{"id":"4e633838-5304-4e75-85f5-b9b9af1d5958","arxiv_id":"2506.03625","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For coprime a,b, at least 0.04 of the primes up to ab-a-b are not representable as au+bv, and for fixed a the optimal asymptotic constant is 1/2 + 1/(2(a-1)).","lead":"This paper counts primes that cannot be written as au+bv with nonnegative u and v, and proves that at least 4 percent of all primes up to ab-a-b are missed. It also determines the exact asymptotic density for each fixed a and verifies a proposed one-half density conjecture for a up to 10.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unreported finite computations are load-bearing: if any of the 'a calculation shows' assertions in Section 2 fails, the universal statements in Theorems 1.2 and 1.5 fail in the affected parameter range.","rationale":"The reader's weakest_assumption identifies the same class of concern, but the scope is broader than stated: Theorem 1.2's Case 2 already contains a one-dimensional unreported computation over 181<=a<=60000, and Theorem 1.5's proof invokes additional 'by calculation' verifications for intermediate ranges of b that the reader did not list. I did not find an internal analytic error in the main proof; rather, the conditional element is the absence of any verifiable trace for several finite but nontrivial ranges. Consequently, the appropriate verdict remains CONDITIONAL rather than a rejection. The paper would be strengthened by supplying the checking code, tables of marginal cases, or a fully explicit certificate for each finite verification.","tokens_in":7667,"tokens_out":24824,"duration_ms":267067,"concrete_test":"Write an independent program that, for each required finite range, computes pi*(a,b) exactly either via the residue formula pi*(a,b)=sum_{v=1}^{a-1} #{p<bv: p≡bv mod a} or by sieving primes up to S and testing membership in <a,b>, then verifies all of the following: Case 2 gives Delta(0.0904,a,S)>0.0401 for a=181..60000 with S=a^2-a-1; Case 3 gives pi*(a,b)>0.04*pi(S) for 16<=a<=180, b<=1000 and for b>1000 using the stated delta; Case 4 gives pi*(a,b)>=pi(S)/2 for 3<=a<=15, b<=180; and Theorem 1.5 gives inequalities (2.7) and (2.9) for a=9,10 over the exact S-ranges and b-ranges stated in its proof. If any check fails, the affected theorem statement needs a counterexample or a corrected bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The analytic skeleton of the paper appears sound: Theorem 1.1's residue-class decomposition and its upper/lower bounds check out, and the large-parameter cases of Theorem 1.2 are justified by the Siegel-Walfisz bound and Lemma 2.5. The load-bearing weakness is the repeated reliance on unreported finite verifications. In Theorem 1.2, Case 2 asserts by calculation that Delta(0.0904,a,S)>0.0401 for every 181<=a<=60000; Case 3 asserts by calculation that the bound holds for 16<=a<=180, b<=1000 and again for b>1000; Case 4 asserts that pi*(a,b)>=pi(S)/2 for 3<=a<=15, b<=180. Theorem 1.5 adds further unreported checks: (2.7) is said to hold by calculation for a=9 with 18595<S<=e^12 and for a=10 with 60180<S<=e^13.5, ranges of roughly 18,000 and 74,000 b-values, and (2.9) is asserted by direct enumeration for a=9,b<=2325 and a=10,b<=6687. If any of these computations is wrong, the corresponding universal claim fails on an explicit finite range; the analytic part covers only the complementary large-parameter ranges. No code, tables, or derivation details are supplied, so the computational premise cannot be independently checked from the text.