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

Updated Oct 8

Arithmetic Functions

adamczewski_2026_erdos126/: Gives the signed-laminar and two-copy matching proof of the square-root prime-support bound for pairwise sums in Erdős Problem 126.

atherfold_2025_almost_sure_bounds_weighted_sums_rademacher/: Gives almost sure upper and lower bounds for weighted partial sums of Rademacher random multiplicative functions.

baker_1998_shifted_primes_without_large_prime_factors/: Shows many primes p up to x have p-a free of prime factors above x^0.2961, improving earlier exponents and the Carmichael number count.

balog_1998_strings_consecutive_integers_no_large_prime_factors/: Proves that for every fixed u above 1 there are infinitely many n followed by a string of about log log log log n consecutive integers all free of prime factors above n to the power one over u, with a companion result for several linear forms.

banks_2005_nonaliquots_robbins_numbers/: Gives the first explicit density bounds for nonaliquot numbers, at least x/48, and shows Robbins numbers have lower density at least 1/3.

banks_2018_counting_integers_smooth_totient/: Fixes a gap in an earlier upper bound for the count of n up to x whose totient is y-smooth, obtaining a stronger and likely optimal estimate.

benli_2023_sums_proper_divisors_missing_digits/: Proves the Erdős--Granville--Pomerance--Spiro preimage conjecture for sets defined by restricting the allowed base-g digits, when g is at least three.

benli_2026_digits_sum_proper_divisors/: Studies typical digits of the sum of proper divisors and sharpens missing-digit preimage bounds after composite inputs are separated from primes.

bhalla_2026_conditional_note_large_prime_factors_polynomial_products/: Derives the degree-scale bound for the greatest prime factor of an irreducible polynomial's running product from a prime-values hypothesis weaker than Bateman--Horn.

bober_2020_smooth_values_polynomials/: Shows every integer quadratic takes values whose largest prime factor is at most any fixed power of the argument, infinitely often.

browkin_1995_integers_not_form/: Answers a question of Sierpiński by proving that no number 2^k times 509203 is of the form n minus Euler's totient of n.

caich_2023_almost_sure_upper_bound_random_multiplicative/: Proves that partial sums of a random multiplicative function are almost surely at most root x times a small power of the iterated logarithm.

cambie_2025_erdos_problem/: Determines the longest sequence of integers up to n along which the largest prime factor strictly decreases, up to a constant factor.

chen_2011_nonaliquot_numbers/: Raises the lower bound for the count of untouchable numbers up to x to 0.06x + o(x), improving the previous bound of x/48.

cohen_1996_iterating_sum_divisors_function/: Tabulates numbers n with the m-th iterate of the sum-of-divisors function equal to kn and tests six statements listed by Erdos, Granville, Pomerance and Spiro.

cuevas_barrientos_2025_greatest_prime_factor_polynomial_values_subexponential_szpiro_families/: Gives a subexponential Szpiro bound for one-parameter elliptic families and pointwise radical and greatest-prime-factor bounds for specified quadratic and cubic polynomial values.

dartyge_maynard_2025_largest_prime_factor_quartic_polynomial_values_cyclic_dihedral/: Gives a positive-proportion fixed-power prime-factor bound for monic irreducible quartics with cyclic or dihedral Galois group.

dekoninck_2011_distance_between_smooth_numbers/: Studies the distance from an integer to the nearest number no rougher than itself, giving heuristics and bounds for sums of that distance and its reciprocal.

erdos_1934_problem_elementary_theory_numbers/: Proves elementarily that, for any k primes, every set of 3 times 2 to the power k-1 positive integers has a pairwise sum with a prime factor outside those k primes.

erdos_1935_normal_number_prime_factors_related_problems/: Shows p-1 normally has about log log p prime factors and bounds how often integers are values or repeated values of Euler's function.

erdos_1936_problem_chowla_related_problems/: Proves Chowla's conjecture that the integers m with d(m+1) > d(m) have density 1/2, and the analog for additive functions f >= 0 with the sum of f(p)/p convergent.

