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

Updated Oct 8

Diophantine Problems and Powers

abram_2014_intersections_multiplicative_translates_3_adic_cantor/: Computes Hausdorff dimensions of finite intersections of multiplicative translates of the 3-adic Cantor set via finite automata.

bajpai_2024_arithmetic_progressions_squarefull_numbers/: Constructs infinitely many coprime four-term progressions of powerful numbers and derives Erdos's conjectures on such progressions from abc.

bajpai_2024_effective_unit_equations_beyond_three_terms/: Solves five-term S-unit equations effectively when S has at most three places and thereby answers Newman's question with explicit bounds.

bauer_2007_question_erdos_graham/: Constructs counterexamples showing three disjoint blocks of four consecutive integers can have square product infinitely often; the authors believe three blocks is the minimum.

bennett_2006_powers_products_consecutive_terms_arithmetic_progression/: Shows that for 4 <= k <= 11 no product n(n+d)...(n+(k-1)d) with n and d positive and coprime is a perfect power, with finiteness results beyond.

bennett_2012_squares_blocks_consecutive_integers_problem_erdos/: Constructs, for each r >= 5, an infinite family of r disjoint blocks of five consecutive integers whose product is a square, answering a question of Erdos and Graham in the negative for blocks of five.

bennett_2020_conjecture_erdos_supersingular_primes_short/: Gives an effective prime-exponent bound and finitely many positive perfect-power solutions for each fixed sufficiently large length of a progression with coprime first term and difference; this is finiteness, not nonexistence.

bennett_2024_computing_four_term_arithmetic_progressions_powerful_numbers/: Gives a 111-digit coprime four-term progression of powerful numbers and explains the elliptic-curve search that found it.

beukers_1999_irreducibility_polynomials_arithmetic_progressions_equal_products/: Proves that for fixed lengths and differences, two arithmetic progressions have equal products of terms only finitely often, apart from listed exceptions.

beyer_de_ryke_2026_density_deficit_cube_full_sums/: Shows that the positive integers that are not a sum of one, two or three cube-full numbers form a set of positive lower natural density.

blomer_2006_estimates_representation_numbers_quadratic_forms/: Gives estimates and asymptotics for the moments of the representation numbers of a binary quadratic form, uniformly in the discriminant, and deduces that the number of integers up to x that are sums of two powerful numbers is x over log x to the power one minus two to the minus one third, up to powers of log log x.

borwein_1990_questions_erdos_graham_numbers_form_sum_g_n_2_g_n/: Answers the first Erdős–Graham question on sums of distinct terms k over two to the k by an explicit identity giving infinitely many n with n over two to the n so representable, reduces the all-n question to a termination conjecture for the iteration a to 2 times a mod n, and studies uniqueness and multiplicity of such representations.

browning_2013_incomplete_kloosterman_sums/: Interval criteria for multiplicative inverses, using the 2012 arXiv version.

browning_heath_brown_2018_counting_rational_points_quadric_surfaces/: Proves a coefficient-uniform upper bound for the number of primitive integer zeros of a nonsingular quaternary quadratic form in a box, with explicit dependence on the discriminant.

browning_munshi_wang_2026_beyond_square_root_barrier_cubic_forms_perazzo_type/: Proves the asymptotic N(B) ~ (sigma_infty/zeta(3)) B^3 log B for the weighted count of integer points on the Perazzo-type cubic fourfold x_1 y_1^2 + x_2 y_2^2 + x_3 y_3^2 = 0.

browning_verzobio_2026_sums_three_powerful_numbers/: Proves upper bounds beating the trivial B^{1/p+1/q} for the number of primitive solutions of a + b = c with a, b, c respectively p-full, q-full, and r-full, for large exponents.

bui_2024_problem_erdos_graham_granville_selfridge_integral/: Shows that t_n <= n^c has the same density as P^+(n) <= n^c for the Erdős-Graham-Selfridge quantity t_n, unconditionally disproving Granville's expectation that t_n exceeds a power of n.

burr_1996_complete_sequences_sets_integer_powers/: Shows that a set of integers with positive upper density and gcd one has a finite subset whose powers form a complete sequence.

chan_2025_note_three_consecutive_powerful_numbers/: Rules out three consecutive powerful numbers whose middle term is a cube and whose outer terms each have one prime to an odd power.

cilleruelo_2007_lattice_points_circles_squares_arithmetic_progressions/: Survey with new proofs linking squares in arithmetic progressions, sumsets of squares, lattice points on short circular arcs and Sidon sets of squares.

