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

Updated Oct 8

Factorials and Binomial Coefficients

alexeev_2025_decomposing_factorial_into_large_factors/: Determines the asymptotic of the largest threshold t(N) for writing N factorial as a product of N factors, answering a question of Erdos and Graham.

andrejic_2016_distinct_residues_factorials/: Links primes with all factorial residues distinct to Kurepa's left factorial and verifies none exist below 10^11.

berend_1998_arithmetical_properties_middle_binomial_coefficients/: Shows the central binomial coefficients are dense in the p-adic integers for odd p and weakly well-distributed modulo every odd prime.

bergman_2011_common_divisors_multinomial_coefficients/: Gives explicit lower bounds for the gcd of two binomial coefficients with the same upper argument, but not for their largest common prime factor.

bhat_2010_remark_factorials_that_are_products_factorials/: Improves the bound on how far the largest factor can sit below n when n factorial equals a product of smaller factorials.

bloom_2025_integers_small_digits_multiple_bases/: Shows that for large distinct coprime bases there are infinitely many integers whose digits are almost all small in every base at once.

brindza_1991_diophantine_problems_involving_powers_factorials/: Proves that for each fixed r a sum of r distinct large factorials is never powerful, and bounds every solution of (p-1)! + a^(p-1) = p^k by an absolute constant.

bui_2023_power_savings_counting_solutions_polynomial_factorial/: Gives a power-saving bound C(P,s) N^{33/34} on the number of n in [N, 2N) with s n! = P(x) for some integer x, for a fixed integer polynomial P of degree at least 2 and a fixed nonzero integer s, improving Berend and Osgood's o(N) and covering the Brocard-Ramanujan equation.

bui_2026_binomial_coefficients_divisors_avoiding_interval/: Resolves an Erdős-Graham question by finding binomial coefficients with no divisor close to n, while proving such divisors exist once k is large.

corvaja_zannier_2010_integral_points_divisibility_between_values_polynomials_entire_curves_surfaces/: Corvaja and Zannier prove that certain systems of divisibilities between polynomial values, in general position, have solutions that are not Zariski-dense, via integral points on blow-ups of the plane; the card records why the method gives no case of Problem 699.

croot_et_al_2023_conjecture_graham_p_divisibility_central_binomial_coefficients/: Proves that for any fixed sufficiently large distinct primes infinitely many central binomial coefficients have small valuation at each of them, a low-multiplicity weakening of Graham's coprime-to-105 question (Problem 376), and records how far the method reaches toward Problem 699.

dusart_2010_estimates_some_functions_over_primes_without_r_h/: Records Dusart's explicit short interval containing a prime.

ecklund_1974_new_function_associated_prime_factors/: Studies g(k), the least n above k+1 whose binomial coefficient has all prime factors above k, and bounds it between k^{1+c} and exp(k(1+o(1))).

ecklund_et_al_1978_prime_factorization_binomial_coefficients/: Determines exactly when the prime-at-least-k part of a binomial coefficient is smaller than its prime-below-k part; the card uses that split for the range j ≤ 3i/2 of Problem 699.

ecklundjr_1969_prime_divisors_binomial_coefficient/: Proves that n choose k for n at least twice k has a prime divisor at most the larger of n over k and n over two, with the exception 7 choose 3.

erdos_1934_theorem_sylvester_schur/: Proves that every binomial coefficient (n choose k) with n at least 2k has an individual prime divisor exceeding k, without supplying a prime shared by two such coefficients.

erdos_1937_uber_diophantische_gleichungen_der_form_und/: Shows factorials are generally not sums or differences of two equal powers, and that sums or differences of two factorials are rarely perfect powers.

erdos_1955_consecutive_integers/: Shows a block of about k over log k consecutive integers above k must contain a prime factor exceeding k, and counts how many do.

erdos_1975_prime_factors/: Shows the central binomial coefficient is usually divisible by high powers of small primes, yet can avoid any two given odd primes.

erdos_1978_number_theoretic_problems_binomial_coefficients/: Poses problems on greatest common divisors and prime factors of pairs of binomial coefficients, and introduces the function f(n) with elementary bounds.

erdos_1982_another_property_239_related_questions/: Studies factorings of n factorial into increasing factors above n and bounds the least possible largest factor between two n plus constants times n over log n.

erdos_1982_miscellaneous_problems_number_theory/: Bounds the number of distinct prime exponents in n factorial, studies factorings of n factorial, and collects additive and prime-factor problems.

erdos_1988_prime_factors_binomial_coefficients_related_problems/: Proves that a bounded sequence with the consecutive-integer property must be a permutation of 1 to k, and classifies the permutations that occur.

erdos_1993_estimates_least_prime_factor_binomial_coefficient/: Estimates the least prime factor of N choose k, proving that the least N above k plus 1 for which every prime factor exceeds k is larger than a constant times k squared over log k. Defines the deficiency of such a binomial coefficient, lists the 17 known with deficiency above one, and conjectures that the least prime factor is at most max(N/k, 29).

erdos_1996_number_divisors/: Gives an asymptotic expansion for the divisor count of n factorial and determines the limit points of the ratio of successive such counts.

granville_1996_explicit_bounds_exponential_sums_scarcity_squarefree/: Proves the central binomial coefficient is never squarefree for n>4, and that squarefree binomial coefficients sit near the row edges.