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies π*(a,b), the number of primes below S=ab−a−b that are not representable as au+bv with u,v≥0, for coprime positive integers a and b. Theorem 1.1 establishes that for each fixed a≥3 there is a constant C_a such that π*(a,b) exceeds (1/2+1/(2(a−1)))·S/log S for all sufficiently large b, and proves that this constant is best possible. Theorem 1.2 claims the uniform lower bound π*(a,b) ≥ 0.04π(S) for all b>a≥1 with gcd(a,b)=1. The authors also conjecture that the optimal constant is 1/2, and Theorem 1.5 verifies both conjectures for all a≤10. The proofs combine a residue-class decomposition of the nonrepresentable primes, explicit prime-counting bounds (Bennett–Martin–O'Bryant–Rechnitzer, Siegel–Walfisz, Rosser–Schoenfeld), and a case split that leaves several finite parameter ranges to be checked by unreported computation.","tokens_in":7923,"tokens_out":20657,"duration_ms":217203,"significance":"If the theorems are correct, they give a striking uniform result: for any two coprime generators a<b, the primes not representable by ⟨a,b⟩ always constitute at least 4% of all primes below the conductor S, and for each fixed a the asymptotic proportion approaches the sharp constant (1/2+1/(2(a−1))). The analytic reduction is elegant: the main work is a careful lower bound for π*(a,b) via primes in arithmetic progressions, with the universal claim reduced to a finite family of inequalities. The paper also confirms the proposed conjectures for a≤10, strengthening the evidence for the conjectured constant 1/2. The principal weakness is that a substantial portion of the proof is delegated to unreported finite computations, so the full-range claims are not independently verifiable from the text alone.","major_comments":[{"comment":"In the proof of Theorem 1.2, Case 2 (p. 9, just after (2.2)), the inequality Δ(δ,a,h(a))>0.0401 for every 181≤a≤60000, with h(a)=a^2−a−1 and δ=0.0904, is asserted as \"a calculation.\" This is a finite check over roughly 6×10^4 values of a, and it is load-bearing: it is exactly the step that converts the general bound (2.2) into π*(a,b)>0.0401π(S) for this entire range. No code, table, or algorithmic description is provided, so the reader cannot verify this premise. Please supply a reproducible verification (code, tables, or a rigorous interval-arithmetic certificate) and state how the sum ∑_{1≤v≤δa, gcd(v,a)=1} 1 was evaluated.","section":"Theorem 1.2, Case 2"},{"comment":"In Case 3 of Theorem 1.2 (p. 9), two finite computations are asserted without support: (i) for 16≤a≤180 and b≤1000 with gcd(a,b)=1, the bound π*(a,b)>0.0663π(S); and (ii) for b>1000, the inequality Δ(0.095,a,g(a))>0.0425 with g(a)=1000a−1001, for 16≤a≤180. Both are needed to complete the proof of Theorem 1.2 for 16≤a≤180. The second is a modest check over 165 values of a, but the first is an enumeration over roughly 10^5 pairs (a,b), for which the paper gives no algorithm or data. Without a reproducible computation, the theorem is not verifiable for this parameter range.","section":"Theorem 1.2, Case 3"},{"comment":"In Case 4 of Theorem 1.2 (p. 10), the statement that π*(a,b)≥(1/2)π(S) for 3≤a≤15 and b≤180 with gcd(a,b)=1 is justified only by \"a calculation shows.\" This is a finite enumeration of about 1,300 pairs, but it is unreported and it is the sole support for the theorem in the small-parameter range. A short table of the exceptional pairs, or a description of the exact counting method, would make this checkable, but as written the reader cannot confirm it.","section":"Theorem 1.2, Case 4"},{"comment":"The proof of Theorem 1.5 (pp. 10–11) contains several unreported verifications: the inequality (2.5) for 3≤a≤10 and a<b<50a^2 except (3,4),(3,5),(3,7); the inequality (2.9) for 3≤a≤8 in the complementary b-range; the assertion that (2.7) holds by calculation for a=9 with 18595<S≤e^12 and for a=10 with 60180<S≤e^13.5; and the enumerations (2.9) for a=9,b≤2325 and a=10,b≤6687. The last of these involves about 74,000 b-values and requires exact values of π(S) or certified bounds. No computational details are given, so Theorem 1.5 cannot be independently checked from the manuscript. Please make these computations available or replace them with a fully analytic argument.","section":"Theorem 1.5"}],"minor_comments":[{"comment":"The phrase \"pose following conjecture\" should read \"pose the following conjecture\".","section":"Abstract"},{"comment":"In the last inequality of the lower-bound argument (p. 5), the text \"we use the fact that φ(a)a 2 ≥2a 2 >2(a 2 −1)\" is garbled by typesetting; the intended inequality appears to be φ(a) ≥ 2 > 2(a^2−1)/a^2, which is true for all a≥3.","section":"Proof of Theorem 1.1"},{"comment":"The displayed equality