{"id":"db0aff6b-ffdb-44c0-8219-ca50ecef2261","arxiv_id":"2411.09446","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For coprime integers 2<a<b, there exists a prime p with p≤ab-a-b of the form p=ax+by with x,y≥0, confirming the Ramirez Alfonsin-Skalba conjecture.","lead":"This note proves that for any coprime coin values 2<a<b, a prime number no larger than the Frobenius number ab-a-b can always be paid as ax+by with nonnegative x,y. It confirms a 2020 conjecture about primes in numerical semigroups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 rests on finite verifications that are not supplied; the text is internally inconsistent about whether range (iii) was checked, so the proof is not independently verifiable as written.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: the proof depends on finite verifications that are described but not supplied, and the text is inconsistent about whether (iii) was checked. I re-read the case structure with this in mind. The analytic parts appear internally consistent: the use of the effective Siegel–Walfisz-type bounds in Lemmas 1–4 is appropriate for the stated ranges, the counting argument in Case II correctly uses Sylvester's symmetry, and Cases VII–VIII cover the claimed ranges with plausible inequalities. The 'Minimize' claims and residual checks are not reproducible, however, and the discrepancy between the Conclusion ('(iii) could be done') and footnote 5 ('six days ... verifications of (ii)') prevents a reader from knowing whether the theorem has actually been verified. This does not make the theorem false; it makes the proof incomplete as written. A CONDITIONAL verdict is therefore appropriate, pending the release of a reproducible verifier or analytic replacement for the residual checks. I find no additional fatal flaw, so I would not change the reader's verdict.","tokens_in":6889,"tokens_out":18595,"duration_ms":168688,"concrete_test":"Ask the authors to release a self-contained verifier with exact range specifications and certificates that (i) enumerates all composite coprime pairs with a > 10^4, g <= 5*10^8, and a >= 155 log b; (iii) enumerates all composite coprime pairs with 4 <= a <= 10^4 and b < 10^6, checking for each pair the least x >= 0 such that ax + b is prime and ax + b <= g; and (4/6) recomputes the three 'Minimize' outputs in footnotes 1, 4, and 6 with interval arithmetic or exact rational bounds. Independently run the enumeration for range (iii) with a PARI/GP or C program; if any allowed pair lacks such a prime, Theorem 1 is false, and if the enumeration succeeds, the residual-gap objection is settled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The analytic skeleton of the proof (Cases I–VIII) is coherent: each case reduces the problem to explicit inequalities, and the remaining domains are finite. The load-bearing gap is that the theorem is not established unless the residual computer checks are actually performed and reproducible. In the Conclusion the authors say residual range (i) (a > 10^4 with g <= 5*10^8) and range (iii) (a <= 10^4, b < 10^6) 'could be done' by computer, while footnote 5 reports verifying '(i)' and '(ii)' and spending six days on '(ii)' — but (ii) is supposedly proved theoretically in Cases VII–VIII. Nothing in the text confirms that (iii) was actually run, and no code, pseudocode, or certificate is supplied. The numerical minimizations in footnotes 1, 4, and 6 ('566.0054846', '42.5025', '21647.8') are asserted without justification; these are finite but nontrivial checks, especially Case VI over 4 <= a <= 808 and 10^6 <= b <= 8*10^9. Since Theorem 1 is a universal statement over all coprime pairs (a,b), one missing or wrong finite check leaves the conjecture unproved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1: for coprime integers 2 < a < b, there exists at least one prime p ≤ ab - a - b of the form p = ax + by with x, y ∈ Z_{≥0}, confirming a 2020 conjecture of Ramírez Alfonsín and Skałba. The proof splits into eight cases. Cases I–III handle a > 10^4; Cases IV–VI handle 4 ≤ a ≤ 10^4 with large b; Cases VII–VIII handle the remaining intermediate range by a combination of explicit estimates for primes in arithmetic progressions and a counting argument using Sylvester's symmetry. The residual finite ranges are claimed to be checked by computer, but no code, certificates, or detailed computation logs are supplied.","tokens_in":7201,"tokens_out":6003,"duration_ms":52130,"significance":"If the proof is correct, the paper fully resolves Conjecture 1 of Ramírez Alfonsín and Skałba, a natural prime analogue of the Frobenius problem. The analytic structure is sound and makes effective use of published explicit bounds (Bennett–Martin–O'Bryant–Rechnitzer, Panaitopol) together with a clean counting argument. The paper is significant because it reduces a broad universal statement to a small number of analytic inequalities plus finite verification, and it supplies explicit constants that could be useful for further work. However, the