erdos_1937_note_number_prime_divisors_integers/: Shows that n/2 + o(n) of the integers up to n have more than log log n distinct prime factors, so that log log n is asymptotically their median count.

erdos_1946_distribution_function_additive_functions/: Finds distribution functions for additive functions and their fractional parts, shows that an additive function taking close values on a positive proportion of integers is c log m plus an additive function with sum_p f'(p)^2/p finite, and proves f = c log m for nondecreasing f.

erdos_1952_greatest_prime_factor/: Presents an iterated-logarithmic improvement to Nagell's $x\log x$ bound for prime factors of polynomial products, and a stronger unproved display.

erdos_1955_amicable_numbers/: Proves the amicable numbers have density zero, and states that the method could bound their count below n by a constant times n over log log log n.

erdos_1967_problems_prime_factors_consecutive_integers/: Poses and partly settles elementary questions on prime factors of consecutive integers, proving v_0(n) > 1 for all n outside an explicit finite list.

erdos_1973_uber_die_zahlen_der_form_und/: Proves that the integers not representable as sigma(n)-n have positive lower density, while the analogous question for n-phi(n) was then open.

erdos_1974_distribution_numbers_form_sigma_n_n/: Gives a best-possible bound on the number of integers up to x with sigma(n)/n in a short interval, sharpening earlier counts.

erdos_1978_largest_prime_factors/: Shows the largest prime factors of consecutive integers are almost never close, and deduces that the Aaron numbers have density zero.

erdos_1979_unconventional_problems_number_theory/: Proves that the product-of-exponents function d_0 has infinitely many barriers, a set of positive density, and poses problems on iterated arithmetic functions and sieves.

erdos_1984_two_unconventional_number_theoretic_functions_related/: Studies two arithmetic functions built from the powers, up to n, of the primes dividing n, bounding the averages of f(n)/n and comparing their maxima.

erdos_1985_locally_repeated_values_certain_arithmetic_functions_i/: Proves abundant collisions of n plus its number of distinct prime factors; its totient reference is only a pointer to later work, not a totient-block theorem.

erdos_1985_problems_results_number_theory/: A survey of Erdos problems on divisor functions, consecutive divisors, prime gaps and equidistribution, with several new bounds announced.

erdos_1987_locally_repeated_values_certain_arithmetic_functions/: Shows that the number of n up to x with the same number of prime factors, or the same divisor count, at n and n plus one is O of x over root log log x.

erdos_1987_locally_repeated_values_certain_arithmetic_functions_ii/: Gives a quantitative upper bound for consecutive equal values of Euler's totient function and records the corresponding infinitude conjecture.

erdos_1988_diophantine_equations_many_solutions/: Constructs sums, S-unit equations and Thue-Mahler equations with unexpectedly many solutions, showing known upper bounds are near best possible.

erdos_1990_distribution_values_certain_class_arithmetic_functions/: Studies arithmetic functions with squarefull kernel at consecutive integers, with asymptotics and long blocks of equal or distinct values.

erdos_1990_greatest_prime_factor/: Proves the greatest prime factor of the product of the first x values of an irreducible polynomial exceeds x times a slowly growing factor.

erdos_1990_normal_behavior_iterates_arithmetic_functions/: Determines normal orders for iterates of arithmetic functions and states the image-density conjecture for the sum-of-proper-divisors function that is equivalent to the EGPS preimage conjecture.

erdos_1997_locally_repeated_values_arithmetic_functions_iv/: Shows the number of solutions of m plus omega(m) equals n is unbounded, and bounds the runs of consecutive integers on which omega or Omega stays constant or takes distinct values.

ermoshin_2026_largest_prime_factor_irreducible_cubic_polynomial/: Proves an unconditional fixed-power improvement for every monic irreducible cubic polynomial.

