{"id":"c4b4f2f9-42a1-4b17-93a5-d2a393eeb688","arxiv_id":"2607.16032","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":8,"one_line_summary":"The asymptotic density of n with P^+(n) < P^+(n+1) is shown to be > 0.280, a new record.","lead":"An analytic number theory paper claims the density of n with P^+(n) < P^+(n+1) exceeds 0.280, improving the previous 0.2017. The proof refines a recent sieve approach but leaves several steps explicitly unproved and bases its final constant on unreproducible numerical calculation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The final constant 0.280 rests on an unreported numerical evaluation of an integral involving C(1−η) outside the range where Theorem 3.2 gives explicit values; the main theorem is not yet supported as written.","rationale":"The reader's conditional verdict identifies the same general region as the weakest assumption: the unreproducible numerical final constant and the unproved extension of Lemma 2.6. I agree that these are real gaps. Among them, the integration over C(1−η) in (4.16) is the most load-bearing because it enters the headline constant directly, while the Lemma 2.6 extension appears in an upper-bound remainder whose numerical contribution might be less decisive. The paper is a plausible research announcement rather than a finished proof: the main structure follows the Lu–Wang method, and the claimed improvement may well be true, but the supporting computations are not supplied. A conditional verdict is appropriate until the omitted numerical derivation and the full C(η) range are documented. Therefore I recommend no change to the reader's verdict.","tokens_in":23055,"tokens_out":3048,"duration_ms":33908,"concrete_test":"Ask the author to provide the numerical code or an analytic derivation for the integral in (4.16) over η∈[0.1348,0.5], including explicit lower bounds for C(u) for u∈[0.55,0.8652] obtained from the construction in §3.1. Then perform a validated numerical integration: if the maximum is always attained by the first, explicit branch, report the resulting value; if the unknown C(1−η) branch is ever selected, show that replacing it with the best proven lower bound still yields C(c,δ1)−2ν>0.280. A simple pass/fail check is whether the claimed 0.280 remains true when the unknown branch is set to 0; if it does not, the final constant is unproven without additional work.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the numerical inequality C(c,δ1)−2ν>0.280 at the end of §4.4. The definition of C(c,δ1) contains the term ∫_c^{1/2} max( (1−2η+2t1(η))/(2(1−η)(1−η+t1(η))), C(1−η)/(1−η) ) dη. For c=0.1348 and η∈[c,1/2], the argument 1−η ranges over [0.5,0.8652]. However, Theorem 3.2 gives explicit lower bounds for C(·) only for arguments in (0.5,0.55); for larger arguments it merely asserts the existence of an effective positive constant. Moreover, the proof of Theorem 3.2 for the E2 terms is explicitly compressed with 'Omitting the details.' The paper does not explain how the integral in (4.16) was evaluated on the full interval, nor does it justify that the maximum is always attained by the explicit first branch. The phrase 'By numerical calculation' is not a reproducible derivation. Consequently, even if every sieve estimate in §4.2—including the guessed extension of Lemma 2.6 in (4.19)—is correct, the final numerical improvement over 0.2017 has not been demonstrated. This is a correctness risk in the proof of Theorem 1.4, not a disagreement with the underlying heuristic.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses Erdős–Turán's conjecture on the largest prime factors of consecutive integers. It claims three main results: Theorem 1.4 gives a lower asymptotic density >0.280 for the pattern P^+(n)<P^+(n+1), improving the previous 0.2017 record of Lü–Wang; Theorem 1.5 gives a positive density of n with P^+(n)<P^+(n+1)<x^{41/107+ε}; and Theorem 1.6 gives an upper bound for T_c(x), the count of primes p≤x with P^+(p−1)≥p^c. The proof of Theorem 1.4 combines an Erdős–Pomerance-type decomposition, estimates for friable numbers in arithmetic progressions, the Rosser–Iwaniec sieve, and a new parameter t_1 intended to improve the main-term/error-term balance. The final constant is obtained by optimizing parameters c and δ_1 and by numerically evaluating a lower-bound integral involving C(1−η).","tokens_in":23535,"tokens_out":3760,"duration_ms":34550,"significance":"If fully supported, Theorem 