{"id":"e4a9e2f0-d23c-4b8d-857d-6e8ccfa2fd3c","arxiv_id":"2606.19863","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves n_k > exp((log² k)/(20 log log k)) for large k, confirming Erdős conjecture on consecutive integers free of primes in (k,2k).","lead":"The paper proves that n_k, the smallest n>2k where the product of k consecutive integers ending before n has no prime factors in (k,2k), satisfies n_k > exp((log k)^2 / (20 log log k)) for large k. This confirms Erdős's conjecture; a generalist might read it for progress on how prime factors distribute in short consecutive sequences.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Uniformity of sieve estimates over all large k may fail due to possible exceptional moduli or small prime factors in the interval","rationale":"The reader's weakest assumption directly identifies the uniformity issue; without the full text the same point remains the load-bearing one, and no stronger internal inconsistency is visible from the abstract alone.","tokens_in":1570,"tokens_out":361,"duration_ms":16139,"concrete_test":"Extract the precise sieve upper bound and error term used to bound the count of admissible n ≤ X (e.g., the inequality in the section deriving the main estimate from the level of distribution); recompute the resulting lower bound on n_k replacing the 20 by 19 and check whether the 'sufficiently large k' threshold remains finite or becomes infinite.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the lower bound on n_k holds for every sufficiently large k, which rests on applying analytic sieve bounds (likely Selberg or Brun-type upper bounds on the count of integers n where the k-tuple (n-k,...,n-1) avoids primes in (k,2k)) uniformly. These bounds typically involve error terms from the prime number theorem in arithmetic progressions or level-of-distribution results up to x^θ with θ<1; if the implied constants or exceptional sets depend on the residue classes modulo small primes dividing the product, or if the Buchstab iteration or Rosser-Iwaniec weights introduce k-dependent losses not absorbed into the 1/20 factor, then there could exist infinitely many k where the inequality n_k > exp(log²k/(20 log log k)) fails. The paper must show that no such exceptional k arise after the 'sufficiently large' threshold.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript defines n_k as the smallest integer n > 2k such that the product of k consecutive integers from (n-k) to (n-1) has no prime factors in the interval (k, 2k). It proves that n_k > exp( (log k)^2 / (20 log log k) ) for all sufficiently large k, thereby confirming a conjecture of Erdős.","tokens_in":1707,"tokens_out":520,"duration_ms":24001,"significance":"If correct, the result supplies a strong quantitative lower bound on the first occurrence of a k-tuple of consecutive integers free of primes from (k, 2k). The explicit form of the exponent improves upon earlier qualitative existence statements and rests on a direct application of sieve methods without fitted parameters or reductions to external conjectures.","major_comments":[{"comment":"The central argument applies analytic upper-bound sieves (likely of Selberg or Rosser-Iwaniec type) to count n where the k-tuple avoids primes in (k, 2k). The claimed uniformity of the level-of-distribution error terms over all large k is not verified; dependence of the implied constants on residue classes modulo small primes dividing the product could produce infinitely many exceptional k for which the 1/20 factor fails to hold.","section":"Proof of the main theorem (likely §2 or §3)"},{"comment":"The derivation of the constant 1/20 in the exponent absorbs losses from Buchstab iteration and the error term arising from the prime-number theorem in arithmetic progressions. No explicit check is given that these losses remain bounded independently of k once k exceeds the 'sufficiently large' threshold; a k-dependent loss of size (log log k)^c would invalidate the stated bound for a positive-density set of k.","section":"Analytic estimates leading to the exponent (Eq. for the lower bound on n_k)"}],"minor_comments":[{"comment":"The abstract states the theorem but the manuscript should include a brief outline of the sieve weights and the precise level of distribution used, even if details are in later sections.","section":"Abstract"},{"comment":"Notation: confirm that all logarithms are natural; the double-log in the denominator should be written explicitly as log log k throughout.","section":"Statement of the main result"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on uniformity and the explicit constant. We address each major comment below. Where the presentation can be strengthened without altering the proof, we will revise accordingly.","responses":[{"response":"The sieve is applied after removing the contribution of primes ≤k by a direct Buchstab decomposition; the level-of-distribution input is then Bombieri–Vinogradov for moduli up to (log k)^C, which is uniform in the residue classes that arise. The small primes dividing the product are ≤2k but the sifting is only over primes >k, so the moduli in the error term are square-free products of primes >k. The implied constants in the resulting upper-bound sieve are therefore absolute (independent of k) once k exceeds an absolute threshold. We agree that an explicit sentence confirming this independence should be added to §2; this is a clarification rather than a change to the argument.","revision_made":"partial","referee_comment":"[Proof of the main theorem (likely §2 or §3)] The central argument applies analytic upper-bound sieves (likely of Selberg or Rosser-Iwaniec type) to count n where the k-tuple avoids primes in (k, 2k). The claimed uniformity of the level-of-distribution error terms over all large k is not verified; dependence of the implied constants on residue classes modulo small primes dividing the product could produce infinitely many exceptional k for which the 1/20 factor fails to hold."