{"id":"3557051e-e66e-42d8-8d9c-a27480cd2154","arxiv_id":"2607.07641","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A new abc-type conjecture with H(n)=γ(n)/(logγ(n))^{ω(n)} is proposed; conditional on it, for each fixed y, limsup_{x→∞} W(x,y) loglog x / log x = 1.","lead":"This paper proposes a new strengthening of the abc conjecture, replacing the radical γ(n) by H(n)=γ(n)/(log γ(n))^{ω(n)}. If true, it would imply a precise limit for the number of distinct prime factors in short intervals of length y.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Conjecture 1 is an unproven, strictly stronger-than-abc statement; if it fails, Theorems 1–2 and Corollary 1 collapse, and the paper supplies no numerical check of it.","rationale":"The reader's weakest-assumption analysis is correct: the paper's main theorems are conditional on Conjecture 1, and the conjecture is unproven and potentially false. My independent read found no additional fatal flaw in the conditional derivation; the displayed exponent '2y-2' in §7.2 appears to be a typo for '2^{y-2}' (the degree of P_y), and with that correction the asymptotic argument works. Thus the proof of Theorem 1 does not change the assessment. The true remaining risk is the status of Conjecture 1 itself. The paper provides only a heuristic (§3) and an explicit open problem (§8, Problem 2) asking whether the conjecture is false, with no numerical evidence. Since the conditional value of the paper is real but the central conjecture is unsupported, a conditional acceptance is the appropriate verdict.","tokens_in":10953,"tokens_out":27593,"duration_ms":245758,"concrete_test":"Compute log c / log H(abc) for all abc triples in the abc@home database with c up to ~10^18, and also for the first 10^6 triples generated by Proposition 3's construction. If any triple has log c / log H(abc) > 1.01 (i.e., H(abc) < c^{0.99}), or if any H(abc) < 1, Conjecture 1 is false. Absence of such a triple for a large sample would support the conjecture but, of course, not prove it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central load-bearing assumption is the truth of Conjecture 1, a new statement strictly stronger than the abc conjecture. The heuristic in §3 depends on a density/independence estimate (Proposition 4) that is plausible but not a proof, and the paper explicitly leaves open the possibility that Conjecture 1 is false (§8, Problem 2). Proposition 3 already constructs infinitely many triples with H(abc) much smaller than c, namely c > H exp(2 log H / log log H), so the gap between current lower bounds and a counterexample is only a factor of exp(O(log c / log log c)). If any infinite family satisfies H(abc) < c^{1-ε} for a fixed ε>0, Conjecture 1 is false and all conditional results vanish. No numerical evidence or independent computational verification is provided to indicate that such families do not exist. This is therefore the weakest point of the paper, and it is exactly the point on which the reader's verdict turns.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new abc-type conjecture, Conjecture 1, which replaces the radical γ(n) in the abc inequality by H(n)=γ(n)/(log γ(n))^{ω(n)}. It proves that Conjecture 1 implies the classical abc conjecture, gives a heuristic motivation for Conjecture 1, and derives several conditional consequences. The main result (Theorem 1) states that, assuming Conjecture 1, for each fixed y the sum W(x,y)=Σ_{j=1}^y ω(x+j) satisfies W(x,y)≤(1+δ) log x / log log x for all sufficiently large x. Theorem 2 gives an analogous lower bound for the least solution of a system of congruences, and Proposition 3 constructs infinitely many abc triples with c>H exp(2 log H / log log H).","tokens_in":11172,"tokens_out":43419,"duration_ms":363044,"significance":"The conjectured strengthening is natural and, if true, would give a sharp upper bound for the number of prime factors in short intervals — a result not known to follow from the classical abc conjecture. The polynomial constructions (Lemmas 5 and 6) and the conditional proofs are interesting and mostly transparent. The paper is honest about the conjectural nature: all main theorems are conditional on Conjecture 1, which is strictly stronger than abc and for which no numerical evidence is supplied. The central value is therefore as a conditional contribution, contingent on a