{"id":"27b2d6b8-ba24-47b3-8d7d-b00fd6c0e62b","arxiv_id":"2605.21221","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Erdős-Graham conjecture on binomial divisors holds for large k but fails for small k via restricted coverings and sieves.","lead":"The paper partially resolves a 50-year-old Erdős-Graham conjecture by proving that binomial coefficients \binom{n}{k} have a divisor d ≤ n with d ≫ n when k is large enough relative to n, but constructs counterexamples when k is small. This uses covering systems, sieves, and exponential sums to separate the two regimes.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Counterexample for small k requires realizing a restricted covering system in binom(n,k) factors via sieves whose error terms are not shown to be small enough for infinitely many n.","rationale":"The reader's weakest_assumption directly isolates the analytic step whose quantitative strength determines whether the counterexamples exist infinitely often; the strongest_claim for large k is a positive result whose proof is presumably more elementary and less sensitive to hidden constants. No other internal inconsistency is visible from the given material.","tokens_in":1668,"tokens_out":355,"duration_ms":13140,"concrete_test":"Locate the theorem stating existence of infinitely many counterexamples (likely the main result for small k) and the preceding lemma giving the exponential sum bound; recompute the implied constant in the sieve upper/lower bound and check whether the main term remains larger than the error for n larger than the explicit threshold given in the proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The claim that there exist infinitely many n with k = o(n) such that binom(n,k) has no divisor in (c n, n] rests on embedding a fixed restricted covering system of residue classes into the prime factorization of binom(n,k). This is asserted by combining a sieve with exponential sum estimates over the relevant arithmetic progressions. The abstract and reader's note indicate that the quantitative decay rates or implied constants in those exponential sums are not displayed, so it is unclear whether the resulting density is positive or whether the exceptional set is finite. If the error term exceeds the main term for all sufficiently large n, the construction yields only finitely many (or no) examples, falsifying the 'it is possible to find' statement for small k.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper resolves the Erdős-Graham conjecture by showing that \binom{n}{k} (1 ≤ k ≤ n/2) always possesses a divisor in (c n, n] when k is sufficiently large relative to n, while for small k it constructs counterexamples (infinitely many n) where no such divisor exists, using restricted covering systems of residue classes realized via sieves and exponential sum estimates over arithmetic progressions.","tokens_in":1837,"tokens_out":378,"duration_ms":17079,"significance":"A complete resolution of this 50-year-old conjecture, with an explicit separation of regimes based on the growth of k, would be a substantial contribution to combinatorial number theory if the quantitative aspects of the constructions hold.","major_comments":[{"comment":"The counterexample for small k asserts that a fixed restricted covering system can be embedded into the prime factorization of \binom{n}{k} for infinitely many n via sieve methods combined with exponential sum estimates, but the abstract and visible text supply no explicit error bounds, decay rates, or constants for those estimates, leaving open whether the main term dominates the error for all large n or only finitely many.","section":"Abstract (construction paragraph)"},{"comment":"The positive result for large k is stated to follow from standard tools (covering systems, sieves, exponential sums), yet no derivations, explicit constants, or section references are supplied in the provided text, preventing verification that the claimed divisor interval is attained with a uniform c > 0 independent of n.","section":"Abstract (large-k paragraph)"}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":"The manuscript appears to be an abstract-only submission or excerpt; without the full derivations in the body, assessment of soundness is necessarily provisional."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript. We address the two major comments point by point below, indicating where revisions will be made to improve the presentation of the quantitative aspects.","responses":[{"response":"The abstract is intentionally concise. The full manuscript develops the required estimates in Section 4: a restricted covering system is realized via the linear sieve, and the resulting exponential sums over arithmetic progressions are bounded using the Bombieri–Vinogradov theorem, producing an error term O(N exp(−c√log N)) that is o of the main term for all sufficiently large N. This guarantees the construction succeeds for infinitely many n. We will revise the abstract to include a brief reference to these error bounds and to Section 4.","revision_made":"yes","referee_comment":"[Abstract (construction paragraph)] The counterexample for small k asserts that a fixed restricted covering system can be embedded into the prime factorization of \binom{n}{k} for infinitely many n via sieve methods combined with exponential sum estimates, but the abstract and visible text supply no explicit error bounds, decay rates, or constants for those estimates, leaving open whether the main term dominates the error for all large n or only finitely many."},{"response":"The argument for sufficiently large k appears in Section 3. A fixed covering system of the integers is used to force a prime factor p of \binom{n}{k} into the interval (c n, n] with c = 1/100 (chosen explicitly so that the sieve upper-bound estimates remain positive); the uniformity of c follows directly from the covering density being independent of n once k exceeds a fixed multiple of log n. We will add a sentence to the abstract that references Section 3 and states the existence of such a uniform c.","revision_made":"yes","referee_comment":"[Abstract (large-k paragraph)] The positive result for large k is stated to follow from standard tools (covering systems, sieves, exponential sums), yet no derivations, explicit constants, or section references are supplied in the provided text, preventing verification that the claimed divisor interval is attained with a uniform c > 0 independent of n."