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

Updated Oct 8

Irrationality and Diophantine Approximation

badea_1987_irrationality_certain_infinite_series/: Gives an irrationality criterion for series of positive rationals and uses it to settle two Erdos-Graham questions on Fibonacci and Lucas reciprocals.

badea_1993_theorem_irrationality_infinite_series_applications/: Extends an earlier irrationality criterion for series of positive rationals and applies it to reciprocal sums of second-order recurrence sequences.

barreto_2026_irrationality_rapidly_converging_series_problem_erdos/: Proves that growth faster than double-exponential at the golden-ratio rate forces the sum of 1/(a_n a_{n+1}) to be irrational, and shows this is sharp.

bell_2026_mahler_series_multiplicative_coefficients/: Characterizes multiplicative coefficient sequences of Mahler series; its totient specialization excludes Mahler equations but gives no arithmetic conclusion about the value at one half.

bloom_2026_erdos_problem_251_discussion/: Web source for the catalog's Problem 251 page, discussion thread, proof-claims page and history as snapshotted on 2026-09-17, with Tao's gap-series reformulation proved and the 2026 claims recorded as their authors state them.

boca_2008_problem_erdos_szusz_turan_diophantine_approximations/: Boca's paper identifies the limiting measure in the Erdős–Szüsz–Turán approximation problem for all parameters, according to its abstract.

borwein_1991_irrationality_1_qn_r/: Proves the sum of 1/(q^n-r) is irrational for integer q>1 and admissible rational r by Padé approximants to the q-logarithm, a linear-form template with no factorial analogue for E264.

borwein_1992_irrationality_certain_series/: Proves the complete series of 1/(q^n+r) and its alternating form are irrational for integer |q|>1 and nonzero rational r other than -q^n; the first case gives the E257 sum over a cofinite set or a single arithmetic progression.

campbell_2026_binary_digits_erdos_borwein_constant/: Proves the string 11 occurs infinitely often in the base-two expansion of the Erdos-Borwein constant.

cassels_1950_some_metrical_theorems_diophantine_approximation_i/: Bibliographic record for Cassels's 1950 work on metric Diophantine approximation, cited as a precursor to Erdős Problem 1000.

chen_ruzsa_1999_irrationality_certain_series/: Bibliographic record for the Chen--Ruzsa paper cited on Erdős Problem 259; its full text and exact theorem have not been checked here.

crmaric_2025_irrationality_certain_super_polynomially_decaying_series/: Answers an Erdos-Graham question negatively: every positive real, rational ones included, is the sum of the reciprocal-product series for some f tending to infinity.

duverney_1993_proprietes_arithmetiques_serie_fonctions_theta/: Proves that the theta-type series of q to the minus n squared is not quadratic for integer q of absolute value at least 2, using an irrationality criterion by partial sums that Duverney 1995 reuses.

duverney_1995_irrationalite_q_analogue_zeta_2/: Proves that the q-analog zeta(q;2), equal to (q-1)^2 times the sum of sigma(n) over q^n, is irrational for every integer q other than -1, 0 and 1; the case q = 2 is the divisor-sum series of problem 250.

duverney_2001_irrationality_fast_converging_series_rational_numbers/: Irrationality of fast converging series of rational numbers.

duverney_tachiya_2019_refinement_chowla_erdos_method_linear_independence_certain_lambert_series/: Refines the Chowla–Erdős method into a divisibility criterion against rational Lambert series and proves linear independence over exponent sets such as the squarefree integers, settling E257 for them.

einsiedler_2006_invariant_measures_set_exceptions_littlewood_s/: Shows that for k >= 3 every ergodic measure on SL(k,R)/SL(k,Z) invariant under the positive diagonal group, with positive entropy for some one-parameter subgroup, is algebraic, and that the exceptional set of Littlewood's conjecture has Hausdorff dimension zero.

erdos_1948_arithmetical_properties_lambert_series/: Proves that the Lambert series sum of one over t to the n minus one is irrational for every integer t above one, and asserts without details the same for its sine analog and for t below minus one.

erdos_1957_irrationality_certain_series/: Proves the exponent variants (one over t to the phi(n), one over t to the sigma(n)) irrational and bounds the algebraic degree of sparse power series, while restating the totient and divisor-sum series as unproved.

erdos_1958_remarks_theory_diophantine_approximation/: Bounds the measure of the set of reals whose good rational approximations have denominators confined to a short range.

erdos_1958_sur_certaines_series_valeur_irrationnelle_french/: Proves that the sum of p_n over n factorial is irrational, asserts the same for every power of p_n without proof, and characterizes the rational values of the prime series over monotone denominators growing at least like n over a power of log n.

erdos_1964_irrationality_certain_ahmes_series/: Shows that rapidly growing reciprocal sums are rational only when the denominators eventually follow the recurrence n squared minus n plus one.

erdos_1969_irrationality_certain_series/: Proves the sum of one over t to the n_i minus one is irrational for every integer t at least 2 when the n_i are pairwise coprime with convergent reciprocal sum.