new and as yet unsupported conjecture.","major_comments":[{"comment":"The uniform bound on G(x)=gcd(P(x),Q(x)) is load-bearing for both Theorem 1 and Theorem 2, but the proof is only a sketch. The argument 'bound the largest power of p that can divide at least two of the integers x+a_j' does not by itself control v_p(x+a_j), which is unbounded as x varies. A correct proof should invoke the resultant Res(P,Q) to show G(x) is bounded by a constant depending only on the a_j. Please supply a rigorous proof of (7.4) or replace it with a standard resultant argument. Without this, the proofs of Theorems 1 and 2 contain a gap.","section":"Remark 1, §7.2 and §7.3"},{"comment":"The dyadic summation appears to misstate the count. For fixed A with a∈[A,2A), b∈[B,2B), the expected number of triples with H(ab(a+b))<B^{1-ε} is of order B^{-ε}(AB^2)^{o(1)}, and summing over A≤B gives B^{-ε+o(1)} rather than the claimed B^{-ε/2}. The conclusion that the expected number tends to 0 is unchanged, but the displayed estimate should be corrected. This does not invalidate the heuristic, but it should be made accurate.","section":"§3, heuristic for Conjecture 1"},{"comment":"The paper explicitly leaves open whether Conjecture 1 is false, and all main theorems collapse if it fails. For a conjecture that is strictly stronger than abc, some numerical verification would substantially strengthen the case for it. I recommend checking Conjecture 1 against known abc triples (e.g., the ABC@Home database) or at least providing a quantitative discussion of the heuristic's reliability for large c. As written, the plausibility of the central conjecture rests entirely on the informal §3 heuristic.","section":"§8, Problem 2"}],"minor_comments":[{"comment":"The degree in Lemma 6 should read 2^{k-2}, not '2k−2'; the same superscript appears to be lost in the proof of Theorem 1. Please fix all such exponents.","section":"Lemma 6 and throughout"},{"comment":"The factorization 'N=ab into two integers of size N^{1/2}(log A)^{O(1)}' is asserted without proof. This is standard but should be justified or given a reference.","section":"Proposition 3"},{"comment":"The notation m_j=... and n_j=... for even/odd j is hard to parse. Please state explicitly that m_j=D_j/e for even j and n_j=0, and vice versa for odd j.","section":"Lemma 5"},{"comment":"The bound xy≤n^{1/a+1/b} follows from x^a≤n and y^b≤n, but this is not stated. A one-line justification would help.","section":"§4.2"},{"comment":"The derivation of the expression for n^r and its passage to H(n) is very compressed. Expanding this would improve readability and make the heuristic more convincing.","section":"§3"},{"comment":"The sentence 'the contribution of the smaller values already satisfies the desired inequality' should be made explicit; the reader is left to reproduce the (straightforward) estimate.","section":"Proposition 4"}],"recommendation":"major_revision","confidential_remarks":"The main conditional theorems appear sound, but the gap in Remark 1 is real and must be fixed before publication. The heuristic miscomputation in §3 is secondary but should be corrected. If the author can supply the resultant-based proof of the gcd bound and add some numerical support for Conjecture 1, I would be willing to accept. The paper is on the borderline for a conjecture-driven contribution; the novelty is genuine, but the central conjecture remains unverified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know before reading further: the central conjecture, as stated, is false. The standard high-quality abc triple 2 + 3^10·109 = 23^5 (with gcd 1) has γ(abc) = 15042 and ω(abc) = 4, so H(abc) = 15042 / (log 15042)^4 ≈ 1.76. Conjecture 1 claims c < C(ε) H(abc)^{1+ε} for all triples, but c = 6,436,343, so no finite C(ε) can absorb that. The paper never checks any known abc triple against its H-function; a referee would do this in the first hour.\n\nThat said, the paper is not empty. Proposition 3's construction using the primorial N and the factorization N=ab gives c > H exp(2 log H / log log H), matching the known Stewart–Tijdeman lower-bound shape. Proposition 4's counting bound for n≤X with H(n)≤z is a solid piece of analytic number theory, and the polynomial identities in Lemmas 5–6, based on alternating products with binomial exponents, are genuinely new and could be useful elsewhere. The proof of Theorem 1 is also clever: it balances the degree-2^{y-2} polynomial against the log-power denominator. If Conjecture 1 were true, the argument would work.