final theorem as written depends on asserted computer checks that are not reproducible from the manuscript.","major_comments":[{"comment":"The Conclusion states that residual verifications (i) and (iii) \"could be done by the computer,\" while footnote 5 reports actually verifying (i) and (ii), with six days spent on (ii). This is internally inconsistent: (ii) is supposedly proved theoretically in Cases VII–VIII, and nowhere does the text confirm that (iii) (a ≤ 10^4, b < 10^6) was actually checked. Since Theorem 1 is a universal statement over all coprime pairs, range (iii) is a necessary residual case; if it is not verified, the proof is incomplete. The authors must clarify exactly which ranges were verified and supply code, pseudocode, or certificates so the verifications are independently reproducible.","section":"Conclusion and footnote 5"},{"comment":"The numerical minimizations in footnotes 1, 4, and 6 are load-bearing but are asserted without any supporting evidence. Footnote 1 claims that a function f(g) satisfies f(g) ≥ f(5·10^8) > 0 for all g in [5·10^8, 10^18], with minimum 566.0054846; footnote 4 claims a minimum of 42.5025 over 4 ≤ a ≤ 808, 10^6 ≤ b ≤ 8·10^9; footnote 6 claims a minimum of 21647.8 over 808 < a ≤ 10^4, 10^6 ≤ b ≤ 10^7. These are nontrivial finite (and in footnote 1 continuous) optimization claims. The authors do not name the software, provide code, or give interval-arithmetic certificates. Because Cases II, VI, and VIII rely directly on these assertions, the proof is not independently verifiable as written.","section":"Footnotes 1, 4, and 6"},{"comment":"The derivation of the function f(g) from (2.8) is not shown. The inequality (2.8) contains explicit dependence on b and a; the footnote states only that the inequality b > √g is used and that \"the middle equation of (2.8) turns to be a function of g.\" It is not explained how the terms involving b are bounded (e.g., in the negative term -g/(155 log b log g), replacing b by √g gives a valid lower bound, but this must be stated and checked for every occurrence of b and a). Without a precise definition of f(g) and a justification that it is a valid lower bound for π_{a,b}, the subsequent minimization cannot be assessed.","section":"Case II, equation (2.8) and footnote 1"},{"comment":"In Case VI, Lemma 3 is applied with x = b and x = g, requiring x ≥ 10^6. Since b ≥ 10^6 and g > 10^6, this is satisfied. However, the subsequent inequality (π_{a,b} ≥ ... > 0) is asserted to hold \"with the help of computers\" and the claimed minimum of 42.5025 is not accompanied by any error analysis or exactness guarantee. Because the range 4 ≤ a ≤ 808, 10^6 ≤ b ≤ 8·10^9 contains about 6.4×10^12 pairs, this is a substantial computation; the reader needs to know the algorithm, the machine, the runtime, and ideally a reproducible script or a mathematical proof that the minimum occurs at the stated point.","section":"Case VI, Lemma 3 applicability"}],"minor_comments":[{"comment":"The author name is corrupted as \"Ska/suppress lba\" throughout; it should read \"Skałba.\" This appears to be an encoding artifact, but it must be fixed.","section":"Title, abstract, and introduction"},{"comment":"The displayed inequality is garbled: it should read x/(log x - 1 + (log x)^-0.5) < π(x) < x/(log x - 1 - (log x)^-0.5). The current formatting makes the denominators ambiguous.","section":"Lemma 2"},{"comment":"The inequality \"80(g-b) - 4.2g > 0\" is stated without justification. Since b = (g+a)/(a-1), this is a simple algebraic check, but it should be shown or explained.","section":"Case I, after equation (2.4)"},{"comment":"The citation \"Amer. J. Math. 105 (1882)\" appears incorrect: Sylvester's paper is commonly cited as Amer. J. Math. 4 (1881) or a similar early volume; please verify the correct volume and year.","section":"Reference [9]"}],"recommendation":"major_revision","confidential_remarks":"The paper's analytic skeleton is credible and the case split appears exhaustive, but the reliance on undocumented computer checks is a serious verifiability problem for a theorem that is universal over all coprime pairs. The authors should be encouraged to provide a supplement with code and certificates, or to replace the numerical minimizations with provable monotonicity or elementary bounds. If the verifications are supplied and correct, the paper would likely be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a 2020 conjecture of Ramírez Alfonsín and Skałba: for every coprime pair 2<a<b, there is a prime p ≤ ab−a−b of the form ax+by. That is a real result, not a conditional or asymptotic one, and it goes beyond what was known — [8] only gives positivity for large g via an ineffective constant, and the asymptotic conjecture is a different statement. The proof strategy is sensible: use explicit Bennett–Martin–O'Bryant–Rechnitzer bounds for primes in arithmetic progressions, count representable integers near g via Sylvester symmetry, and split into cases. The analytic inequalities in Cases I–VIII look internally consistent, and the case split covers all pairs. The authors know the literature and cite the relevant prior work, including Ding–Zhai–Zhao on Conjecture 2.