cohn_1998_conjecture_erdos_3_powerful_numbers/: Bibliographic record for Cohn's two-page note constructing infinitely many coprime 3-powerful triples a + b = c with none a perfect cube, strengthening Nitaj's answer to the third question of Problem 939; the text was not read.

conjectures_io_2026_erdos_939_lean_r_powerful_sums/: Records the Conjectures.io Lean submission for Problem 939, kernel-checked against a catalog statement that omitted positivity and paid as a formalization-defect award, whose accepted file also proves the infinitude of solutions for r at least 6 with positive summands.

corralesrodriganez_1997_support_problem_elliptic_analogue/: Answers Erdos's support question affirmatively through a theorem for number fields and proves an elliptic-curve analog for points on an elliptic curve.

corvaja_zannier_2011_abcd_function_fields/: Bounds below the number of zeros outside S of 1 + u + v for S-units u, v on a curve, sharpening the abcd theorem when S is small, and applies this to perfect powers x^a + y^b + 1, curves on x^a + y^a + z^c = 1, and polynomial Diophantine triples.

crowdmath_2020_applications_abc_conjecture_powerful_numbers/: Derives from the abc conjecture several finiteness results about powerful numbers, answering four questions of Cushing and Pascoe.

cushing_2016_powerful_numbers_abc_conjecture/: Assuming the abc conjecture, powerful numbers occur near a factorial only finitely often and coprime progressions hold finitely many powerful triples.

dekoninck_2004_sur_la_proximite_des_nombres/: Shows infinitely many intervals between consecutive squares contain arbitrarily many powerful numbers, and computes the density of intervals containing none.

dekoninck_2005_powerful_numbers_short_intervals/: Proves infinitely many intervals between consecutive kth powers hold many k-full numbers, and that ABC bounds such numbers in shorter intervals.

demjanenko_1975_conjecture/: Claims a proof of Schinzel's conjecture that natural solutions x, y, z > 1 of x to the x times y to the y equals z to the z share the same prime divisors.

doorn_2025_smooth_sums_small_spacings/: Shows every positive integer is a sum of distinct 3-smooth numbers whose largest term is less than six times the smallest.

doorn_2026_three_term_arithmetic_progressions_consecutive_powerful/: Proves infinitely many three-term progressions of powerful numbers have common difference twice the square root of the first term plus one.

dubickas_2021_no_cubic_integer_polynomial_generates_sidon/: Proves Ruzsa's conjecture that no cubic integer polynomial has values forming a Sidon sequence; with the easy linear and quadratic cases, this covers every integer polynomial of degree at most three.

einsiedler_2012_distribution_closed_geodesics_modular_surface_duke/: Gives an ergodic-theoretic proof that closed geodesics of large positive discriminant equidistribute on the modular surface, reproving Duke's theorem.

erdos_1936_representation_integer_as_sum_th_powers/: Shows infinitely many integers have more than exp(c log m / log log m) representations as a sum of k kth powers.

erdos_1937_sum_difference_squares_primes/: Proves that infinitely many integers have more than n^(c/log log n) representations as a sum of two squares of primes.

erdos_1953_arithmetical_properties_polynomials/: Shows that every integer polynomial of degree l >= 3 meeting mild conditions takes infinitely many values free of any nontrivial (l-1)-th power.

erdos_1972_linear_diophantine_problem_frobenius/: Bounds the largest integer not representable by a set of n coprime integers, giving a general bound and the extremal value up to a constant factor.

erdos_1975_product_consecutive_integers_is_never_power/: Proves that no product of at least two consecutive positive integers equals a perfect power, settling a conjecture about 150 years old.

erdos_1976_problems_results_number_theoretic_properties_consecutive/: Survey on prime factors of consecutive integers, including a density theorem on large prime factors and a conjecture on the 2,3-part of n(n+1).

erdos_1976_products_factorials/: Counts distinct products of factorials and shows that whenever some product of factorials with largest factor n! is a square, six factorials suffice.

erdos_1996_d_complete_sequences_integers/: Shows that the numbers 2 to the a times 3 to the b represent every integer as a sum with no summand dividing another, that no other coprime pair of bases does so, and that several triples of primes do, and poses the conjectures and density questions behind problems 123, 845 and 1110.

erdos_2019_number_integers_represented_binary_form/: Shows the count of integers up to u represented by an integral binary form of degree n at least three with nonzero discriminant grows at least on the order of u^{2/n}.

evertse_schlickewei_schmidt_2002_linear_equations_multiplicative_group/: Bounds the number of nondegenerate solutions of a1x1+...+anxn=1 in a multiplicative group of rank r by a function of n and r alone, which also bounds uniformly the number of representations n=2^a+3^b+2^c3^d of Problem 407.