granville_1997_arithmetic_properties_binomial_coefficients_i/: Kummer and Lucas carry criteria, Granville's prime-power congruence, and their reduction of Problem 699 to simultaneous base-p carry conditions.

grebennikov_2024_sequence/: Shows the factorials 1!,2!,... produce at least (sqrt 2+o(1))sqrt p distinct residues mod p, and that seven factorials cover every nonzero class.

habsieger_2019_explicit_bounds_diophantine_equation/: Gives explicit bounds on nontrivial solutions of A!B!=C! and shows that (6,7,10) is the only one with B below 10^3000.

heier_levin_2025_schmidt_nochka_theorem_closed_subschemes_subgeneral_position/: Weighted Diophantine approximation for closed subschemes in subgeneral position, with an account of why it proves no case of Problem 699.

klurman_2017_distribution_factorials_modulo/: Proves the factorials in a short interval occupy at least the square root of 1.5N residue classes mod p, beating the trivial square-root bound.

li_2026_erdos_problem_684_at_density_one/: Determines, for almost all n, when the small-prime part of a binomial coefficient first exceeds a power of n, with Gaussian fluctuations.

li_2026_prime_power_rarefaction_density_one_lower/: Proves that the factorial-excess function is at least (3(k-1)/log 12 - eps) log n for almost all n, with a pointwise upper bound of (k-1) log_2 n + log_2 log n + O_k(1).

li_2026_resolution_erdos_problem_731_under_dyadic/: Pins down the density-tight scale for the least non-divisor of the central binomial coefficient and rules out any dyadically regular equivalent.

luca_2002_diophantine_equation_result_m/: Shows that the ABC conjecture implies that P(x) = n! has only finitely many integer solutions (x, n) with n > 0 for every integer polynomial P of degree at least 2.

matomaki_2022_singmaster_s_conjecture_interior_pascal_s/: Proves Singmaster's conjecture in the interior of Pascal's triangle: for large t at most four binomial coefficients in that region equal t.

mausberg_2026_thirteen_layer_lower_bound_erdos_problem/: Proves an unconditional lower bound of about 0.15516 for the liminf of (f(n)-2n) divided by n over log n in Erdos problem 390.

naciri_2025_brocard_ramanujan_equation_free_integers_prime/: Proves n!+1=x^2 has finitely many solutions when x±1 is k-free or has few prime factors, and lists them for 7-free and prime-power cases.

pomerance_2015_divisors_middle_binomial_coefficient/: Shows n+k divides the central binomial coefficient for almost all n when k is positive, and studies the shifted divisibility densities.

pomerance_2026_remarks_middle_binomial_coefficient/: Shows that for almost all m the product (m+1)...(m+k) divides C(2m,m) for all k up to 0.72 log m, and C(m+k,k) divides it much further.

rousseau_et_al_2024_divisibility_polynomials_degeneracy_integral_points/: Ru–Vojta degeneracy results for polynomial divisibility and their possible use in geometrizing the simultaneous divisibilities arising in Problem 699.

sander_1992_prime_power_divisors_binomial_coefficients/: Shows that binomial coefficients with argument near the middle are divisible by the a-th power of some large prime, answering questions of Erdős and Graham.

schinzel_1958_sur_un_probleme_de_p_erdos/: Answers negatively Erdos's question whether some n-i with 0<=i=2k, by the counterexample k=15, n=99215.

shorey_tijdeman_2007_prime_factors_arithmetic_progressions_binomial_coefficients/: Source record and research digest.

sorenson_2020_algorithm_estimates_erdos_selfridge_function/: Gives an algorithm computing the Erdos-Selfridge function g(k), used for all k up to 375, proves estimates for its approximation M_k/R_k, and, under a uniform distribution heuristic, estimates g(k) and the running time.

sothanaphan_2026_resolution_erdos_problem_728_writeup_aristotle/: Informal writeup of an AI-produced Lean proof giving a logarithmic gap window for the factorial divisibility a!b! dividing n!(a+b-n)!.

trudgian_2013_there_are_no_socialist_primes_less/: Reports computations, by Harvey and by Oliveira e Silva, showing that no prime p between 5 and 10^9 has 2!,...,(p-1)! all distinct modulo p.

van_doorn_rocca_2026_binomial_coefficients_sharing_large_prime_divisor/: Proves that only finitely many triples with i at least 4 fail to have a common prime divisor strictly larger than i, while leaving i = 3 open.

van_doorn_rocca_2026_partial_progress_erdos_problem_699/: Proves that Problem 699 has no counterexamples for i = 1, 2 or i at least 1476, and only finitely many possible counterexamples for fixed i at least 4.

velammal_1995_is_binomial_coefficient_squarefree/: Proves the Erdos conjecture that the central binomial coefficient is never squarefree for every n greater than four.

wang_2026_proposed_solution_erdos_problem_390/: Claims the exact asymptotic constant for the least largest factor in writing n! as a product of distinct integers above n.

xiao_2024_greatest_common_divisors_polynomials_almost_units_applications_linear_recurrence_sequences/: GCD bounds for polynomial values at almost-unit points and the extra coordinate-height hypotheses needed to apply them to Problem 699.

yasufuku_2026_gcd_inequalities_arising_from_codimension_2_blowups/: Unconditional GCD inequalities from codimension-two blowups and the precise hypotheses and penalty terms relevant to Problem 699 at index 3.

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