involving |{p:p=gcd(v,a), p≡bv mod a}| on p. 6 is not obvious; a one-sentence explanation that p must divide gcd(v,a) and that the congruence then forces gcd(v,a)=p would improve readability.","section":"Proof of Theorem 1.1, upper bound"},{"comment":"In the proof of Theorem 1.2, the bound from Lemma 2.1 is applied with x=0, while the lemma as stated requires x>0; the limiting version (or the lemma with x→0+) should be stated explicitly.","section":"Lemma 2.1 application"},{"comment":"The paper uses the phrase \"a calculation\" for both very small checks and large enumerations; please state once, at least in an appendix or footnote, whether all computations were exact integer arithmetic and which software was used.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The analytic core of the paper is coherent and the results are likely correct; the main obstacle to acceptance is the repeated reliance on unreported finite computations that are load-bearing for the universal statements. I recommend requesting a supplement with the verification code, tables, or a rigorous computational certificate, together with a clear description of the algorithms. If the computations can be supplied and checked, the paper would be a solid contribution. The paper fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things worth knowing. The paper gives the first unconditional positive-proportion lower bound for the number of primes not representable by a two-generator numerical semigroup, pi*(a,b) >= 0.04 pi(S), and for fixed a it proves an asymptotic lower bound with optimal constant (1/2+1/(2(a-1))) S/log S. Both are new. The authors also confirm their conjectures for a <= 10.\n\nWhat they do well: the proof of Theorem 1.1 is clean. The residue-class decomposition (write n not representable as p = au+bv with u<0, 0<=v<=a-1) plus the lower and upper estimates for primes in arithmetic progressions works as written. The constant is genuinely best possible, shown by the matching upper bound. The analytic part of Theorem 1.2, using pi(delta S) minus representable primes and bounding with Montgomery-Vaughan's large sieve estimate, also checks out. The citation pattern is honest and the target result is not assumed anywhere.\n\nThe soft spot is exactly what the reader flagged. Several 'a calculation shows' statements are load-bearing for the universal form of the theorems. In Theorem 1.2, Cases 2-4: delta = 0.0904 for 181 <= a <= 60000, the ranges 16 <= a <= 180, b <= 1000 or b > 1000, and 3 <= a <= 15, b <= 180. In Theorem 1.5, for a=9, 18595 < S <= e^12 and S <= 18595 (b <= 2325), and for a=10, 60180 < S <= e^13.5 and S <= 60180 (b <= 6687). No code, tables, or derivations are supplied. If any of those checks fails, the corresponding universal claim fails on an explicit finite range. The analytic parts only cover the complementary large-parameter ranges. This is not a deep flaw in the method, but it is an external checkability problem: these are finite ranges and the paper should ship the verification.\n\nThe other caveats are minor. The 0.04 is a crude constant; the authors say so themselves. The conjectures are natural and the a<=10 evidence is fine, but the paper does not try to push beyond. The analytic estimates are standard, which is a feature here rather than a bug.\n\nWho it's for: number theorists working on Frobenius-type problems and primes in arithmetic progressions; also anyone who wants a clean example of using explicit prime bounds. I'd send it to a serious referee. The referee should ask for the finite computations; once those are available, the paper is acceptable. The result itself deserves to be in the literature.","headline":"First unconditional positive-proportion lower bound for primes missing from <a,b>, with a solid analytic core; the universal statements lean on unreported finite checks that should be made reproducible before publication.","tokens_in":8499,"tokens_out":8471,"would_cite":true,"duration_ms":83731,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11D07","11N13","11Y35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Nonrepresentable primes are at least 4% of primes below