fan_2025_hardy_ramanujan_inequality_sifted_sets_its_applications/: Proves a Hardy–Ramanujan inequality for weighted sifted sets and deduces that the sum of proper divisors rarely has an atypical number of prime factors, a weighted special case of Problem 955.

ford_1998_distribution_totients/: Determines the true order of the counting function of Euler totient values and of totients of each possible multiplicity.

ford_2010_common_values_arithmetic_functions/: Proves unconditionally that Euler's totient and the sum-of-divisors function take infinitely many common values, settling a conjecture of Erdos.

ford_2020_solutions_phi_n_phi_n_k/: Shows phi(n)=phi(n+k) has infinitely many solutions for every multiple of some even k at most 3570, and sigma(n)=sigma(n+k) for a positive proportion of k.

gabdullin_2024_numbers_form/: Shows a positive proportion of integers are of the form k+f(k) for f the divisor, prime-divisor or totient function, with upper bounds 0.94x and 0.93x.

garaev_2011_number_common_values_arithmetic_functions_below/: Shows that for every A > 0 and large x there are at least exp((log log x)^A) integers up to x that are common values of Euler's phi and sigma.

gpt_5_5_pro_2026_totient_fibre_extremes/: A five-page note whose title page credits GPT-5.5 PRO, proving that the largest ratio of a maximal to a minimal totient preimage over totient values up to x is (e^gamma + o(1)) log log x; the written form of the argument the site accepted for Problem 694.

graham_1999_solutions_phi_n_phi_n_k/: Studies shifted equal-totient solutions for fixed shifts, including a fixed-k exceptional bound and a conditional construction of equal-totient arithmetic progressions.

grimmelt_merikoski_2025_greatest_prime_factor_uniform_equidistribution_quadratic_polynomials/: Gives an exponent-1.312 interval prime-factor bound for an^2+h under a prime-sum hypothesis, including an unconditional specialization to n^2+1.

grytczuk_2001_conjecture_erdos_concerning_inequalities_euler_totient/: Shows phi(n) > phi(n - phi(n)) holds for a set of lower density at least 0.54 and that the reverse inequality holds infinitely often with a gap.

gyory_1986_prime_factors_sums_integers_i/: Proves a conjecture of Erdős and Turán that the number of distinct prime factors of the product of pairwise sums grows at least logarithmically.

ha_2019_many_solutions_s_unit_equation_1/: Constructs arbitrarily large prime sets S for which a + 1 = c has at least of order exp(s^{1/4}/log s) solutions with all prime factors of ac in S.

harper_2013_bounds_suprema_gaussian_processes_omega_results/: Gives explicit non-asymptotic lower bounds for Gaussian suprema tails and deduces new omega results for a random multiplicative function's partial sums.

kinlaw_2020_equation_phi_n_phi_n_1/: Proves that the reciprocal sum of the integers n with phi(n) = phi(n+1) is less than 7.8358.

konyagin_2007_two_s_unit_equations_many_solutions/: Constructs, for each positive beta < 2 - sqrt 2, arbitrarily large sets S of s primes for which the S-unit equation a+b=c has at least exp(s^beta) solutions, and arbitrarily large S for which a+1=c has at least exp(s^{1/16}), improving earlier lower bounds.

kruer_kohlmeyer_2026_doubling_law_distinct_totient_values/: Working exposition of the Lean proof, accepted by the bounty site Conjectures.io in September 2026, that the count of distinct totient values up to x satisfies V(2x)/V(x) -> 2, the first question of Problem 416.

lai_2021_largest_prime_divisor/: Improves the lower bound for the limsup of P(n!+1)/n from 11/2 to 1+9 log 2, about 7.238, and extends it to polynomial shifts.

lau_2013_mean_values_random_multiplicative_functions/: Shows that partial sums of a random multiplicative function are almost surely at most the square root times a power of the double logarithm.

lau_2026_number_prime_factors_consecutive_integers/: Shows there are infinitely many n with at most C log k prime factors of n+k for every k at least 2, improving the previous O(k) bound.

