2n-1} are the g-ary digits of w; the paper extends Rabinowitz and Gilbert's 1991 binary family, which it presents as an answer to the Erdős-Graham request behind the Graham-Pollak recurrence.

stoll_2006_problem_erdos_graham_concerning_digits/: Extends the Graham-Pollak floor recurrence generating binary digits of sqrt 2 to families of recurrences producing base-g digits of arbitrary positive reals.

tan_zhang_2026_sharp_diameter_bounds_nonnegative_cyclotomic_multiples/: A focused E0774 digest with a complete local Markdown reading copy.

tang_2025_average_first_passage_times_character_sums/: Proves that the sum over odd primes up to x of the first time a Legendre character sum drops below a linear barrier is asymptotic to c x/log x.

tang_2025_erdos_479_congruences/: An unpublished exposition of the power-of-two family in Problem 479.

tao_2014_254a_notes_1_elementary_multiplicative_number_theory/: A theorem-indexed source review of Tao's 254A Notes 1 blog post, a web page with no PDF.

tao_2019_almost_all_orbits/: Proves almost all Collatz orbits attain almost bounded values in logarithmic density, the strongest known partial result on problem 1135.

terras_1976_stopping_time_problem/: The founding density result on problem 1135: almost all n in natural density have finite 3x+1 stopping time, with a limiting stopping-time distribution.

various_1999_some_pauls_favorite_problems/: Preserves the conference booklet, its local interpolation question, and its original random-walk questions.

vinogradov_1948_estimate_trigonometric_sums_prime_numbers/: Proves general bounds for exponential sums over primes when a polynomial coefficient has a good rational approximation whose denominator is at least a fixed power of P.

voight_2021_quaternion_algebras_over_global_fields/: Classifies quaternion algebras over global fields by even ramification sets, proving the rational case via Hilbert reciprocity and Hasse–Minkowski.

wang_2026_exact_formula_erdos_problem_1005/: A 2026 arXiv preprint, with declared AI assistance, claiming that the Problem 1005 function equals van Doorn's upper bound floor(n/4) + d, with d = 1, 2, 2, 4 for n = 0, 1, 2, 3 mod 4, for all sufficiently large n and, with a computer verification, its exact value for every n >= 4, the bound failing only at fifteen listed n up to 91; a proof claim on the site's tab, recorded as a lead, not refereed.

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