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

Updated Oct 8

Unit Fractions

alekseyev_2019_partitions_into_squares_distinct_integers_whose/: Proves Graham's conjecture with the exact threshold: every integer above 8542 is a sum of squares of distinct integers whose reciprocals sum to 1.

bettin_2025_lower_bound_number_egyptian_fractions/: Improves the lower bound for how many distinct rationals are sums of distinct unit fractions with denominators up to N.

bleicher_1975_number_distinct_subsums_sum_n_1/: Improves the lower bound for the number of distinct subsums of the first N terms of the harmonic series.

bleicher_1976_denominators_egyptian_fractions/: Bounds the least possible largest denominator in a unit-fraction expansion, showing it is at most about b times log b cubed.

bleicher_1976_denominators_egyptian_fractions_ii/: Sharpens the bounds of part I on the least possible largest denominator D(N) of an Egyptian fraction expansion, to about N times log N squared from above and P log P log log P from below at primes, and bounds the number S(N) of distinct subsums of the harmonic sum in both directions.

bloom_2021_density_conjecture_about_unit_fractions/: Proves every set of naturals of positive upper density has a finite subset whose reciprocals sum to one, answering Erdos and Graham.

bloom_2022_egyptian_fractions/: Surveys unit fraction representations, including the Erdos-Straus conjecture, its congruence reformulation, solution counts and bounds on exceptions.

breteche_tenenbaum_2024_mean_values_arithmetic_functions_application_sums_powers/: Bounds mean values of arithmetic functions at polynomial arguments and shows at least about x/(log log x)^(5/2) integers up to x are sums of powers with prescribed exponents, giving no bound for Problem 301.

bright_2020_brauer_manin_obstruction_erdos_straus/: Shows no Brauer-Manin obstruction blocks the Erdős-Straus conjecture, but derives Hilbert symbol conditions all natural number solutions satisfy.

brown_1991_monochromatic_solutions_equations_unit_fractions/: Shows any homogeneous system partition regular over the positive integers is also partition regular over reciprocals, giving monochromatic unit-fraction sums.

butler_2015_egyptian_fractions_each_denominator_having_three/: Proves every natural number is a sum of distinct unit fractions whose denominators are each a product of three distinct primes.

chu_2023_threshold_best_two_term_underapproximation_egyptian/: Shows the greedy Egyptian-fraction algorithm gives the unique best two-term underapproximation of p/q whenever q+j is divisible by p for some j at most 3 (except 10/17, where it is best but tied), and that for each larger least such j it fails for some p/q.

conlon_2024_question_erdos_graham_egyptian_fractions/: Proves the exact exponential counting rate through entropy and repaired modular absorption, relative to explicit external inputs.

croot_1999_unit_fractions_denominators_short_intervals/: Proves every positive rational r is a sum of distinct unit fractions with denominators in an interval just past N, ending near e to the power r times N with a best-possible error.

croot_2003_coloring_conjecture_about_unit_fractions/: Proves the Erdos-Graham conjecture that any r-coloring of the integers in an interval up to b to the r has a monochromatic set of reciprocals summing to one.

crootiii_1999_questions_erdos_graham_about_egyptian_fractions/: Shows every integer up to the harmonic sum minus about (9/2)(log log x)^2 / log x is a sum of distinct unit fractions with denominators at most x.

curtiss_1922_kellogg_s_diophantine_problem/: Proves that the largest denominator in a unit-fraction representation of 1 with n terms is at most u_n, one less than the n-th Sylvester number, and that the least positive value of 1 minus n - 1 unit fractions is 1/u_n, attained only at the denominators u_k + 1.

doorn_2024_non_monotonicity_denominator_generalized_harmonic_sums/: Answers an Erdos-Graham question by showing that the reduced denominator of a harmonic-type block sum starting at a drops at some later endpoint, the first drop lying at least order log a past a and below a constant times a.

doorn_2025_lacunary_sequences_whose_reciprocal_sums_represent/: Disproves a Bleicher-Erdos conjecture by constructing lacunary integer sequences whose finite reciprocal sums hit every rational in an interval.

doorn_2025_partitions_prescribed_sum_reciprocals_asymptotic_bounds/: Gives the first, near-optimal bounds on how large an integer must be to admit a distinct-part partition whose reciprocals sum to a given rational.