lebowitz_lockard_et_al_2021_distribution_mod_p_eulers_totient_sum_proper_divisors/: Proves that the sum of proper divisors of composite n is equidistributed modulo every prime up to a fixed power of log x, with companion asymptotics for Euler's totient, a residue-class tool for Problem 955.

li_2026_rank_amplification_shifted_equal_values_euler_totient_function/: Gives a moving-rank decomposition for shifted equal totients and a sharper upper bound for the number of consecutive equal-totient solutions.

li_2026_resolution_erdos_problem_1061_sum_divisors/: Claims a resolution of Erdős Problem 1061, showing the count of pairs with sigma(a)+sigma(b)=sigma(a+b) exceeds x times any fixed power of log x.

li_2026_square_annular_dynamics_coalescence_frontiers_n/: Recasts the Erdos-Graham coalescence question for iterates of n plus the divisor count as synchronization of finite square-annular transfer maps.

lichtman_2022_primes_arithmetic_progressions_large_moduli_shifted/: Proves there are infinitely many primes whose shift by a fixed integer has no prime factor above the 0.2844 power of the prime.

lu_2025_largest_prime_factors_consecutive_integers/: Proves, in its 2018 preprint version, that the integers n whose largest prime factor is smaller than that of n plus one have lower density at least 0.2017, and the same for the reverse inequality.

luca_2002_problems_makowski_schinzel_erdos/: Shows sigma(phi(n))/n tends to infinity on a density-one set and proves Erdos's conjecture that phi(n - phi(n)) < phi(n) almost always.

luca_2011_arithmetic_function_arising_carmichael_s_conjecture/: Proves that the number F(n) of m with phi(m) = phi(n) normally lies between K(n)^{1/2 - epsilon} and K(n)^{3/2 + epsilon}, K(x) = (log x)^{(log log x)(log log log x)}, that phi(n) + 1 is almost always squarefree, and that values with very many preimages have many small prime factors.

luca_pomerance_2015_range_sum_of_proper_divisors_function/: Proves via a second-moment collision bound that the even values of the sum of proper divisors have positive lower density, and restates as its Conjecture 1 the preimage conjecture of Erdős, Granville, Pomerance and Spiro that is exactly Problem 955.

maciejewski_2026_bounded_box_reductions_subbarao_warren_problem/: Within a bounded box of source kernels, isolates the obstruction to a sixth unitary perfect number in a set H_even, whose finiteness reduces to one prime branch, and bounds H_even up to 50000 by computation.

mahler_1935_uber_den_grossten_primteiler_spezieller/: Shows that for A one of plus or minus 1 and plus or minus 2, D squarefree and coprime to A and x coprime to A, the largest prime factor of Dx^2 - A exceeds (log log x)/(1 + epsilon) for all large x.

maier_1984_third_iterates_phi_sigma_functions/: Proves Schinzel's conjecture for the third iterate of the sum-of-divisors function and gives analogous lower bounds for the third iterate of Euler's function.

maier_1988_number_distinct_values_euler_s_function/: Determines the order of the count of distinct totient values below x as x/log x times exp of (C + o(1)) times (log log log x)^2.

mangerel_2022_additive_functions_short_intervals_gaps_conjecture/: Proves short-interval averaging theorems for additive functions and partial cases of Erdos's conjecture that almost monotone ones are logarithms.

ono_2000_distribution_partition_function_modulo_m/: Proves that for every prime m at least 5 a positive proportion of primes l give congruences for the partition function along the progressions (m^k l^3 n + 1)/24 with n coprime to l, so every prime divides some partition number and every prime at least 5 divides a positive proportion of them.

openai_2026_asymptotic_formula_number_totients/: Claims V(x) ~ (x/log x) G_m A(1;theta) for the number of distinct totients up to x, with a bounded positive phase coefficient given as a uniform limit of finite arithmetic sums, and V(cx)/V(x) -> c for every fixed c > 0, by prefix-collision counting over Ford's normal structure; Problems 416, 417, 51.

