REVIEW 4 major objections 4 minor 1 cited by
Note on a conjecture of Ram\'{\i}rez Alfons\'{\i}n and Ska{\l}ba
T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read For every coprime pair 2<a<b, a prime p≤ab-a-b equals ax+by for nonnegative x,y.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.'
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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 $g$.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (4)
- [Conclusion and footnote 5] 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.
- [Footnotes 1, 4, and 6] 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.
- [Case II, equation (2.8) and footnote 1] 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.
- [Case VI, Lemma 3 applicability] 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.
minor comments (4)
- [Title, abstract, and introduction] 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.
- [Lemma 2] 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.
- [Case I, after equation (2.4)] 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.
- [Reference [9]] 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.
Circularity Check
No circularity: Theorem 1 is derived from independent explicit bounds and finite verification; the target conjecture is never used as an input.
full rationale
The proof of Theorem 1 does not use the conjecture π_{a,b} > 0 as an assumption anywhere. Each case derives a lower bound for π_{a,b} from external published results: the effective Siegel–Walfisz bounds of Bennett–Martin–O'Bryant–Rechnitzer (Lemmas 1, 3, 4), Panaitopol's explicit prime-counting inequalities (Lemma 2), and Sylvester's symmetry property for representable integers. The constants in these lemmas are explicit, and the reductions to finite ranges are carried out through elementary inequalities; the remaining domains are asserted to be checkable by computer, and footnote 5 describes some of those checks. Even though the finite verifications are not fully reproducible as written, that is a completeness or correctness issue, not circularity. The authors' citations to their own prior work (Ding, Ding–Zhai–Zhao, Ding–Komatsu) concern Conjecture 2 and generalizations, and they are not load-bearing for the proof of Conjecture 1. The phrase 'we will follow the proof of Ramírez Alfonsín and Skalba [8, Theorem 1] with explicit calculations' refers to the proof strategy of an external paper, not to an assumption of the result being proved. There is no self-definitional step, no fitted parameter renamed as a prediction, and no load-bearing self-citation chain. The central derivation is self-contained modulo the stated finite checks, so the circularity score is 0.
Assumptions & free parameters
assumptions (4)
- standard math Sylvester's symmetry: for 0≤s≤g exactly one of s and g-s is representable as ax+by.
- standard math Explicit bounds for primes in arithmetic progressions from [1, Theorems 1.3, 1.9, Corollary 1.7] are valid in the quoted ranges.
- standard math Panaitopol's bounds on π(x) [6, Theorem 1] are correct and apply for x≥59.
- ad hoc to paper The finite verifications (i) and (iii) and the numerical minimizations in footnotes 1, 4, and 6 were actually performed and are error-free.
Cite this review
Pith. "Pith review of Note on a conjecture of Ram\'{\i}rez Alfons\'{\i}n and Ska{\l}ba." pith.science (2026). https://pith.science/paper/W6YLRGQW
@misc{pith2026241109446,
author = {Pith},
title = {Pith review of: Note on a conjecture of Ram\'\irez Alfons\'\in and Ska\lba},
year = {2026},
howpublished = {\url{https://pith.science/paper/W6YLRGQW}},
note = {Machine review of arXiv:2411.09446}
}
abstract
Let $2< a<b$ be two relatively prime integers and $g=ab-a-b$. It is proved that there exists at least one prime $p\le g$ of the form $p=ax+by~(x,y\in \mathbb{Z}_{\ge 0})$, which confirms a 2020 conjecture of Ram\'{\i}rez Alfons\'{\i}n and Ska{\l}ba.
Forward citations
Cited by 1 Pith paper
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Primes of the form $ax+by$
For coprime 3≤a<b, at least 0.005(ab-a-b)/log(ab-a-b) primes below the Frobenius number are representable as ax+by, and for fixed a the asymptotic fraction is 1/2-1/(2(a-1)).
Reference graph
Works this paper leans on
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[8]
J. L. Ram´ ırez Alfons´ ın, M. Ska/suppress lba,Primes in numerical semigroups, C. R. Math. Acad. Sci. Paris 358 (2020), 1001–1004
work page 2020
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[1]
M. A. Bennett, G. Martin, K. O’Bryant, A. Rechnitzer, Explicit bounds for primes in arithmetic progressions, Illinois J. Math. 62 (2018), 427–532
work page 2018
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[2]
Ding, On a conjecture of Ram´ ırez Alfons´ ın and Ska/suppress lba,J
Y. Ding, On a conjecture of Ram´ ırez Alfons´ ın and Ska/suppress lba,J. Number Theory 245 (2023), 292–302
work page 2023
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[3]
Y. Ding, T. Komats, On a conjecture of Ram´ ırez Alfons´ ın and Ska/suppress lba III,to appear in Integers, arXiv:2311.03997 (2023)
work page Pith review arXiv 2023
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[4]
Y. Ding, W. Zhai, L. Zhao, On a conjecture of Ram´ ırez Alfons´ ın and Ska/suppress lba II,to appear in J. Th´ eor. Nombres Bordeaux,arXiv:2309.09796v2 (2023)
arXiv 2023
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[6]
Panaitopol, Inequalities concerning the function π(x): applications, Acta Arith
L. Panaitopol, Inequalities concerning the function π(x): applications, Acta Arith. 94 (2000), 373–381
work page 2000
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[7]
J. L. Ram´ ırez Alfons´ ın,The Diophantine Frobenius Problem, Oxford Lecture Series i n Mathe- matics and its Applications, vol. 30, Oxford University Press, 2005
work page 2005
Show all 9 references
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[9]
J. J. Sylvester, On subvariants, i.e. Semi–Invariants to Binary Quantics of an Unlimited Order, Amer. J. Math. 105 (1882), 79–136. (Tianhan Dai) School of Mathematical Sciences, Yangzhou Un iversity, Yangzhou 225002, People’s Republic of China Email address: 3408606588@qq.com ...
Reviewed August 12, 2026 · model on record in the stance chip above.
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