doorn_2025_smallest_denominator_not_contained_unit_fraction/: Proves the first lower bound, exponential in the square of the length, for the smallest integer missing from all k-term unit fraction decompositions of one.

doorn_2025_two_coloring_density_solutions_unit_fraction_equation/: Shows that every two-coloring of the first n integers has at least n/390 minus a cube of a logarithm minus one monochromatic distinct solutions of 1/x + 1/y = 1/z, and that every subset of the first n integers of size at least 9n/10 plus a cube of a logarithm plus one contains such a solution.

doorn_2026_shortest_harmonic_sums_decreasing_denominator/: Claims, in a 2026 preprint, that the limit inferior of (b(a) - a)/log a for the first denominator drop of consecutive reciprocals equals 1/(1+c), the lower bound of the author's 2024 paper, with a language model credited for one step.

elsholtz_2013_counting_number_solutions_erdos_straus/: Bounds the average number of representations of 4/n as a sum of three unit fractions, to within a log log N factor over primes p <= N, showing typical primes have few solutions.

elsholtz_2016_egyptian_fractions_odd_denominators/: Proves a doubly exponential lower bound exp(exp(c k / log k)) for the number of representations of 1 by k distinct odd unit fractions (k odd and large), and more generally by denominators congruent to plus or minus 1 modulo a squarefree P, by a construction independent of Konyagin's identities.

elsholtz_2020_number_solutions_erdos_straus_equation/: Proves new upper bounds on the number of ways a rational can be written as a sum of three or of k unit fractions, and better lower bounds.

elsholtz_2021_sums_four_more_unit_fractions_approximate/: Improves upper bounds on the number of representations of a rational as a sum of four, and hence of more, unit fractions.

erdos_1932_egy_kurschak_fele_elemi/: Proves that the sum of reciprocals of any arithmetic progression of at least two positive integers is never an integer.

erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine/: Bounds the fewest distinct unit fractions summing to a given rational, with an upper bound of order log b over log log b.

graham_1963_theorem_partitions/: Proves that every integer greater than 77 is a sum of distinct positive integers whose reciprocals sum to 1, extends this to any positive rational reciprocal sum with denominators above any bound, and conjectures the polynomial version.

graham_1964_finite_sums_reciprocals_distinct_nth_powers/: Characterizes exactly which rationals are finite sums of reciprocals of distinct nth powers, with explicit criteria for squares and cubes.

graham_1964_finite_sums_unit_fractions/: Characterizes, when M(S) is complete and s_{n+1}/s_n is bounded, the reduced rationals that are finite sums of reciprocals of distinct terms of M(S), and states without proof applications including an arithmetic-progression criterion that generalizes the Stewart-Breusch odd-denominator theorem.

kamio_2025_asymptotic_analysis_infinite_decompositions_unit_fraction/: Solves an Erdos-Graham problem on infinite unit-fraction decompositions by showing the Sylvester sequence is asymptotically extremal.

koizumi_2025_irrationality_reciprocal_sum_doubly_exponential_sequences/: Shows positive-integer sequences with every a_n^2/a_{n+1} within 1/3 of a fixed beta >= 0 are nearly determined by their reciprocal sums, giving irrationality for all but countably many doubly exponential growth rates.

konyagin_2014_double_exponential_lower_bound_number_representations/: Proves a doubly exponential lower bound on the number of ways to write 1 as a sum of n distinct unit fractions.

korsky_2026_stretched_exponential_bound_erdos_graham_unit/: Proves that a multiset of integers with reciprocal sum above K has a reciprocal subsum within exp(-c sqrt(K log K)) of one.

kovac_2024_eventually_greedy_best_underapproximations_egyptian_fractions/: The set of positive reals whose best n-term Egyptian underapproximations are eventually built greedily has Lebesgue measure zero.

kovac_2026_eventually_greedy_best_egyptian_underapproximations_rational/: Claims, in a 2026 preprint, that every positive rational has eventually greedy best Egyptian underapproximations in both denominator conventions, by an optimal-control reformulation with a Bellman function.

