Primes

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alkan_2022_generalization_hardy_littlewood_conjecture/: Proves pi_P(x+y) < pi_P(x)+pi_P(y) when x, y >= x_0 and Kx/log x <= y <= x for prime sets P with pi_P(x) = c pi(x)+O(x/(log x)^3), 0 < c <= 1, and derives infinitely many violations from the prime k-tuples conjecture.

axler_2018_new_estimates_some_functions_defined_over_primes/: Holds Axler's 2018 Integers paper for the two statements Wang–Crapis import through it: Dusart's short-interval theorem as quoted on p. 13 and the printed digits of the Meissel–Mertens constant on p. 16.

axler_2019_some_results_conjecture_hardy_littlewood/: Proves pi(m+n) <= pi(m)+pi(n) for integers m >= n >= 2 whenever n >= c_0 m/(log m)^2 with c_0 = 0.70881..., or m+n <= 39,708,229,123, and records that the prime k-tuples conjecture would force infinitely many violations.

baker_2001_difference_between_consecutive_primes/: Proves that the interval from x minus x to the power 0.525 up to x contains a prime for all large x.

banks_2016_limit_points_sequence_normalized_prime_gaps/: Shows that among any nine nonnegative reals some difference is a limit point of the normalized prime gaps, so that the limit points meet [0,T] in measure at least (1 - o(1))T/8 as T tends to infinity.

banks_2023_ratios_consecutive_prime_gaps/: Gives a Hardy-Littlewood-based heuristic predicting that the primes with d(n+1)/d(n) at least c have relative density 1/(c+1).

blecksmith_1999_cluster_primes/: Defines cluster primes, asks whether there are infinitely many, and proves by Brun's sieve that for each fixed s fewer than x/(log x)^s of them are at most x once x is large, so the sum of their reciprocals converges.

chahal_et_al_2025_second_hardy_littlewood_conjecture/: Proves pi(x+y) <= pi(x)+pi(y) for large x whenever 3CR(2x)(log x)^2/log log x <= y <= x, for any prime number theorem error bound CR(x), reaching y of order sqrt(x)(log x)^2 under the Riemann hypothesis.

chen_2022_conjecture_erdos/: Proves Erdos's 1950 conjecture that any set of more than log x integers up to x admits an integer with many representations as a prime plus a member.

chen_2023_conjecture_erdos_p_2_k/: Refutes an Erdos conjecture by showing the odd integers not of the form a prime plus a power of two are not finitely many progressions plus a zero-density set.

chojecki_2026_note_erdos_problem_1201/: Deduces from the Matomaki-Radziwill short-interval theorem that for every ε,η>0 some k gives P^+(n(n+1)...(n+k)) > n^{1-ε} on a set of n of lower density at least 1-η.

clark_jarvis_2001_dense_admissible_sequences/: Computes or bounds the largest admissible set in each interval length x up to 1426, finding it below pi(x) through 1120 and at most pi(x) through 1426, finds 657 admissible points in length 4916, one more than pi(4916), and finds admissible sets beating Erdős's bound 2pi(x/2) at three lengths.

cramer_1936_order_magnitude_difference_between_consecutive_prime/: Introduces the probabilistic heuristic suggesting prime gaps are O((log p)^2) and proves conditional bounds limiting how often large gaps occur.

ding_2025_two_romanoff_type_problems_erdos/: Proves for almost all real y > 1 that primes plus integer parts of powers of y have positive lower density, with an explicit bound of the right order.

doorn_2026_optimal_bounds_erdos_problem_matching_integers/: Determines exactly how many members of any m-set of positive integers can always be matched to distinct multiples in any open interval of length twice its largest member.

dusart_1999_kth_prime_lower_bound/: Compiles the explicit analytic specialization and prime-range deduction with exact external inputs.

dusart_2002_sur_la_conjecture_pi_x_y_pi_x_pi_y/: Proves pi(x+y) <= pi(x)+pi(y) for 2 <= x <= y <= (7/5)x log x log log x, partly by computer-checked prime-index inequalities, so failing pairs up to X have density at most 5/(7 log X log log X).

elsholtz_2001_inverse_goldbach_problem/: Shows that if A+B agrees with the primes up to finitely many elements, with |A|,|B| >= 2, then for large x the counting functions A(x) and B(x) lie, up to constant factors, between x^{1/2}/(log x)^5 and x^{1/2}(log x)^4, and rules out three such summands.

elsholtz_2003_cluster_primes/: On cluster primes.

elsholtz_2015_additive_decompositions_sets_restricted_prime_factors/: Shows that certain sets defined by restricted prime factors, including smooth numbers, admit no ternary asymptotic sumset decomposition, settling the ternary version of a conjecture of Sarkozy for small exponents, and sharpens the bounds on hypothetical summands of the primes.