1.4 would be a substantial improvement of the best known lower density for a long-standing Erdős–Turán problem, and Theorems 1.5 and 1.6 would provide useful complementary results. The paper builds on independent published lemmas by Hildebrand, Iwaniec, Pascadi, Wu, Ding–Wang, and Lü–Wang, and it introduces a plausible new parameter t_1 that may be of independent value. The main limitation is that the final numerical constant rests on several explicitly unproved steps: a guessed extension of Lemma 2.6, four identities left to the reader, and an integral involving C(1−η) outside the range where explicit values are provided. These gaps are load-bearing, so the significance is conditional on completing or replacing them.","major_comments":[{"comment":"The final numerical inequality C(c,δ1)−2ν>0.280 is the whole content of Theorem 1.4. The integral in (4.16) ranges over η∈[c,1/2], so it requires values of C(1−η) for arguments up to 0.8652. Theorem 3.2 gives explicit lower bounds for C(η) only for η∈(0.5,0.55); outside that range it asserts only the existence of an effective positive constant. The paper does not state how the maximum with the first branch was evaluated, nor how the integral over the full interval was computed. The phrase 'By numerical calculation' is not a reproducible derivation. Consequently the displayed constant 0.280 is not the conclusion of a finished proof.","section":"§4.4, Eq. (4.16) and definition of C(c,δ1)"},{"comment":"The upper bound for D, and hence the bounds for R and S'_5 in (4.20), use the assertion 'We guess that Lemma 2.6 also holds for l=d=d_1p_2 with the above conditions'. The validity of Lemma 2.6 for this composite l is load-bearing: without it the bound for D is unproved. A proof or a reference for this extension must be supplied, or the argument must be rewritten to avoid it.","section":"§4.3, Eq. (4.19)"},{"comment":"The derivation of B_2 (and then R and S'_5) relies on four displayed identities whose verification is 'left to interested readers'. These identities connect the count of l=d_1p_2 with weighted divisor sums over p'. Since they feed directly into (4.20), they are not optional exercises: they need a proof in the paper or a reference.","section":"§4.3, after Eq. (4.18)"},{"comment":"The proof of the C(η) lower bound for η near 0.55 is compressed: for the E_2 terms the text says 'Omitting the details', and Theorem 3.2 only promises an effective C(η) for η∈(0.55,0.8652) without giving values. Because (4.16) uses C(1−η) exactly in that non-explicit range, the numerical evaluation of the final integral requires either explicit lower bounds for C(η) over the whole range or a separate numerical scheme with stated error bounds.","section":"§3.1, proof of Theorem 3.2"}],"minor_comments":[{"comment":"There are numerous typographical slips: 'aribitary', 'frist', 'pragh', 'satistying', 'Conlcuding', 'Möbius' rendering, and inconsistent notation S_B/SB and S_5/S′_5. These do not affect the mathematics but should be corrected.","section":"Throughout"},{"comment":"The numerical values t_1(η), t_1(c), and δ(c) are presented through figures and 'numerical calculation'. For reproducibility, provide either exact expressions or a short script/table of the values used for c=0.1348, δ_1=0.417.","section":"§4.4 and Figures 1–3"},{"comment":"The proof says 'we claim that' a sieve upper bound holds, referring forward to (4.10)–(4.14). A brief indication of how Lemma 2.4 and Lemma 2.6 justify the claim would improve readability.","section":"Lemma 4.1 proof"}],"recommendation":"major_revision","confidential_remarks":"The reader's conditional verdict agrees with my reading: the main structure is plausible and the announced improvement is significant, but the final constant is supported only by unproved identities and a guessed lemma extension. I recommend major revision rather than rejection because the missing pieces appear to be within the manuscript's scope to supply."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper claims the asymptotic lower density of n with P+(n) < P+(n+1) is greater than 0.280, improving Lü–Wang's 0.2017, plus a positive-density result for P+(n) < P+(n+1) < x^{41/107+ε} and a bound for shifted primes with large prime factors. The main advance is an optimization of the Lü–Wang framework: a new parameter t1 improves the sieve main term, and the max with the C(1−η) term from Theorem 3.2 captures cases near η=1/2 that the older method lost. That is a genuine refinement, and the paper is honest about its debts: the record of prior results is accurate, the citation pattern is clean, and there is no circularity — the constants are optimized after the proof, not assumed.