\n\nThe soft spot is not that the conjecture is merely unproved; it is actually false. The heuristic in §3 is informal and does not consider the many known abc triples with very small H. Consequently, Theorems 1, 2, and Corollary 1 are all conditional on a false premise. The author's Problem 2 asks whether Conjecture 1 is false, and the answer is yes. A modified statement—replacing log γ by log c, or weakening the exponent—might survive, but the paper does not provide one.\n\nWho should read this? Someone interested in the anatomy of abc refinements will find the technical machinery worth studying, and the counterexample is instructive as a cautionary tale. As a paper, it cannot be accepted with Conjecture 1 in its current form.\n\nMy recommendation: send it to a referee. Not because the conjecture is fixable but because the unconditional content and the polynomial construction deserve expert attention. The referee can state the counterexample clearly and possibly point the author toward a viable reformulation. My own verdict is reject as is, but with encouragement to revise.","headline":"Conjecture 1 is false as stated; the standard abc triple 2 + 3^10*109 = 23^5 gives H(abc) ≈ 1.76 while c ≈ 6.4 million, so no constant C(ε) can work. The conditional theorems are vacuous, though the unconditional sections still have value.","tokens_in":11647,"tokens_out":9936,"would_cite":false,"duration_ms":87424,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N25","11N37","11N56"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes a stronger abc-type conjecture based on a log-refined radical, and shows it forces the prime-factor count in short intervals to be asymptotically log x/loglog x.","keywords":["abc conjecture","radical","index of composition","number of prime factors","short intervals","Mason–Stothers theorem","Chinese remainder theorem","Mersenne numbers"],"falsifier":"The most direct falsifier would be an explicit infinite family of coprime triples (a,b,c) for which c/H(abc)^{1+ε} → ∞ for some fixed ε>0; the paper's own Mersenne-number example suggests a candidate family. A computation of c/H(abc)^{1.01} on known abc triples with large ratios would immediately show whether violations accumulate, and since Conjecture 1 is stronger than abc, any one such unbounded family would settle the conjecture false regardless of the unspecified constant C(ε).","tokens_in":10802,"feed_emoji":"🔢","tokens_out":4307,"duration_ms":40601,"temperature":0.7,"pith_summary":"The paper proposes a strengthened form of the abc conjecture in which the radical is replaced by a smaller quantity, H(abc)=γ(abc)/(log γ(abc))^{ω(abc)}. It argues that H better captures how 'rare' an integer is, and shows that this new conjecture implies the classical abc conjecture. The main result is conditional: assuming the new conjecture, any fixed window of consecutive integers near x contains at most (1+δ) log x / loglog x distinct prime factors in total, for all large x. This makes the previously known lower bound sharp, giving the exact limsup of W(x,y) loglog x / log x = 1. The paper also derives consequences for sums of powers, Mersenne numbers, and systems of congruences.","feed_headline":"Sharper abc conjecture yields exact prime-factor growth","feed_subtitle":"If the new H-based conjecture holds, the total number of distinct prime factors in fixed windows is asymptotically log x / loglog x.","key_machinery":"The central object is H(n)=γ(n)/(log γ(n))^{ω(n)}: the radical divided by a power of its logarithm, which is much smaller than γ(n) whenever n has many distinct prime factors. The paper motivates Conjecture 1 by a heuristic counting argument (Section 3) showing that triples with unusually small H(abc) are extremely rare. The proofs of Theorems 1 and 2 rely on a family of Mason–Stothers-style polynomial identities P_k(z)=∏_{2|j}(z+j)^{binom{k-1}{j-1}}, Q_k(z) similarly with odd j, and R_k(z)=P_k(z)-Q_k(z). Evaluating these at z=x, their integer values factor in a way controlled by the numbers x+j; applying Conjecture 1 to the reduced identity yields the short-interval bound and the CRT lower","core_discovery":"The central claim is Conjecture 1: for every ε>0 there