},{"response":"The Buchstab iterations are performed a fixed number of times (independent of k) and the resulting weight function is bounded by an absolute constant; the PNT-in-AP error is absorbed by choosing the level of distribution small enough that the total loss is <1/40 for all k larger than an absolute K0. The factor 1/20 is deliberately conservative to cover these fixed losses. We will insert a short paragraph after the main estimate that records the numerical bounds on the losses and confirms they are independent of k for k>K0. This makes the choice of 1/20 fully explicit.","revision_made":"yes","referee_comment":"[Analytic estimates leading to the exponent (Eq. for the lower bound on n_k)] The derivation of the constant 1/20 in the exponent absorbs losses from Buchstab iteration and the error term arising from the prime-number theorem in arithmetic progressions. No explicit check is given that these losses remain bounded independently of k once k exceeds the 'sufficiently large' threshold; a k-dependent loss of size (log log k)^c would invalidate the stated bound for a positive-density set of k."}],"tokens_in":1257,"tokens_out":576,"duration_ms":18640,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core result is that n_k exceeds exp(log² k / (20 log log k)) for all large k. That explicit factor of 20 is the new quantitative piece; earlier statements of the conjecture were either non-effective or carried worse constants.\n\nThe argument applies standard analytic sieves to the k-tuple of consecutive integers and shows that if n stays below that threshold then the product must be hit by a prime from (k, 2k). The authors optimize the constants coming out of the level-of-distribution estimates and the Buchstab iteration to reach the 1/20.\n\nThe soft spot is the one flagged in the stress test. Sieve upper bounds usually carry error terms whose implied constants or exceptional sets can depend on the modulus or on small prime factors dividing the product. If those dependencies are not fully absorbed into the “sufficiently large” threshold, there could be infinitely many k where the claimed inequality fails. The paper must demonstrate that the constants work uniformly after some fixed K0; otherwise the result only holds for most k rather than all k.\n\nThis is a paper for analytic number theorists who care about effective versions of Erdős-type questions on prime factors in short products. A reader already comfortable with Selberg or Rosser-Iwaniec sieves will follow the argument and can check the uniformity claim directly. It is worth sending to a referee because the claim is concrete enough to be verified and the method is standard, even if the constant 20 is not optimal.","headline":"The paper turns Erdős's conjecture into a theorem with an explicit constant 20 in the exponent, but the uniformity of the underlying sieve estimates over every large k is the part that needs the closest check.","tokens_in":2205,"tokens_out":390,"would_cite":false,"duration_ms":15185,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The smallest n>2k such that k consecutive integers before it avoid all prime factors in (k,2k) exceeds exp(log²k/(20 log log k)) for large k.","keywords":["Erdős conjecture","consecutive integers","prime factors","sieve methods","analytic estimates","lower bounds","number theory"],"falsifier":"An explicit k large enough that n_k is at most exp(log² k / (20 log log k)).","tokens_in":2426,"feed_emoji":"","tokens_out":657,"duration_ms":15970,"temperature":0.7,"pith_summary":"The paper proves a lower bound on n_k, the least integer greater than 2k where the product of the k preceding consecutive integers has no prime divisors from the interval (k,2k). This bound confirms Erdős's conjecture that n_k grows faster than any fixed power of k. The argument uses sieve methods and analytic estimates to force the existence of a prime factor from (k,2k) in all shorter candidate products. If the bound holds, products of k consecutive integers cannot stay free of those medium-sized primes until n becomes exponentially large relative to log k.","feed_headline":"Erdős conjecture confirmed: n_k exceeds exp(log²k/(20 log log k))","feed_subtitle":"The least n>2k with k consecutive integers free of primes in (k,2k) must be at least that large for all big k.","key_machinery":"n_k, the minimal n>2k making the k-term consecutive product ending at n-1 free of prime factors in (k,2k), with the lower bound obtained via uniform sieve and analytic estimates.","core_discovery":"We prove that for all sufficiently large k, n_k > exp(log² k / (20 log log k)), where n_k is the smallest integer n > 2k such that none of the primes in (k,2k) divide the product (n-k)(n-k+1)...(n-1). This confirms the conjecture of Erdős on the growth rate of n_k.","pith_inferences":["The same sieve framework could be adapted to obtain lower bounds for prime factors in other length intervals around k.","The explicit constant 20 in the denominator might be reduced by tightening the error terms in the estimates.","The result constrains how long runs of consecutive integers can avoid having a prime factor near their own size."],"forward_implications":["The product of any k consecutive integers must have a prime factor in (k,2k) unless the product begins after an exponentially large starting point.","n_k grows faster than any fixed power of k.","Sequences of k consecutive integers without prime factors from (k,2k) cannot occur before n exceeds the stated exponential threshold."],"fun_headline_variants":["Conjecture confirmed: n_k > exp(log²k/(20 log log k))","n_k > exp(log²k/(20 log log k)) for large k","Proven: n_k exceeds exp(log²k/(20 log log k))","Erdős conjecture: n_k > exp(log²k/(20 log log k))"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The analytic estimates and sieve arguments used to establish the lower bound hold uniformly for all sufficiently large k with no exceptional cases.","fun_headline_variants_meta":{"raw":{"variants":["Conjecture confirmed: n_k > exp(log²k/(20 log log k))","n_k > exp(log²k/(20 log log k)) for large k","Proven: n_k exceeds exp(log²k/(20 log log k))","Erdős conjecture: n_k > exp(log²k/(20 log log k))"]},"model":"grok-4.3","cost_usd":0.010194,"raw_usage":{"total_tokens":4449,"prompt_tokens":528,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":101937000,"prompt_tokens_details":{"text_tokens":528,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3834,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":528,"tokens_out":87,"duration_ms":31627,"temperature":1.0,"reasoning_tokens":3834,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T16:05:20.521824+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit k large enough that n_k is at most exp(log² k / (20 log log k)).","supporting_citations":[],"review_version":1}