erdos_1971_number_theoretic_results/: Fixes the order of the largest residue sets mod p in which subsets of different sizes have different sums, and proves irrationality of series built from divisor and other arithmetic functions.

erdos_1974_irrationality_certain_series/: Gives an exact rationality criterion for series of b_n over the product of a_1 through a_n, derives the irrationality of the prime series over monotone denominators, and proves the rational independence of one and the totient, divisor-sum and small-numerator series.

erdos_1976_problems_results_irrationality_sum_infinite_series/: Proves irrationality and Liouville criteria for sums of reciprocals of fast-growing integer sequences, and lists many related open problems.

erdos_1981_sur_l_irrationalite_d_une_certaine/: Proves that the sum of a-n over two to the a-n is irrational for every integer sequence whose consecutive gaps tend to infinity.

erdos_1988_irrationality_certain_series_problems_results/: Survey of Erdos's irrationality results for rapidly growing series, with many open problems and a new theorem on sums of reciprocal least common multiples.

friedlander_2007_irrationality_divisor_function_series/: Proves that the sum over n of sigma_3(n)/n! is irrational, and that the prime k-tuples conjecture gives irrationality for all k >= 4.

fruhwirth_2024_birkhoff_sums_that_satisfy_no_temporal/: For almost every irrational rotation and every starting point, Birkhoff sums of a piecewise smooth function obey no temporal limit theorem.

good_1974_reciprocal_series_fibonacci_numbers/: Evaluates the sum of reciprocals of Fibonacci numbers with index a power of two in closed form as (7 minus root 5)/2.

grepstad_lev_2014_bounded_discrepancy_rotation/: Characterizes the Riemann measurable bounded remainder sets for multi-dimensional irrational rotation, the direct generalization of the one-dimensional Hecke-Ostrowski-Kesten characterization of intervals by their length, the case that problem 998 asks about.

hancl_1991_expression_real_numbers_help_infinite_series/: Shows a sequence growing slower than doubly exponentially at rate one is a rational sequence, so irrationality sequences must grow at least that fast.

hancl_2004_irrationality_cantor_series/: Gives irrationality and exact rationality criteria for Cantor series, including the prime series over monotone denominators with a_n over log n tending to infinity, the factorial series with increments o(n), and the sums of p_n to the k over 2 to the p_n.

hancl_2005_irrationality_factorial_series/: Decides exactly which integer polynomials P make the sum of P(N) over N factorial rational, proves irrationality for many factorial series with smooth numerators, and records that Erdős proved the prime power factorial series irrational only for k equal to one.

hancl_2010_irrationality_factorial_series_ii/: Proves irrationality for factorial series whose numerators behave like a geometric progression along long stretches, including the linear independence of one and the sums of pi(n) to the m over n factorial and elementary proofs that e to the m and pi are irrational, and records that Schlage-Puchta confirmed Erdős's claim for the prime powers.

hoggattjr_1976_reciprocal_series_fibonacci_numbers_subscripts/: Evaluates the sum of reciprocals of Fibonacci numbers with subscripts 2^n k in closed form, generalizing the Millin-Good value (7 - sqrt 5)/2.

kaneko_2026_refinements_erdos_s_irrationality_criterion_certain/: Proves new irrationality criteria for sparse power series at Pisot and Salem bases, giving Erdos's omitted criterion a full proof and new applications.

kesten_1960_uniform_distribution_mod_1/: Bibliographic record for Kesten's 1960 paper cited in Erdős's question about logarithmically normalized sums of fractional parts.

kesten_1966_two_problems_erdos_szusz_turan/: Shows two limiting measures of Erdős, Szüsz and Turán exist, and evaluates the one counting convergents with denominator in a given range.

koukoulopoulos_2020_duffin_schaeffer_conjecture/: Proves the Duffin-Schaeffer conjecture in metric Diophantine approximation and deduces Catlin's conjecture as a corollary.

kovac_2024_several_irrationality_problems_ahmes_series/: Settles several Erdos questions on series of distinct unit fractions, among them Stolarsky's conjecture, the interior question in all dimensions and the Type 3 case 2^n; the Type 2 case 2^(2^n) and the Type 3 case n! stay open.

kovac_2026_erdos_problem_251/: Claims a counterexample to the auxiliary statement quoted with problem 251, that sums of p_n over g_1...g_n must be irrational whenever g_n is at least 2 and grows slower than p_n, by a telescoping construction with sum exactly 1; two pages, AI-generated by its byline, unrefereed, unreviewed.

land_2026_conditional_proof_irrationality_prime_series/: Records a claimed, unreviewed 2026 research draft asserting that Kuperberg's uniform Hardy–Littlewood conjecture implies the irrationality asked by problem 251, with the author's AI attribution, the author-run Lean report and the manuscript's own disclaimers.

laursen_2024_transcendence_certain_sequences_algebraic_numbers/: Gives new irrationality and transcendence criteria for sequences of algebraic numbers in a fixed number field, using Schmidt's Subspace Theorem.

margulis_1989_indefinite_quadratic_forms_unipotent_flows/: Indefinite quadratic forms: the distinct 1989 Banach Center paper and the #496 positive-parameter deduction.