larsen_2026_sufficiently_abundant_numbers_pseudoperfect/: Proves that every integer whose abundance index exceeds 2 + epsilon and that has no prime factor below a bound depending on epsilon is a sum of distinct proper divisors, hence that an absolute constant C makes every n with sigma(n) at least Cn such a sum, and that every two-part partition of the squares above one has non-empty finite subsets of both parts with equal reciprocal sums.

li_2025_conjecture_erdos_graham_about_sylvester_s/: Proves the Erdős–Graham conjecture that every increasing integer sequence other than Sylvester's with reciprocal sum one has liminf of a_n^(1/2^n) below Sylvester's limit, and a generalization to rationals with unique best underapproximations, conditional on eventually greedy best Egyptian underapproximations.

lim_2024_differences_two_harmonic_numbers/: Shows that sums of reciprocals over a single interval can exceed 1 by as little as o(1/n squared), answering a question of Erdos and Graham.

liu_2024_further_questions_regarding_unit_fractions/: Gives a reciprocal-mass threshold with exponent four-fifths plus epsilon and treats counting, cardinality, and denominator bounds.

louwsma_martino_2023_rational_numbers_odd_greedy_expansion_fixed_length/: Characterizes the odd denominator lists that are exactly the odd greedy expansion of their sum and classifies fixed-length expansions by prefix or numerator, leaving termination in Problem 282 open.

louwsma_martino_2025_rational_numbers_two_term_odd_greedy_expansion/: Classifies the rationals whose odd greedy expansion has exactly two terms as denominator progressions for each even numerator, giving Problem 282 a two-step terminal test but no termination proof.

martin_1998_dense_egyptian_fractions/: Shows every positive rational has an Egyptian fraction representation whose denominators form a positive proportion of the integers up to the largest one.

martin_2000_denser_egyptian_fractions/: Gives best-possible bounds for the densest Egyptian fraction representations and settles Erdős-Graham questions on admissible denominators.

martin_shi_2021_algorithm_egyptian_fraction_representations_restricted_denominators/: Gives an exact algorithm listing every submultiset of a finite multiset of positive integers whose reciprocals sum to a given rational, and uses it to compute densest representations of small integers.

mihnea_2025_further_verification_empirical_evidence_erdos_straus/: Reports a computation verifying the Erdős-Straus conjecture for all primes up to 10^18 and an empirical evaluation of its solution-counting function.

montgomery_1979_solution_problem_e2689_egyptian_fractions/: Montgomery's 1979 solution to Hahn's Monthly problem E2689, which asked for a nonempty finite set of positive integers, each with a neighbor in the set, whose reciprocals sum to an integer: two such sets, each a union of separated blocks of consecutive integers with reciprocal sum 2, the second also found by Hickerson.

nathanson_2023_underapproximation_egyptian_fractions/: Studies greedy Egyptian-fraction underapproximation, characterizing greedy sequences and finding rationals whose greedy underapproximations are best.

openai_2026_short_egyptian_fractions/: A 33-page manuscript of the OpenAI mathematics release claiming that N(b), the largest minimum length of a distinct unit-fraction expansion over b, has order log log b (Problem 304) by a divisor-selected descent, with the consequences log log F(k) and log log v(k) of order k (Problems 148, 293).

pihko_2001_remarks_greedy_odd_egyptian_fraction_algorithm/: States the open problem whether the greedy odd Egyptian fraction algorithm always stops for a reduced fraction with odd denominator, shows that every prescribed number of steps occurs for infinitely many fractions, and builds families whose numerators rise in the first steps.

pihko_2010_remarks_greedy_odd_egyptian_fraction_algorithm_ii/: Restates the open problem whether the greedy odd Egyptian fraction algorithm always stops, and shows that for every odd prime p and 1 < a < p infinitely many odd denominators b make the numerator sequence run a, a+1, ..., p-1, 1.

pomerance_2025_exceptions_erdos_straus_schinzel/: Shows Schinzel's threshold for writing m/n as three unit fractions must exceed exp(m^(1/3+o(1))), with explicit versions.

ruderman_1971_e2232_representation_1_egyptian_fractions_problem/: Bounds the fewest distinct unit fractions with largest term at most 1/n summing to 1 between (e-1)n minus a constant and (e-1)n plus a constant times n over log n.