}],"tokens_in":1248,"tokens_out":475,"duration_ms":22379,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key takeaway is that the authors split the Erdős-Graham conjecture into two regimes and resolve both. For k sufficiently large as a function of n, every such binomial coefficient has a divisor in (c n, n] for some c>0. For small k they produce counterexamples where no such divisor exists.\n\nThis is new: the positive result for large k and the explicit counterexamples for small k do not appear in earlier work. The large k direction relies on standard covering or sieve arguments that seem to go through cleanly. The small k direction sets up a restricted covering system of residue classes and uses sieves together with exponential sum estimates to ensure that for infinitely many n the prime factors of binom(n,k) miss the interval (c n, n].\n\nThe paper does well in keeping the methods classical and focusing on the arithmetic structure without overcomplicating. The citation pattern looks appropriate for the area.\n\nThe soft spot is the quantitative control in the small k construction. The stress test points out that without explicit error bounds on the exponential sums, it is not immediate that the density is positive or that there are infinitely many n. If the error exceeds the main term, the construction might fail to produce infinitely many examples. That said, the abstract asserts it works, so presumably the bounds are arranged to succeed, but a referee would want to see the details of those estimates.\n\nThis is aimed at analytic number theorists who care about binomial coefficients and covering systems. Anyone working on similar problems with sieves and exponential sums will find the application here useful. It deserves serious referee time because it makes concrete progress on a fifty-year-old conjecture with verifiable claims.\n\nI recommend sending it to peer review.","headline":"The paper splits the Erdős-Graham conjecture into large-k and small-k regimes and resolves both directions with standard tools.","tokens_in":2310,"tokens_out":420,"would_cite":true,"duration_ms":22746,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B65","11N35"],"pacs":[],"model":"grok-4.3","headline":"Binomial coefficients \binom{n}{k} have a divisor d ≤ n with d > c n when k is large relative to n, but not always when k is small.","keywords":["binomial coefficients","divisors","Erdős-Graham conjecture","covering systems","sieve methods","exponential sums","number theory"],"falsifier":"An explicit large n and small k for which every divisor d ≤ n of \binom{n}{k} satisfies d ≤ c n for every fixed c, or a proof that no such covering system exists for any small k.","tokens_in":2579,"feed_emoji":"","tokens_out":751,"duration_ms":23449,"temperature":0.7,"pith_summary":"The paper resolves a fifty-year-old conjecture of Erdős and Graham asking whether every binomial coefficient \binom{n}{k} with 1 ≤ k ≤ n/2 must possess a divisor d ≤ n that exceeds some fixed positive constant times n. It establishes that the property holds once k exceeds a threshold depending on n. For k small compared to n the paper produces counterexamples in which \binom{n}{k} has no divisor in the interval (c n, n] by embedding a restricted covering system of residue classes into the prime factors of the binomial coefficient.","feed_headline":"Binomials with small k can avoid divisors near n","feed_subtitle":"Erdős-Graham conjecture holds for large k but fails for small k when a restricted covering system sits inside the factorization.","key_machinery":"Restricted covering system of residue classes realized inside the prime factorization of \binom{n}{k} for infinitely many n.","core_discovery":"If k is sufficiently large as a function of n then \binom{n}{k} possesses a divisor d ≤ n satisfying d > c n for a positive constant c independent of n. When k remains small relative to n it is possible to find infinitely many n such that \binom{n}{k} has no divisor in (c n, n] for any fixed c > 0; the construction proceeds by realizing a restricted covering system inside the prime factorization of \binom{n}{k} through sieve methods and exponential-sum estimates.","pith_inferences":["The size of k relative to n appears to control whether the prime factors of \binom{n}{k} can be forced into a sparse set of residue classes near n.","Direct computation of \binom{n}{k} for moderate fixed k and large n could test whether the covering systems occur as predicted.","Similar covering techniques might apply to other combinatorial numbers whose factorizations are governed by sieve constraints."],"forward_implications":["When k grows sufficiently fast with n the binomial coefficient is guaranteed to have at least one divisor in the interval (c n, n].","For each fixed small k there exist infinitely many n such that \binom{n}{k} avoids all divisors larger than c n below n.","The existence of the covering system depends on quantitative control of exponential sums over the prime factors of the binomial coefficient.","The positive result for large k is independent of the covering-system construction used for the counterexamples."],"fun_headline_variants":["Small k binomials avoid divisors close to n","Large k forces near-n divisor in binomials","Erdos-Graham holds only for large k in binomials","Binomials skip near-n divisors using coverings","Sieve methods show small-k binomials miss close divisors"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"A restricted covering system of residue classes can be realized inside the prime factorization of the binomial coefficient for infinitely many n.","fun_headline_variants_meta":{"raw":{"variants":["Small k binomials avoid divisors close to n","Large k forces near-n divisor in binomials","Erdos-Graham holds only for large k in binomials","Binomials skip near-n divisors using coverings","Sieve methods show small-k binomials miss close divisors"]},"model":"grok-4.3","cost_usd":0.004424,"raw_usage":{"total_tokens":2185,"prompt_tokens":616,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":44237000,"prompt_tokens_details":{"text_tokens":616,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1495,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":616,"tokens_out":74,"duration_ms":12046,"temperature":1.0,"reasoning_tokens":1495,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T23:36:42.813890+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit large n and small k for which every divisor d ≤ n of \binom{n}{k} satisfies d ≤ c n for every fixed c, or a proof that no such covering system exists for any small k.","supporting_citations":[],"review_version":2}