erdos_1948_new_questions_distribution_prime_numbers/: Shows the primes are infinitely often locally convex and infinitely often locally concave, in both additive and multiplicative senses.

erdos_1949_applications_brun_s_method/: Uses Brun's sieve to bound the least prime in arithmetic progressions, above and below, for a positive proportion of residues, and to find long runs of widely spaced primes.

erdos_1950_integers_form_related_problems/: Studies representations as a power of two plus a prime, and builds an arithmetic progression of odd numbers containing no such number.

erdos_1955_remarks_number_theory_hebrew/: Three notes: continuum many reals with pairwise far-apart power sequences [a^n] without choice, a shift covering many integers, and products of two factors.

erdos_1976_problems_results_consecutive_integers/: Proves lower bounds on how many of k consecutive integers past a power of k have a prime factor above k, and conjectures the Dickman-constant asymptotic.

erdos_1980_matching_natural_numbers_up_n_distinct/: Bounds the shortest interval containing distinct multiples of 1 up to n, showing the length is superlinear and at most about n times the square root of log n.

erdos_1980_small_sieve/: Proves that sifting the integers up to x by primes of bounded reciprocal sum always leaves at least a positive proportion of them unsifted.

erdos_1985_my_problems_number_theory_i_would/: Erdos's selection of his most-wanted number theory problems, including prime-gap distribution questions and the squared-totative-gap conjecture.

florez_2019_distribution_generalized_greatest_common_divisor_visibility/: Computes mean values of arithmetic functions of a generalized gcd and characterizes which patterns of visible and invisible lattice points occur.

fourn_2025_percolative_properties_random_coprime_colouring/: Shows the random visibility coloring of Z^d (d >= 2) almost surely has exactly one infinite visible cluster and no infinite invisible cluster, with partial extensions to other lattices and Cayley graphs.

gafni_2025_rough_numbers_between_consecutive_primes/: Shows almost all prime gaps contain an integer whose least prime factor is at least the gap length, confirming a prediction of Erdos.

goldston_2009_primes_tuples_i/: Proves that liminf (p_{n+1} - p_n)/log p_n = 0, and that the Elliott-Halberstam conjecture gives p_{n+1} - p_n <= 16 infinitely often.

granville_1990_note_sums_primes/: Assuming the prime k-tuplets conjecture, constructs infinite sets of odd primes whose weighted sums, divided by a fixed gcd, are all prime.

granville_1995_harald_cramer_distribution_prime_numbers/: Surveys Cramér's probabilistic model of the primes and its history, explains Maier's theorem against it, and argues from a sieve-corrected model that the largest prime gap up to x should be at least about 2e^{-gamma} log^2 x rather than Cramér's log^2 x.

green_2017_arithmetic_kakeya_conjecture_katz_tao/: Gives several equivalent forms of the Katz-Tao arithmetic Kakeya conjecture, proves a finite field variant of it, and records lower bounds.

hensley_1974_primes_intervals/: Proves that the largest admissible tuple in an interval of x integers exceeds the number of primes up to x by at least a constant times x/(log x)^2 for large x, so the prime k-tuples conjecture is incompatible with the inequality pi(x+y) <= pi(x) + pi(y).

herzog_1971_patterns_visible_nonvisible_lattice_points/: Characterizes exactly which prescribed patterns of visible and nonvisible lattice points can be realized by a translate in any dimension.

johnston_yang_2022_some_explicit_estimates_error_term_prime_number_theorem/: Gives explicit unconditional prime number theorem error terms such as |pi(x)-li(x)| <= 9.59x(log x)^0.515 exp(-0.8274 sqrt(log x)) for x >= 2.

kuperberg_2023_sums_singular_series_large_sets_tail/: Averages singular series over large sets of shifts, states the uniform Hardy–Littlewood prime-tuples conjecture (Conjecture 1.3) assumed by the 2026 conditional claims on problem 251, and bounds the tail of the distribution of primes in short intervals under it.

lebowitz_lockard_2025_increasing_sequences_decreasing_prime_factors/: Bounds the longest increasing sequence of integers up to x whose smallest prime factors decrease: at most about 2 sqrt(x)/log x unconditionally, and of order at least sqrt(x)/(log x)^2 under a Cramér-type prime gap conjecture.

martineau_2022_coprime_percolation_visibility_graphon_local_limit/: Identifies the local limit of the coprimality coloring of the integer lattice around a uniform point, and records percolation properties of the limit coloring that follow from Vardi's theorems.