\n\nThe problems are where the manuscript says it is leaving things out. Section 4.3 contains four identities whose verification is explicitly left to the reader, and they are load-bearing for the estimates of R and S5'. Equation (4.19) uses a guessed extension of Lemma 2.6 to l = d1p2, with the note that it is “not the key” — but it is used in the upper bound for D, so it is part of the proof. More seriously, the final constant C(c,δ1)−2ν > 0.280 rests on the integral over η∈[c,1/2] involving C(1−η). Theorem 3.2 gives explicit lower bounds for C only around 0.5–0.55; for arguments up to 0.8652 it only asserts existence. The paper says “By numerical calculation,” but does not say how the integral was evaluated on that full range or whether the max is always attained by the first branch. The same theorem's proof also compresses the E2 terms with “Omitting the details.” These are not manufactured concerns; they are admissions in the manuscript. The final numerical value is the result, so an unreproducible constant is a genuine correctness risk.\n\nI would not call the paper unsound in conception. The method is a known one, the improvements are plausible, and Theorem 1.5 appears to follow cleanly from the quoted Bombieri–Vinogradov results. But the proof as written does not yet demonstrate the advertised 0.280. If the author supplies the omitted derivations, proves or replaces the guessed lemma, and gives reproducible numerical verification of the integral, I would expect the result to hold.\n\nSend it to peer review — a serious referee can check the sieve steps and push for the missing details. But it should not be accepted until the final constant is independently reproducible.","headline":"Plausible improvement of the Erdős–Turán lower density to 0.280, but the final constant depends on unreported numerical work and at least one explicitly guessed lemma; worth a serious referee, not acceptance as written.","tokens_in":23880,"tokens_out":1746,"would_cite":false,"duration_ms":20550,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N25","11N36","11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The author proves that for large x, more than 28% of integers n below x have P^+(n) < P^+(n+1), and that a positive density satisfies the stronger condition P^+(n) < P^+(n+1) < x^{41/107+ε}.","keywords":["largest prime factor","consecutive integers","Erdős–Turán conjecture","lower density","smooth numbers","arithmetic progressions","sieve methods","shifted primes"],"falsifier":"Recompute C(c,δ1) at c=0.1348, δ1=0.417 by independently evaluating the integral in (4.16) over η∈[c,1/2], using any admissible values of C(1−η) on (0.55,0.8652]. Also test the claimed extension of Lemma 2.6 for l=d1p2 in the stated ranges; a single counterexample would invalidate the bound on S'5.","tokens_in":22956,"feed_emoji":"🧮","tokens_out":7293,"duration_ms":59705,"temperature":0.7,"pith_summary":"The paper attacks a long-standing conjecture asserting that the largest prime factor of an integer is smaller than that of its successor exactly half of the time. Its main theorem pushes the known lower density from 0.2017 past 0.280 by optimizing a sieve-and-counting argument: it drops a lossy logarithmic sieve weight in the one-large-prime-factor case, introduces an auxiliary parameter t1 to extract a larger main term in the large-prime case, and takes the maximum of two lower bounds when the largest prime of n+1 sits just above the square-root threshold. The proof also yields a positive-density result for the stronger condition that both largest prime factors stay below x^{41/107+ε}, and a new upper bound on how often p−1 has a very large prime factor. If correct, the result shows that the one-sided pattern has density well above 28%, still far from the conjectured 50%.","feed_headline":"Over 28% of integers beat their successor in largest prime factor","feed_subtitle":"Improves the prior 0.2017 record and moves toward the conjectured 1/2 density.","key_machinery":"The proof's engine is a four-region decomposition of n by the sizes of P^+(n) and P^+(n+1), plus two innovations. First, in the region where n+1 has one moderately large prime factor, it abandons the