is a constant C(ε) such that every coprime triple a+b=c satisfies c < C(ε) H(abc)^{1+ε}, where H(n)=γ(n)/(log γ(n))^{ω(n)}. Since H(abc) < γ(abc) for every nontrivial triple, this is strictly stronger than the ordinary abc conjecture. The paper's main theorem states that if Conjecture 1 holds, then for every fixed y and δ>0, W(x,y)=Σ_{j≤y}ω(x+j) ≤ (1+δ) log x / loglog x for all x ≥ x_0(δ,y). Consequently, for each fixed y, the limsup of W(x,y) loglog x / log x equals 1, matching a standard lower bound. The proof applies Conjecture 1 to a polynomial identity P_y(x)-Q_y(x)=R_y(x) constructed from binomial coefficients, whose prime factors","pith_inferences":["If Conjecture 1 is ever established, it would give a near-tight uniform bound for the total number of prime factors of products of any fixed set of translates, not just single integers—a whole family of 'normal order' statements at once.","The Mersenne-number discussion suggests that the true obstruction to a small H(n) is forced arithmetic divisibility (e.g., d|k implies 2^d-1 divides 2^k-1). A weaker but possibly unconditional version of Theorem 1's conclusion might be derived from existing anatomy-of-integers results, since the polynomial construction only needs a bound of the same shape.","The paper leaves open whether abc itself implies Conjecture 1 (Problem 1). A natural test is whether the best available refinements of abc can be reshaped into the H-based form; the answer would clarify whether Conjecture 1 is a genuine strengthening or merely a repackaging of known heuristics.","Conjecture 1, if true, would also settle sharp lower bounds for the least solution of CRT systems with prescribed moduli, connecting the 'atypicality' of integers to a single unified measure H."],"forward_implications":["Conjecture 1 implies the classical abc conjecture, since H(abc) < γ(abc) for all admissible triples except (1,1,2) (Proposition 1).","Under Conjecture 1, each fixed window of y consecutive integers has at most (1+δ) log x / loglog x distinct prime factors in total, and this is asymptotically sharp: the limsup equals 1 (Theorem 1 and Corollary 1).","For sums of two powers, n=x^a + y^b with gcd(x,y)=1, Conjecture 1 gives ω(n) ≤ (1/a + 1/b + δ) log n / loglog n; similarly ω(n(n+k)) ≤ (1+δ) log n / loglog n for each fixed k.","For CRT systems x+a_i ≡ 0 (mod n_i) with pairwise coprime n_i, the least positive solution satisfies x_0 ≥ C(δ,A) (N/H(N))^{1-δ}, where N=n_1⋯n_k (Theorem 2)."],"fun_headline_variants":["New abc conjecture sharpens prime factor growth bound","Stronger abc conjecture sets exact prime-factor limit","Prime factor count in windows hits predicted maximum","abc variant pins down optimal prime factor density","Exact asymptotic for prime factors from stronger abc"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Everything rests on the truth of Conjecture 1, an unproven strengthening of abc that the paper supports only with a heuristic and with lower-bound examples, and whose falsity would remove Theorems 1, 2, and Corollary 1.","fun_headline_variants_meta":{"raw":{"variants":["New abc conjecture sharpens prime factor growth bound","Stronger abc conjecture sets exact prime-factor limit","Prime factor count in windows hits predicted maximum","abc variant pins down optimal prime factor density","Exact asymptotic for prime factors from stronger abc"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000506,"raw_usage":{"total_tokens":2284,"prompt_tokens":701,"completion_tokens":1583,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":445,"completion_tokens_details":{"reasoning_tokens":1514}},"tokens_in":445,"tokens_out":1583,"duration_ms":10303,"temperature":1.0,"reasoning_tokens":1514,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T08:03:39.683946+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The most direct falsifier would be an explicit infinite family of coprime triples (a,b,c) for which c/H(abc)^{1+ε} → ∞ for some fixed ε>0; the paper's own Mersenne-number example suggests a candidate family. A computation of c/H(abc)^{1.01} on known abc triples with large ratios would immediately show whether violations accumulate, and since Conjecture 1 is stronger than abc, any one such unbounded family would settle the conjecture false regardless of the unspecified constant C(ε).","supporting_citations":[],"review_version":2}