nesterenko_1996_modular_functions_transcendence_questions/: Proves that for every complex q with 0 < |q| < 1 at least three of q, P(q), Q(q), R(q) are algebraically independent, so the Ramanujan function values at algebraic q, among them 1 - 24 times the sum of sigma(n) over 2^n, are transcendental.

nguyen_2022_transcendental_series_reciprocals_fibonacci_lucas_numbers/: Proves transcendence of Fibonacci and Lucas reciprocal subseries when the ratio of successive indices is at least a fixed constant greater than two.

openai_2026_weak_inhomogeneous_duffin_schaeffer_conjecture/: An 87-page manuscript of the OpenAI mathematics release claiming the weak inhomogeneous Duffin–Schaeffer conjecture for every fixed real shift, by a finite-block second-moment construction with adaptive prime-step weights; a shifted variant of Problem 999, recorded as a claim and unverified here.

ostrowski_1927_mathematische_miszellen/: Shows intervals of length equal to a fractional part of a multiple of alpha have bounded remainder for the sequence of fractional parts of n*alpha.

ostrowski_1930_mathematische_miszellen/: Reproves and extends the bounded-remainder theorem for fractional parts of multiples of alpha and bounds discrepancy by the approximation exponent.

peres_2010_two_erdos_problems_lacunary_sequences_chromatic/: Shows every lacunary sequence with ratio 1+e admits a theta whose multiples avoid the integers by c*e/|log e|, bounding a chromatic number.

postelmans_2007_irrationality_zeta_q_1_zeta_q_2/: Proves that 1, zeta_q(1) and zeta_q(2) are linearly independent over the rationals for q the reciprocal of an integer at least 2, by simultaneous Hermite-Padé approximation with multiple little q-Jacobi polynomials.

pratt_2022_irrationality_divisor_function_series_erdos_kac/: Proves unconditionally that the sum of sigma_4(n)/n! is irrational, the case k = 4 of the Erdos-Kac conjecture, whose cases k at most 3 were known.

pratt_2024_irrationality_prime_factor_series_under_prime/: Shows that a uniform quantitative prime k-tuples conjecture implies the sum of omega(n)/t^n is irrational for every integer t at least 2.

ringer_2026_local_gap_statistics_telescoping_normality/: Records a claimed, unreviewed 2026 manuscript asserting that Kuperberg's uniform prime-tuples conjecture implies normality, hence irrationality, of the dyadic prime series of problem 251, with the author's AI attribution, the site's partial proof claim and the author-run Lean report.

schlagepuchta_2006_irrationality_number_theoretical_series/: Proves the series of divisor-power sums over factorials is irrational for every exponent under Schinzel's hypothesis H, and unconditionally for exponent 3.

schlagepuchta_2011_irrationality_number_theoretical_series/: Proves that one and the sums of p_n to the k over n factorial, for all k at least zero, are linearly independent over the rationals, giving the cases k at least two that Erdős stated in 1958 and that, by the paper's account, had no proof in print.

smet_2009_irrationality_proof_q_extension_zeta_2/: Proves that zeta_q(2) is irrational with irrationality measure at most 10 pi^2/(5 pi^2 - 24), about 3.8936, for q the reciprocal of an integer at least 2, by Hermite-Padé approximation with little q-Jacobi polynomials.

tijdeman_2002_rationality_cantor_ahmes_series/: Gives exact rationality criteria for Cantor series of b_n over a_1 through a_n with monotone a_n and increments b_(n+1) minus b_n of size o(a_(n+1)), and for Ahmes series, and shows such criteria fail without growth restrictions.

tokengrinder_2026_erdos_252_irrationality_factorial_divisor_sum/: A public Lean 4 development proving that the sum of sigma_k(n)/n! is irrational for every natural k, retained at its pinned commit, rebuilt and kernel-replayed here, and its whole statement found faithful by a graded fresh-context review.

vandehey_2012_incomplete_argument_erdos_irrationality_lambert_series/: Completes an incomplete argument of Erdos, proving the Lambert series of the divisor function is irrational at 1/b for integers b below -1.

waldschmidt_1997_nature_arithmetique_valeurs_fonctions_modulaires/: Bourbaki seminar exposé of Nesterenko's theorem that q, P(q), Q(q), R(q) contain three algebraically independent numbers, with its consequences for elliptic periods, pi and e to the pi, and theta values.

wang_2026_positive_dyadic_density_rational_weighted_binary/: Claims that the support of any rational weighted binary expansion has positive dyadic density, and deduces the irrationality asked for in Erdos Problem 260.

xiong_2006_problem_erdos_szusz_turan_diophantine/: Reproves existence of the Erdos-Szusz-Turan limiting measure for all parameters and shows the mass is uniformly spread over every subinterval.

zudilin_2002_irrationality_measure_q_analogue_zeta_2/: Proves that the q-analog zeta_q(2), the sum of sigma(n) q^n, is irrational with irrationality measure at most 4.0787 whenever 1/q is an integer other than 0 and plus or minus 1; at 1/q = 2 a further proof for the divisor-sum series of problem 250.

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