matomaki_2016_multiplicative_functions_short_intervals/: Shows that the short-interval average of a bounded multiplicative function matches its long average in almost all intervals of any growing length.

maynard_2015_small_gaps_between_primes/: A multidimensional refinement of the GPY sieve shows that gaps between primes m apart are bounded for every m, with liminf of p_{n+1} - p_n at most 600.

maynard_2016_large_gaps_between_primes/: The Rankin bound for the largest prime gap below x is improved to hold with an arbitrarily large constant, answering a question of Erdos.

mcnew_2018_convex_hull_prime_number_graph/: Gives improved counts and gap bounds for the primes on the convex hull of the prime number graph, resolving conjectures of Pomerance and Tutaj.

merikoski_2020_limit_points_normalized_prime_gaps/: At least one third of positive reals are limit points of normalized prime gaps, and gaps between such limit points are bounded by an absolute constant.

openai_2026_additive_indecomposability_primes/: An 80-page manuscript of the OpenAI mathematics release claiming Ostmann's inverse Goldbach conjecture: no set that differs from the primes in finitely many elements is $A+B$ with $|A|,|B|\ge2$, by sieve, character-sum and tree-comparison arguments; the negative answer claimed for Problem 431.

openai_2026_positive_lower_density_large_prime_gaps/: A manuscript of the OpenAI mathematics release claiming that for every fixed $C>0$ the indices $n$ with $p_{n+1}-p_n>C\log p_n$ have positive lower density, by adjacent-interval sieve weights built on Bombieri and Vinogradov; claims Problem 968 through $p_n/n$, bears on Problems 234 and 5.

openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12/: Claims every finite-order Hecke L-function over Q(sqrt(-3)), hence every Dirichlet L-function and zeta, is zero-free for Re s > 11/12, by a mean-square bound for sextic-twisted Möbius sums (Poisson summation, cubic theta, quadratic large sieve); proposed input for Problems 770, 985, 969, 769, 1204 and 855.

openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_7_8/: A 199-page manuscript claiming that every finite-order Hecke L-function over Q(sqrt(-3)) and every Dirichlet L-function has no zero in Re s > 7/8, by comparing two representations of a completed cubic-theta character sum; it names no Erdős problem and deduces a polylogarithmic least-nonresidue bound.

openai_2026_uniform_exclusion_landau_siegel_zeros/: Claims an absolute constant c>0 with (1-beta) log q >= c for every real zero beta of every primitive nonprincipal real Dirichlet L-function of conductor q>=3, by comparing Hadamard and prime-divisibility bounds for an interpolation determinant; names no Erdős problem, touches pages via Siegel-zero inputs.

pintz_2016_polignac_numbers_conjectures_erdos_gaps_primes/: Uses Zhang's bounded gap theorem to prove Polignac numbers have positive lower density and that some interval [0,c] consists of limit points of normalized prime gaps.

pollack_2017_bounds_first_several_prime_character_nonresidues/: Shows every nontrivial Dirichlet character to a large modulus m has more than a fixed power of m prime nonresidues below the Burgess-Norton bound.

pollack_et_al_2013_sets_monotonicity_euler_totient_function/: Proves that any subset of [1,x] on which Euler's totient is monotone has size o(x), with a strong bound in the nonincreasing case and, in the nondecreasing case, a fixed fraction below the number of totient values; bounds the shifted collisions phi(n)=phi(n+k) uniformly over growing ranges of k; and records the computations behind the conjecture that the nondecreasing maximum equals pi(x)+64 for every x >= 31957.

pomerance_1979_prime_number_graph/: Proves by convex hulls that infinitely many n satisfy p_n^2 > p_{n-i}p_{n+i} for all 0 < i < n, that infinitely many n satisfy 2p_n < p_{n-i}+p_{n+i} for all 0 < i < n, and conjectures that the second defect is unbounded.

ramachandra_1976_grimm_s_problem_relating_factorisation_block/: Proves that n+1, ..., n+g have at least g distinct prime factors in all for g up to exp(c (log n)^{1/2}), a weakened form of Grimm's conjecture in that range, and that for k >= 2 at most pi(k) of u+1, ..., u+k are k-smooth once u is at least exp(C (log k)^2), using linear forms in logarithms.

richards_1974_incompatibility_two_conjectures_concerning_primes_discussion_use_computers_attacking_theoretical_pro/: Sketches the midpoint-sieve proof that some admissible set in an interval of length x exceeds pi(x) by (log 2 - o(1))x/(log x)^2, so the prime k-tuples conjecture contradicts pi(x+y) <= pi(x)+pi(y).