openai_2026_joint_dickman_law_consecutive_integers/: An 84-page manuscript of the OpenAI mathematics release claiming the joint Dickman law in ordinary natural density for the largest prime factors of n and n+1, by amplifying a mixed bin-character correlation into a divisor graph controlled by short-interval estimates; it addresses Problems 928 and 371.

openai_2026_poisson_dirichlet_law_prime_predecessors/: A 74-page release manuscript claiming the Poisson-Dirichlet law for the ordered prime factors of p-1 over primes p up to x (the Ford-Konyagin-Luca conjecture) by two marked correlation estimates and size-biased sampling; it touches Problems 821 and 1057 only through the smooth-prime counts implied.

openai_2026_weighted_dilation_graphs_smooth_shifted_primes_totient_fibers/: A 68-page release manuscript claiming Erdős's conjecture on the largest totient fibers, $g(n)>n^{1-\varepsilon}$ for infinitely many $n$ , derived by a product-and-pigeonhole step from a claimed count of $x^{1-o(1)}$ primes $p\in(2x,5x]$ with $P^+(p-1)\le x^\delta$ for every fixed $\delta>0$ , which the manuscript derives from a transference theorem for weighted dilation graphs, a Type II estimate and a sieve; bears on Problem 821 and, as background, Problem 1057.

pascadi_2026_large_sieve_exceptional_maass_greatest_prime_factor/: Proves large sieve inequalities for exceptional Maass forms with exponential-phase and dispersion coefficients, and from them an infinitely-often exponent-1.3 bound for the greatest prime factor of n^2+1, with an eventual exponent-1.30008 dyadic-product bound inside the proof.

pasten_2024_largest_prime_factor_improvements_subexponential/: Proves the largest prime factor of n^2+1 is at least a constant times (log log n)^2/log log log n, nearly squaring Chowla's bound.

pasten_2026_improvement_largest_prime_factor_n_squared_plus_one/: Improves the pointwise iterated-log lower bound for the largest prime factor of n^2+1, gives a related radical inequality, and counts the large prime factors of n^2+1.

pollack_2014_arithmetic_properties_sum_proper_divisors_sum/: Reproves in quantitative form that s(s(n))/s(n) exceeds s(n)/n by more than (log log x)^(-1/4) for only o(x) integers n <= x, and shows that s(beta(n))/beta(n), with beta(n) the sum of the distinct prime divisors of n, has Davenport's distribution function.

pollack_2015_palindromic_sums_proper_divisors/: Proves that the integers whose sum of proper divisors is a palindrome in a fixed base form a density-zero set, a special case of Problem 955, with parallel results for other arithmetic functions.

pollack_2015_remarks_fibers_sum_divisors_function/: Shows every positive real is a limit of fractions m/n with sigma(m)=sigma(n), and that typical sigma-fibers share one largest prime factor.

pollack_2016_problems_erdos_sum_divisors_function/: Improves bounds on aliquot reversals, gives a heuristic density for nonaliquot numbers, bounds the count of primitive friendly pairs, and shows the count of n with at least k friends has a limiting density.

pollack_2018_divisor_sum_fibers/: Proves a uniform preimage bound for finite sparse target sets, constructs large localized fibers of the sum-of-proper-divisors function, and bounds the solutions of sigma(n) = a (mod n) uniformly in a.

pollack_roy_2021_powerfree_sums_proper_divisors/: Proves for each k at least 4 that the sum of proper divisors of n is k-free exactly when n is, for almost all n, so the n with s(n) k-free have density 1/zeta(k); k = 2, 3 would follow from an affirmative answer to Problem 955.

pollack_troupe_2021_sums_proper_divisors_follow_erdos_kac_law/: Proves an Erdős–Kac law for the number of distinct prime factors of the sum of proper divisors, and gives conditions under which the same law holds for related functions such as n minus Euler's totient.