ab-a-b.","keywords":["numerical semigroup","Frobenius problem","primes in arithmetic progressions","distribution of primes","Siegel-Walfisz theorem","nonrepresentable primes","lower bound","positive density"],"falsifier":"Recompute pi*(a,b) by direct enumeration for every coprime pair with 16<=a<=180 and b<=1000, and for 3<=a<=15 with b<=180, checking the inequality pi*(a,b) >= 0.04 pi(S). Likewise enumerate a=9, b<=2325 and a=10, b<=6687 to check (2.5) and (2.9). Any single pair that violates one of these inequalities would disprove Theorem 1.2 or the a<=10 part of Theorem 1.5 as stated.","tokens_in":7397,"feed_emoji":"🔢","tokens_out":12283,"duration_ms":115746,"temperature":0.7,"pith_summary":"Take two coprime positive integers a and b. Every integer larger than S=ab-a-b is a sum of nonnegative multiples of a and b, but below S some integers are not. This paper counts the primes among those missed integers and proves two lower bounds: a uniform one, pi*(a,b) >= 0.04 pi(S), and, for each fixed a>=3 and all large enough b, an asymptotic lower bound with the sharp leading constant (1/2+1/(2(a-1))) S/logS. The paper also conjectures the stronger uniform bound pi*(a,b) >= 1/2 pi(S) and verifies it for all a<=10. If these results are correct, then for every coprime pair the primes that cannot be written as au+bv always make up a non-negligible share of all primes up to S.","feed_headline":"Nonrepresentable primes are at least 4% of primes below ab-a-b","feed_subtitle":"For every coprime pair, at least 4 percent of the primes below ab-a-b are missed by the form au+bv.","key_machinery":"The key machinery is a one-to-one counting identity: an integer n is not representable as au+bv with u,v>=0 if and only if n = au+bv for some u<0 and 0<=v<=a-1. This turns pi*(a,b) into a sum over v=1,...,a-1 of the number of primes p < bv with p ≡ bv (mod a). The paper bounds those residue-class counts with the Siegel-Walfisz theorem for large moduli and with explicit bounds for primes in arithmetic progressions for moduli up to 1200, then combines them with the elementary identity sum_{1<=v<=a-1, gcd(v,a)=1} v = a $\\varphi$(a)/2 to obtain the clean lower bound ab/(2 log S) minus lower-order terms. This same machinery powers the delta-truncation argument that yields the uniform 0.04 bound on pi(S).","core_discovery":"The central discovery is that nonrepresentable primes cannot be rare: for every coprime pair with b>a>=1, pi*(a,b) >= 0.04 pi(S), and for each fixed a>=3 there is a threshold C_a such that pi*(a,b) > (1/2+1/(2(a-1))) S/logS whenever b>=C_a and gcd(a,b)=1. The proof converts each nonrepresentable number into a unique expression au+bv with u<0 and 0<=v<=a-1, so counting missed primes becomes counting primes in arithmetic progressions p ≡ bv (mod a) below the numbers bv. Summing these counts over v coprime to a, and using the identity that the residues v pair to sum to a $\\varphi$(a)/2, yields the lower bounds. The factor (1/2+1/(2(a-1))) is shown best possible by a matching upper bound, and the paper proves the conjectural bound pi*(a,b) >= 1/2 pi(S) for all 1<=a<=10, with equality only in the small cases (1,b), (2,3), (2,5), and (3,5).","pith_inferences":["Editorial: Because the counting identity is purely additive, the same argument could be applied to other sparse sets with known distribution in arithmetic progressions, such as squarefree numbers or integers with a fixed number of prime factors.","Editorial: The finite 'a calculation shows' steps in the proofs of Theorems 1.2 and 1.5 are all checkable by direct enumeration; publishing those tables would remove the only non-analytic, externally invisible premise from the argument.","Editorial: The fixed-a asymptotic implies a small but lasting bias: for each a, primes below S are missed at a rate exceeding one half by 1/(2(a-1)) of S/logS, and Conjecture 1.4 is precisely the statement that this over-half rate never drops below one half for any pair."],"forward_implications":["For every coprime pair (a,b), the number of nonrepresentable primes satisfies pi*(a,b) >= 0.04 pi(ab-a-b).","For each fixed a>=3, as b grows the ratio pi*(a,b)/(S/logS) tends to (1/2+1/(2(a-1))), since