fang_2017_quantitative_form_erdos_birch_theorem/: Makes the Erdős–Birch theorem effective. For coprime p and q above one it bounds a threshold B and an exponent range K, both by explicit towers in p and q, such that every integer at least B is a sum of distinct numbers p to the a times q to the b with b at most K.

ghidelli_2019_arbitrarily_long_gaps_between_values_positive_definite_cubic_biquadratic_diagonal_forms/: Proves that the values of a positive definite diagonal cubic form in three variables, and of a non-exceptional diagonal quartic in four, have arbitrarily long gaps, with explicit gap lengths below N.

gyory_2004_diophantine_equation/: States perfect-power exclusions for four or five positive terms in a primitive arithmetic progression, with a later proof correction and related finiteness results.

gyory_2009_perfect_powers_products_consecutive_terms_arithmetic_progression/: Proves that for 3 < k < 35 the product of k consecutive terms of a coprime positive arithmetic progression is never a perfect power, through the almost-perfect-power equation with prime exponents n >= 7 and n = 5.

hasler_2024_sums_distinct_powers_3_4/: Proves that the number of integers up to x that are sums of distinct powers of 3 and distinct powers of 4 is at least a constant times x to the 0.97777, and that the lower density of this set is at most 1015/1458. Leaves the positive-density question open.

heath_brown_1997_density_rational_points_cubic_surfaces/: Proves that if F is a nonsingular integral cubic form in four variables whose surface F = 0 contains three rational coplanar lines, then the integer zeros of F of Euclidean length at most P lying on no rational line of the surface number O(P^{4/3+epsilon}), with the corollary that at most O(x^{4/9+epsilon}) positive integers up to x have two or more distinct representations as a sum of two cubes of nonnegative integers.

heath_brown_2009_sums_differences_three_kth_powers/: Counts integer solutions of F(x) = N for a fixed non-singular ternary form outside low-degree polynomial families, with exponents 9/10 and 10/k for the height in a dyadic shell, and counts primes p with p^k + h (k-1)-free.

heathbrown_2000_arithmetic_applications_kloosterman_sums/: Survey lecture showing how Kloosterman sum bounds solve mn congruent to a mod p, for p not dividing a, with m and n at most 2 p^(3/4) log p.

hegyvari_2025_elementary_question_erdos_graham/: Answers an Erdos-Graham question by bounding the intersection of the sets {r(k-r)} for two moduli and showing it is unbounded.

hwang_2024_frobenius_problem_numerical_semigroups_generated_binomial/: Solves the Frobenius problem for the numerical semigroup generated by the binomial coefficients with fixed upper index n, for every n > 1, dividing them by p when n is a power of a prime p.

kulkarni_2005_class_diophantine_equations_involving_bernoulli_polynomials/: Shows equations relating Bernoulli polynomials to the products x(x+1)...(x+n-1) have finitely many bounded-denominator rational solutions outside explicit exceptions.

lagarias_2009_ternary_expansions_powers_2/: Bounds how often iterates of doubling omit the digit 2 in base three, giving uniform counting bounds and Hausdorff dimensions of the exceptional sets.

luca_2001_conjecture_erdos_stewart/: Proves the Erdős–Stewart conjecture that n factorial plus one is a product of powers of the two primes following n only for n at most 5, by p-adic linear forms in two logarithms, the Erdős–Obláth theorem and a computer search.

luca_2014_squares_factorials_products_factorials/: Bounds the number of integers up to X that are the largest of three distinct factorials whose product is a square, and of no fewer, by X over an exponential in a fourth root of log X, sharpening the Erdős–Graham bound little-o of X, and bounds the related set of integers whose block product has its largest prime to a power above one. Also treats products of factorials that are a factorial.

mahler_1935_lattice_points_curves_genus_1/: Shows cubic curves of genus 1 can carry arbitrarily many lattice points, with infinitely many integers having more than (log k)^{1/4} representations as sums of two cubes of positive integers.

mahler_1936_note_hypothesis_k_hardy_littlewood/: Disproves Hardy and Littlewood's Hypothesis K for cubes by exhibiting infinitely many N with at least a constant times N^(1/12) representations.

melfi_2004_certain_positive_integer_sequences/: Surveys practical numbers, sum-free sequences and complete power sequences, and disproves the only-if half of a conjecture of Burr, Erdos, Graham and Li.

narumi_2025_number_k_full_integers_between_three/: Computes the density of integers n with prescribed k-full integers in the two intervals between n^k, (n+1)^k and (n+2)^k, giving infinitely many triples of successive k-th powers that are consecutive k-full integers.