ruzsa_1982_small_sieve_ii_sifting_composite_numbers/: Shows that sifting by integers above 1 (not only primes) of bounded reciprocal sum can leave only x to the epsilon survivors, and bounds the least reciprocal sum of a covering system with distinct moduli up to x.

ruzsa_1995_few_multiples_many_primes/: Constructs, for each rho at least 3 and all large n, sets of n primes such that some interval of length rho times the largest prime contains fewer than C(rho) (n log n)^{1-1/[rho]} of their multiples.

segal_1962_x_y_x_y/: Shows pi(x+y) <= pi(x)+pi(y) for all x,y >= 2 is equivalent to p_n >= p_(n-q)+p_(q+1)-1 for n >= 3 and 1 <= q <= (n-1)/2, locates the least failing sum at a prime, and reports a machine check giving it for x+y <= 101,081.

shorey_2016_arithmetic_properties_blocks_consecutive_integers/: Surveys bounds for prime factors and powerfree parts of products of consecutive integers and shows the explicit abc-conjecture implies Erdős-Woods.

stadlmann_2022_mean_square_gap_between_primes/: Proves that the sum of squared gaps between consecutive primes up to x is at most x to the power 1.23 plus epsilon, improving the previous exponent 1.25.

tao_2023_convergence_alternating_series_erdos_assuming_hardy/: Shows the alternating series of (-1)^n n over the nth prime converges, assuming a strong quantitative Hardy-Littlewood prime tuples conjecture.

tao_2023_infinite_partial_sumsets_primes/: Proves there are infinite sets of natural numbers whose pairwise sums in one direction are all prime.

tao_2024_monotone_nondecreasing_sequences_euler_totient_function/: Proves that the largest subset of integers up to x on which Euler's totient is nondecreasing has size asymptotic to the number of primes up to x.

tschebotareff_1926_density_primes_substitution_class/: Tschebotareff's 1926 proof that the primes in a given substitution class have density equal to the class size over the order of the Galois group.

vardi_1998_prime_percolation/: Builds a random model of Gaussian primes and locates the critical step size for an unbounded walk, supporting the conjecture that no bounded-step walk exists.

vardi_1999_deterministic_percolation/: Shows the visible-lattice-point graph of coprime pairs has a unique infinite component of positive asymptotic density, using an almost-everywhere sieve.

warlimont_1991_problem_posed_i_z_ruzsa/: Determines the exact asymptotic constant log(2^5 3^6/23^3) for a relaxed version of Ruzsa's small-sieve covering-cost problem.

zhang_2014_bounded_gaps_between_primes/: Proves that consecutive primes differ by less than 70 million infinitely often, the first bounded prime gap result.

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This folder holds sources whose primary subject is Primes.

Sources with other primary subjects

Explicit links to this subject's problems support these cross-references.

• green_2026_100_open_problems
  • E1202
• romanoff_1934_uber_einige_satze_der_additiven
  • E0244
• erdos_1957_unsolved_problems
  • E0855
• erdos_1967_problems_prime_factors_consecutive_integers
  • E0890
• openai_2026_joint_dickman_law_consecutive_integers
  • E1201
• teravainen_2018_binary_correlations_multiplicative_functions
  • E1201
• tijdeman_1973_integers_many_small_prime_factors
  • E0240
• erdos_1976_problems_results_number_theoretic_properties_consecutive
  • E1201
• alexeev_2026_short_proofs_combinatorics_probability_number_theory
  • E1141
• erdos_1981_applications_graph_theory_combinatorial_methods_number
  • E0860
• erdos_1978_problems_results_combinatorial_analysis_combinatorial_number
  • E1143
• banks_2014_consecutive_primes_tuples
  • E0006
• erdos_1986_problems_number_theory
  • E1143
• ford_2018_long_gaps_between_primes
  • E0004
• granville_2020_sieving_intervals_siegel_zeros
  • E0004
  • E0855
• konyagin_2022_construction_schinzel_many_numbers_short_interval_without_small_prime_factors
  • E0855
• erdos_1965_recent_advances_current_problems_number_theory
  • E0005
• erdos_1980_old_new_problems_results_combinatorial_number_theory
  • E0428
• erdos_1980_survey_problems_combinatorial_number_theory
  • E1200
  • E1201
  • E1202
  • E1212
• erdos_1995_my_favourite_problems_number_theory_combinatorics
  • E0049
• guy_2004_unsolved_problems_number_theory
  • E0004
  • E0006
  • E0017
  • E0233
  • E0850
  • E0855
  • E0860
  • E1055
  • E1059
  • E1142
• erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk
  • E0004
• erdos_1997_some_my_favorite_problems_results
  • E0004
  • E0005