polya_1918_zur_arithmetischen_untersuchung_der_polynome/: Shows the largest prime factor of f(n) tends to infinity for products of two essentially different linear factors (Thue's result) and for irreducible quadratics, hence integers built from a fixed finite set of primes have growing gaps.

pomerance_1981_distribution_amicable_numbers/: Proves that the count of amicable numbers up to x is at most x exp(-(log x)^{1/3}) for large x, so their reciprocals converge.

pomerance_2015_amicable_numbers/: Improves the upper bound for the count of amicable numbers up to x to x/exp((log x)^{1/2}) for large x.

pomerance_2018_first_function_iterates/: Extends the Bosma-Kane geometric-mean theorem for s(2n)/2n to the next aliquot iterate and bounds the number of s-preimages.

rotkiewicz_1964_nombres_naturels_n_k_pseudopremiers/: Shows that 23 is the least k above 1 for which some pseudoprime n has kn also pseudoprime, and builds pseudoprimes n and pn in prescribed residue classes; it states nothing about the largest prime factors of n and n plus one.

schinzel_nd_two_theorems_gelfond_applications/: Makes Gelfond's irrationality measures for ratios of logarithms explicit and applies them to linear recurrences, Diophantine equations and prime factors.

steinerberger_2025_iterated_arithmetic_function_problem_erdos_graham/: Reduces the r equals 2 case of iterating n plus Euler phi of n to one Euler-phi equation and finds its six explicit families of solutions.

stewart_2013_divisors_lucas_lehmer/: Gives an effective lower bound for the largest prime divisor of Lucas and Lehmer cyclotomic factors whose direct integer specialization proves that P(2^n-1)/n tends to infinity.

stewart_nd_greatest_prime_factor/: Proves that the greatest prime factor of a^n-b^n divided by n grows without bound on a density-one set of exponents that includes the primes.

tao_2019_structure_correlations_multiplicative_functions_at_almost/: Establishes the Chowla conjecture for odd orders and for pairs at almost all scales, via a structure theorem for unweighted multiplicative correlations.

tao_2025_quantitative_correlations_problems_prime_factors_consecutive/: Proves results on consecutive prime factors and related irrationality; v2 Remark 1.4 also states the exact arbitrary-sequence result in problem 258.

tenenbaum_1990_sur_une_question_erdos_schinzel_i/: Estimates how often polynomial values have a divisor in a short interval and records the positive-power history for the quadratic x^2+1 case.

tenenbaum_1990_sur_une_question_erdos_schinzel_ii/: Proves a general subpower exponential lower bound for the greatest prime factor of products of values of irreducible integer polynomials of degree greater than one.

teravainen_2018_binary_correlations_multiplicative_functions/: Proves a discorrelation estimate for logarithmically averaged binary correlations of multiplicative functions equidistributed in fixed-modulus progressions.

tijdeman_1973_integers_many_small_prime_factors/: Gives effective gap estimates for consecutive integers built from a fixed finite set of primes, and shows the estimates are nearly best possible.

troupe_2015_number_prime_factors_values_sum_proper/: Proves that the number of prime factors of s(n), the sum of proper divisors of n, has normal order log log n.

troupe_2020_divisor_sums_representable_as_sum_two/: Shows the count of n up to x with s(n) a sum of two squares has order x/(log x)^{1/2}, the order of the count of n up to x that are sums of two squares.

turturean_2026_positive_density_equality_set_erdos_problem/: Proves a positive-density set of integers where the least prime congruent to one modulo n equals the least integer whose totient is divisible by n.

wang_crapis_2026_complete_answer_erdos_problem_690/: Compiles the all-k non-unimodality route for Problem 690 with explicit external premises and forty-two pending finite certificates.

zeraoulia_2026_fixed_scale_limit_points_distinct_totients/: Self-published July 2026 preprint claiming, for each fixed c>1, that c is a subsequential limit of V(cn)/V(n) and that the cluster set of V(cx)/V(x) is a closed interval containing c; explicitly not a proof of the limit.

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