nitaj_1995_conjecture_erdos_3_powerful_numbers/: Bibliographic record for Nitaj's two-page note constructing infinitely many coprime 3-powerful triples a + b = c, which answers the third question of Problem 939; the full text is not held here.

odoni_1981_problem_erdos_sums_two_squarefull_numbers/: Answers Erdos negatively: sums of two squarefull numbers up to x exceed the expected constant times x(log x)^(-1/2) by a growing factor.

openai_2026_selmer_converse_elliptic_curves_at_prime/: Claims the Selmer converse in coranks zero and one for every elliptic curve over Q at every prime, with finite Tate-Shafarevich group, by tame deformations and unitary theta congruences; at p = 3 it claims the Sylvester cube-sum cases l = 4, 7, 8 mod 9, the rank input Walsh needs for Problem 939.

openai_2026_squarefree_values_quartics_power_free_values_polynomials/: Claims the predicted positive density of (d-2)-power-free values for irreducible integer polynomials of degree 4 to 8 under the local condition, by number-field factorization and determinant cuts, so including the squarefree values of n^4+2 of Problem 978; with Browning's theorem it claims to cover every d >= 4.

pandey_2024_squarefree_numbers_short_intervals/: Proves that every large interval of length X^{1/5-eta} contains a squarefree number, improving the Filaseta-Trifonov bound.

pipeline_math_2026_tiling_complement/: Constructs a tiling complement for all integer thirteenth powers, with a reviewed reconstruction relative to two identified literature premises.

price_2026_infinite_r_powerful_sums/: Public one-page manuscript, attributed by its poster to GPT-5.5 Pro, with an elementary binomial construction of infinitely many r-powerful sums of r-2 positive jointly coprime r-powerful numbers for every r at least 6, and its Lean autoformalization.

salberger_2023_counting_rational_points_projective_varieties/: Proves the dimension growth conjecture N(X;B) << B^{dim X + epsilon} for integral projective varieties of degree at least 2, uniformly in X for degree at least 4.

saye_2022_two_conjectures_concerning_ternary_digits_powers/: Checks by computer that no power 2^n with n up to 2·3^45 is a counterexample to the Erdős or Sloane conjectures on the ternary digits of powers of two.

she_2025_nonexistence_consecutive_powerful_triplets_around_cubes/: Rules out three consecutive powerful numbers centered at a cube whose outer terms are each a prime square times a cube, and deduces that x to the sixth minus one is never two prime squares times a nonzero cube. Extends Chan 2025 on the Erdős–Mollin–Walsh conjecture.

stewart_2008_cubic_thue_equations_many_solutions/: Proves that for every cubic binary form with integer coefficients and nonzero discriminant, the Thue equation F(x, y) = m has at least a constant times the square root of log m integer solutions for infinitely many m, raising Mahler's exponent 1/4 and Silverman's 1/3 to 1/2 by finding twists of rank at least two.

tao_2026_products_consecutive_integers_unusual_anatomy/: Obtains asymptotics for integers in bad or very bad intervals and near-matching bounds for the factorial equation a1!a2!a3! equal to a square.

tengely_2020_diophantine_equation_erdos_graham/: Bounds the largest term and enumerates solutions of the Erdős-Graham equation expressing n/2^n as a sum of terms a_i/2^{a_i}.

tijdeman_1988_sums_products_powers_given_prime_numbers/: Solves three exponential equations in 2 and 3 completely, proves that every rational number beyond an unspecified constant has at most four distinct representations as 2^a 3^b + 2^c + 3^d, and bounds the number of representations as a sum of n products of powers of given primes.

uchiyama_1984_diophantine_equation/: Proves new cases in which the equation x^x y^y = z^z has no nontrivial solutions, and shows solutions of each fixed index below 1/4 are effectively finite.

walker_1976_consecutive_integer_pairs_powerful_numbers_related/: Characterizes all pairs of consecutive powerful numbers via Pell equation solutions having a prime-divisibility property.

walsh_2024_question_erdos_powerful_numbers_elliptic_curve/: Builds infinitely many coprime solutions of a+b=c in 3-powerful numbers from rational points on the elliptic curves Y^2 = X^3 - 432p^2 of positive rank.

wang_2021_sums_cubes_ratios_conjectures/: Studies the six-variable equation x_1^3 + ... + x_6^3 = 0 and, conditionally on Ratios Conjectures for the relevant Hasse–Weil L-functions, sharpens the count of its solutions and its link to sums of three cubes.

wooley_2015_sums_three_cubes_ii/: Proves that the number of integers up to X representable as a sum of three positive cubes is >> X^0.91709477.

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