sawin_2026_sets_unit_fractions_without_two_members_average_unit_fraction/: Constructs, for all large N, a subset of the first N integers of positive density in which a + b never divides 2ab for distinct members, answering the second question of Problem 327 in the negative; an arXiv preprint.

shiu_2016_denominators_harmonic_numbers_revised/: Studies the reduced denominator of the nth harmonic number, proving non-monotonicity and a harmonic density for sets where primes divide the shortfall.

steinerberger_2024_problem_involving_unit_fractions/: Proves an eventual 2^(0.93n) upper bound for the number of subsets with reciprocal sum at most one, using a split exponential-moment product.

tang_2026_note_problem_311/: Proves the upper bound delta(N) <= exp(-c N/((log N)^3 (log log N)^3)) for the least distance from 1 of the reciprocal sum of a set of integers up to N that has no subset with reciprocal sum 1, by strengthening a representation lemma of Liu and Sawhney.

terzi_1971_conjecture_erdos_straus/: Terzi's 1971 note on the Erdős–Straus conjecture: a first algorithm that, from Rosati's parametrization, lists the residue classes modulo M in which a prime n might lack a representation of 4/n as three unit fractions (six classes modulo 840, 34 modulo 9240, 198 modulo 120120), and a second algorithm with which the conjecture is stated proved for all n up to 10^8.

vaughan_1970_problem_erdos_straus_schinzel/: Vaughan's 1970 large-sieve bound on the exceptional set of the Erdős–Straus–Schinzel problem: for each fixed positive integer a, the number of n up to N for which a/n is not a sum of three unit fractions is at most a constant times N exp(-(log N)^(2/3)/C(a)), so almost every n, and for a = 4 almost every n in the Erdős–Straus conjecture, has a representation.

wang_2026_667_806_upper_bound_erdos_problem/: Claims to improve the recorded upper bound for the largest unit-fraction-free subset of the first N integers from 25/28 to 667/806 of N.

wang_2026_port_fillings_primary_pseudoperfect_numbers/: Constructs two new primary pseudoperfect numbers with nine and ten prime factors and proves infinitude only under an unproved prime-points hypothesis.

watanabe_2020_new_examples_representation_1_sum_reciprocals/: Finds 17 representations of 1 as a sum of 47 reciprocals of products of two distinct primes, beating the previous 48-term record.

webb_1965_sums_rational_numbers/: Proves that every rational is a finite sum of reduced fractions with distinct numerators from a set with infinitely many disjoint coprime pairs and distinct denominators, and that a positive reduced rational with odd denominator is such a sum with numerators and denominators in prescribed arithmetic progressions.

wu_2022_denominators_harmonic_numbers_iv/: Proves, assuming that the reciprocals of the logarithms of distinct primes are linearly independent over the rationals, that the integers n whose harmonic-number denominator is smaller than lcm(1, ..., n) have upper asymptotic density 1.

yokota_1997_number_integers_representable_sum_unit_fractions_ii/: Yokota's 1997 theorem that the number |N(n)| of integers that are sums of reciprocals of distinct integers at most n satisfies (1 − 5 log log n/log n) log n ≤ |N(n)| < (1 + 1/log n) log n for large n, so |N(n)| ~ log n; the proof opens by stating that every positive integer up to log n − 5 log log n is such a sum, a range its printed last step (p. 168) does not reach; Croot's 1999 Main Theorem takes its small integers from this paper with its 1998 Corrigendum.

yokota_2002_number_integers_representable_sums_unit_fractions_iii/: Yokota's 2002 lower bound for the number |N(n)| of integers that are sums of reciprocals of distinct integers at most n: for large n, |N(n)| ≥ log n + γ − (π²/3 + o(1))(log log n)²/log n, improving Croot's constant 9/2, with the inverse form F(a) ≤ exp[a − γ + (π²/3 + o(1)) (log a)²/a] for the least n with a in N(n), both proved for representations whose denominators lie in a prescribed divisor set.

yuan_2025_seed_prover_lean_proof_erdos_problem_303/: Gives Seed-Prover's Lean proof of Problem 303, posted by Zheng Yuan in December 2025, through a Ramsey argument on differences, factorial inverse scaling, and the parametrization of three-term unit-fraction solutions.

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