the lower and upper bounds agree to leading order.","For all a<=10, the conjectural bounds pi*(a,b) > (1/2+1/(2(a-1))) S/logS and pi*(a,b) >= 1/2 pi(S) hold, with equality in the latter only at the four listed small pairs.","If Conjecture 1.4 holds for all a,b, then at least half of the primes below the conductor are always missed by the semigroup, matching the exact integer count of missed integers at one half of S."],"supporting_citations":[{"why":"Supplies Lemma 2.1, the short-interval upper bound for primes in arithmetic progressions used to control representable primes below delta S.","marker":"[5]"},{"why":"Supplies the Siegel-Walfisz theorem (Lemma 2.2), the asymptotic distribution of primes in progressions used in the large-modulus part.","marker":"[2]"},{"why":"Gives Lemma 2.4, explicit upper and lower bounds for primes in progressions with modulus up to 1200, used in the fixed-a theorem and in small cases.","marker":"[1]"},{"why":"Gives Lemma 2.5, bounds on pi(x) that convert S/logS estimates into lower bounds for a fraction of pi(S).","marker":"[7]"},{"why":"Introduced the complementary problem of primes representable by the semigroup <a,b> and provides the motivation for the conjecture studied here.","marker":"[6]"},{"why":"Established the representable-primes analogue for almost all pairs, a precedent the paper extends to nonrepresentable primes.","marker":"[3]"}],"fun_headline_variants":["At least 4% of primes below ab-a-b are missed by au+bv","4% of primes below ab-a-b can't be written as au+bv","For coprime a,b, at least 4% of primes below ab-a-b are missed","Guaranteed 4% of primes below ab-a-b are not of form au+bv","Nonrepresentable primes: at least 4% below ab-a-b"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The universal 0.04 bound and the a<=10 confirmation rest on several unreported finite computations in the ranges 16<=a<=180 with b<=1000, 3<=a<=15 with b<=180, and a=9 with b<=2325 and a=10 with b<=6687; if any of those 'a calculation shows' lines is wrong, the corresponding small cases fail, though the analytic argument still covers all large a and b.","fun_headline_variants_meta":{"raw":{"variants":["At least 4% of primes below ab-a-b are missed by au+bv","4% of primes below ab-a-b can't be written as au+bv","For coprime a,b, at least 4% of primes below ab-a-b are missed","Guaranteed 4% of primes below ab-a-b are not of form au+bv","Nonrepresentable primes: at least 4% below ab-a-b"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001403,"raw_usage":{"total_tokens":5658,"prompt_tokens":920,"completion_tokens":4738,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":4628}},"tokens_in":536,"tokens_out":4738,"duration_ms":43127,"temperature":1.0,"reasoning_tokens":4628,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:59:18.444627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute pi*(a,b) by direct enumeration for every coprime pair with 16<=a<=180 and b<=1000, and for 3<=a<=15 with b<=180, checking the inequality pi*(a,b) >= 0.04 pi(S). Likewise enumerate a=9, b<=2325 and a=10, b<=6687 to check (2.5) and (2.9). Any single pair that violates one of these inequalities would disprove Theorem 1.2 or the a<=10 part of Theorem 1.5 as stated.","supporting_citations":[{"cited_title":"Bennett, G","cited_arxiv_id":null,"evidence_quote":"Gives Lemma 2.4, explicit upper and lower bounds for primes in progressions with modulus up to 1200, used in the fixed-a theorem and in small cases."},{"cited_title":"Ram\\'irez Alfons\\'in, M","cited_arxiv_id":null,"evidence_quote":"Gives Lemma 2.5, bounds on pi(x) that convert S/logS estimates into lower bounds for a fraction of pi(S)."},{"cited_title":"Montgomery, R.C","cited_arxiv_id":null,"evidence_quote":"Introduced the complementary problem of primes representable by the semigroup <a,b> and provides the motivation for the conjecture studied here."},{"cited_title":"Ding, On a conjecture of Ram\\'irez Alfons\\' in and Ska ba, J","cited_arxiv_id":null,"evidence_quote":"Established the representable-primes analogue for almost all pairs, a precedent the paper extends to nonrepresentable primes."}],"review_version":1}