\n\nThe soft spot is the one the reader flags. The theorem is not finished by the analytic part; it rests on finite verifications, and the text does not supply them. The Conclusion says ranges (i) and (iii) 'could be done' by computer; footnote 5 then reports verifying (i) and (ii), with six days on (ii) — but (ii) is supposed to be handled theoretically in Cases VII–VIII. Nothing confirms (iii) was actually run, and the claimed software minima in footnotes 1, 4, 6 are asserted without code or certificates. As written, the proof is not independently verifiable. The fix is easy in principle: attach the code, state exactly which ranges were checked, and correct the footnote/Conclusion mismatch. If (iii) was in fact checked, that should be said plainly.\n\nI do not see circularity or a load-bearing mathematical error; the analytic skeleton is sound. But a universal statement proved by finite cases needs those cases to exist. The paper deserves serious refereeing — the approach is legitimate and, if confirmed, the result cleanly resolves an open conjecture — but the referee should demand reproducible computation or a theoretical argument for (iii).","headline":"Settles a 2020 conjecture modulo unstated finite checks; the analytic casework is solid, but the proof is not complete as written because the residual computations are neither supplied nor consistently described.","tokens_in":7655,"tokens_out":4554,"would_cite":false,"duration_ms":36346,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11D07","11N13"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every coprime pair 2<a<b, a prime p≤ab-a-b equals ax+by for nonnegative x,y.","keywords":["numerical semigroup","representable primes","primes in arithmetic progressions","short intervals","coprime pairs","explicit prime counting","nonnegative integer combinations","coin problem"],"falsifier":"Run an exhaustive search over the two residual families: pairs with $a>10^4$ and $g \\le 5\\cdot 10^8$ (under $a \\ge 155\\log b$), and pairs with $a \\le 10^4$ and $b<10^6$. If any such coprime pair with $2<a<b$ has no prime of the form $ax+by$ at or below $g$, Theorem 1 is false; publishing the search results for both families would confirm the computational step the text leaves as 'could be done.'","tokens_in":6716,"feed_emoji":"🔢","tokens_out":15241,"duration_ms":126204,"temperature":0.7,"pith_summary":"This paper proves that for any two coprime integers $2<a<b$, at least one prime $p\\le ab-a-b$ can be written as $ax+by$ with $x,y$ nonnegative integers. That is the full statement of a 2020 conjecture: the set of all nonnegative combinations of $a$ and $b$ always contains a prime, not merely for all sufficiently large products. The proof splits the parameter range into analytic regimes and small finite ranges, using effective bounds for primes in arithmetic progressions and for primes in short intervals, together with computer checks. If correct, the result closes the existence question for every admissible pair.","feed_headline":"A prime below ab-a-b always has the form ax+by","feed_subtitle":"Proof of the 2020 conjecture: every coprime pair 2<a<b has a representable prime.","key_machinery":"The carrying object is the difference $\\pi(g;a,b)-\\pi(b;a,b)$, the number of primes congruent to $b$ modulo $a$ in $(b,g]$; any such prime lies in the semigroup because $p=a\\cdot((p-b)/a)+b\\cdot 1$. For the middle ranges the paper switches to a second mechanism: a short interval at the top, $(g-g/\\log g, g]$, where the number of primes is controlled by explicit inequalities for $\\pi(x)$ and the number of representable integers is controlled by a lattice-point count. The classical symmetry of representability modulo $g$ ensures that the nonrepresentable integers in that top interval are exactly the mirror image of the representable integers near the bottom.","core_discovery":"The central claim is Theorem 1: for every pair of relatively prime integers with $2<a<b$, the count $\\pi_{a,b}$ of primes $p\\le g=ab-a-b$ of the form $ax+by$ is positive. The proof works by counting primes in a single residue class: any prime $p\\equiv b \\pmod a$ with $p>b$ automatically equals $a\\cdot((p-b)/a)+b\\cdot 1$, so the needed prime is found as soon as such a prime exists at or below $g$. In the large-$a$ cases the paper uses explicit effective versions of the prime number theorem in arithmetic progressions to show that this residue class contributes more primes than the error terms can remove. In the intermediate cases it instead counts all primes in a short top interval $(g-g/\\log g, g]$ and subtracts the integers in that interval that are representable as $ax+by$, using the classical symmetry that exactly one of $s$ and $g-s$ is representable. The remaining bounded ranges are handed to finite computer verifications.","pith_inferences":["One implication the paper