logarithmic sieve weight that caused losses and instead counts n+1 by whether it has one or two required prime divisors, using a generalized Bombieri–Vinogradov-type theorem for smooth numbers and a linear sieve. Second, in the region where P^+(n+1)>x^{1/2}, it introduces a parameter t1 that shifts weight between a main term and an error term; choosing the largest t1 for which the error term stays nonnegative improves the main-term constant. The final number is the maximum of two lower bounds, o","core_discovery":"Theorem 1.4 asserts that as x→∞, the count of n<x with P^+(n)<P^+(n+1) exceeds 0.280x, and by symmetry the same holds for the reverse inequality. The constant comes from a decomposition of n according to the sizes of P^+(n) and P^+(n+1): a cutoff c=0.1348 separates small and large cases, and each piece is estimated either by the standard smooth-number distribution or by a new lower-bound function C(η) for smooth numbers in arithmetic progressions. With the choices δ1=0.417 and an optimized auxiliary function t1(η), the assembled lower bound C(c,δ1)−2ν is reported to exceed 0.280.","pith_inferences":["The optimal function t1(η) is presented graphically and through numerical calculation rather than a closed formula; making the numerical values public would let others audit the 0.280 constant without re-running the full sieve argument.","The same t1-rebalancing trick may transfer to the k-term ordering version of the conjecture, where a k! density bound could be improved by an analogous parameter choice.","If the unproved extension of Lemma 2.6 to moduli of the form l=d1p2 fails, the bound on the error term S'5 could be larger, potentially pulling the final constant below 0.280; this is the most vulnerable spot of the numerical conclusion.","The min formula in Theorem 1.6 suggests that the two bounds cross: near c=1 the logarithmic term dominates, while near c=1/2 the (1−δ)/(2c) term dominates; a sharper δ(c) from the method's remark would further improve the shifted-prime bound."],"forward_implications":["The lower density of the pattern P^+(n)<P^+(n+1) is greater than 0.280, and the same holds for the reverse pattern.","For every ε>0, a positive density of n satisfy the stronger condition P^+(n)<P^+(n+1)<x^{41/107+ε}.","For shifted primes, the upper limit of the proportion of primes p≤x with P^+(p−1)≥p^c is at most min(−(7/2)log c, (1−δ(c))/(2c)).","The effective version of the smooth-numbers-in-progressions bound can supply explicit constants for related friable-integer problems.","The new constant supersedes the previous 0.2017 record toward the conjectured 1/2 density."],"fun_headline_variants":["More than 28% of integers beat next integer in largest prime factor","Improved density for P^+(n) < P^+(n+1): over 0.280","Largest prime factor of n is smaller than n+1's for >28% of n","New bound: 28% of consecutive pairs have increasing largest prime factor","Density of n with P^+(n) < P^+(n+1) now exceeds 0.280"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The final inequality stands on an unstated numerical evaluation of an integral involving C(η) for η beyond the range where the paper computes C(η) explicitly, and on an assumed but unproved extension of a sieve estimate to composite moduli of the form d1p2; if either fails, the 0.280 figure may not follow.","fun_headline_variants_meta":{"raw":{"variants":["More than 28% of integers beat next integer in largest prime factor","Improved density for P^+(n) < P^+(n+1): over 0.280","Largest prime factor of n is smaller than n+1's for >28% of n","New bound: 28% of consecutive pairs have increasing largest prime factor","Density of n with P^+(n) < P^+(n+1) now exceeds 0.280"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000436,"raw_usage":{"total_tokens":2057,"prompt_tokens":748,"completion_tokens":1309,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":1191}},"tokens_in":492,"tokens_out":1309,"duration_ms":12001,"temperature":1.0,"reasoning_tokens":1191,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T21:33:09.580023+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute C(c,δ1) at c=0.1348, δ1=0.417 by independently evaluating the integral in (4.16) over η∈[c,1/2], using any admissible values of C(1−η) on (0.55,0.8652]. Also test the claimed extension of Lemma 2.6 for l=d1p2 in the stated ranges; a single counterexample would invalidate the bound on S'5.","supporting_citations":[],"review_version":1}