leaves implicit is that the proof could be converted into an explicit bound for the smallest representable prime, since the worked inequalities already identify intervals where the lower bounds are positive.","The text says the residual range with a ≤ 10^4 and b < 10^6 only 'could be done' by computer and does not display the output; an independent exhaustive check of that range would settle whether the theorem is fully established.","The short-interval counting trick for the top of the interval might extend toward three-generator semigroups, though the symmetry that makes it work for two generators has no direct analogue there.","A fully machine-checkable proof certificate for the finite verifications would remove the only part of the argument that currently rests on unshown computation."],"forward_implications":["Every coprime pair $(a,b)$ with $a>2$ has a numerical semigroup containing a prime no larger than the largest nonrepresentable integer.","The existence part of the conjecture becomes unconditional: no lower bound on $ab-a-b$ is needed, and the earlier asymptotic lower bound is supplemented by positivity for all admissible pairs.","In most ranges the proof finds a prime congruent to $b$ modulo $a$, so it gives a concrete search strategy: scan primes in $(b, ab-a-b]$ in that residue class.","The residual parameter sets are finite and explicitly bounded, so independent code can check the computational core of the proof without rerunning the analytic estimates.","Combined with the known asymptotic density result, the theorem completes a qualitative picture: such primes are present for every pair and, for large $a$, occupy about half of the prime positions below $g$."],"supporting_citations":[{"why":"Supplies the explicit effective bounds for primes in arithmetic progressions used as Lemmas 1, 3, and 4 in the main analytic cases.","marker":"[1]"},{"why":"Supplies the explicit two-sided inequalities for the prime-counting function used as Lemma 2 to bound primes in short intervals.","marker":"[6]"},{"why":"States the 2020 conjecture being proved and provides the prior asymptotic lower bound for the number of representable primes.","marker":"[8]"},{"why":"Provides the formula $g=ab-a-b$ and the representability symmetry on which the counting of representable integers in the middle cases rests.","marker":"[9]"}],"fun_headline_variants":["Conjecture confirmed: representable prime always exists below ab-a-b","Coprime pair guarantee: prime of form ax+by lies below ab-a-b","Proof settles 2020 conjecture on primes in numerical semigroups","Every coprime pair has a prime p=ax+by with p ≤ ab-a-b","New theorem: Frobenius number is always preceded by a representable prime"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the claim that every pair in the residual finite ranges actually has a representable prime; the text verifies some ranges explicitly but only says that others 'could be done' by computer, without showing the output.","fun_headline_variants_meta":{"raw":{"variants":["Conjecture confirmed: representable prime always exists below ab-a-b","Coprime pair guarantee: prime of form ax+by lies below ab-a-b","Proof settles 2020 conjecture on primes in numerical semigroups","Every coprime pair has a prime p=ax+by with p ≤ ab-a-b","New theorem: Frobenius number is always preceded by a representable prime"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000302,"raw_usage":{"total_tokens":1685,"prompt_tokens":835,"completion_tokens":850,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":451,"completion_tokens_details":{"reasoning_tokens":749}},"tokens_in":451,"tokens_out":850,"duration_ms":8368,"temperature":1.0,"reasoning_tokens":749,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:39:45.159412+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an exhaustive search over the two residual families: pairs with $a>10^4$ and $g \\le 5\\cdot 10^8$ (under $a \\ge 155\\log b$), and pairs with $a \\le 10^4$ and $b<10^6$. If any such coprime pair with $2<a<b$ has no prime of the form $ax+by$ at or below $g$, Theorem 1 is false; publishing the search results for both families would confirm the computational step the text leaves as 'could be done.'","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the explicit effective bounds for primes in arithmetic progressions used as Lemmas 1, 3, and 4 in the main analytic cases."},{"cited_title":"Panaitopol, Inequalities concerning the function π(x): applications, Acta Arith","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit two-sided inequalities for the prime-counting function used as Lemma 2 to bound primes in short intervals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the 2020 conjecture being proved and provides the prior asymptotic lower bound for the number of representable primes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the formula $g=ab-a-b$ and the representability symmetry on which the counting of representable integers in the middle cases rests."}],"review_version":1}