Primes of the form $ax+by$

Hui Zhu; Yong-Gao Chen

REVIEW 3 major objections 5 minor 1 cited by

Primes of the form $ax+by$

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For every coprime $3 \le a < b$, more than $0.005\,s(a,b)/\log s(a,b)$ primes up to $ab-a-b$ have the form $ax+by$, and for fixed $a$ the proportion tends to $1/2 - 1/(2(a-1))$.

desk verdict Genuinely new quantitative results on primes in numerical semigroups, but the final constant in Theorem 1.2 leans on unshown finite computations that should be pinned down before publication. read the letter →

arxiv 2506.03620 v1 pith:MMVIRB2T submitted 2025-06-04 math.NT

Yong-Gao Chen , Hui Zhu This is my paper

classification math.NT MSC 11D0711N1311Y35

keywords Frobeniusnumberprimedistributionnumericalsemigrouplargesieveprimesinarithmeticprogressionsexplicitprime-countingboundsEulertotientfunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves two quantitative statements about primes of the form $ax+by$ with $x,y\ge 0$, for coprime integers $3\le a0.005\,s(a,b)/\log s(a,b)$ for every such pair, a uniform positive proportion of all primes up to $s(a,b)$; Theorem 1.1 gives the sharper asymptotic $\pi(a,b)=(1/2-1/(2(a-1))+o(1))\pi(s(a,b))$ as $b\to\infty$ with $a$ fixed. Together these establish the long-standing expectation that $\pi(a,b)$ is always positive and that for large $a$ about half the primes below $s(a,b)$ are representable, without relying on earlier partial proofs.

What carries the argument

The load-bearing machinery is the disjoint residue strip decomposition $S_k(a,b)=\{xa+kb: x\ge0,\ xa+kb\le S\}$ together with the reflection identity $p\notin T(a,b)\iff S-p\in T(a,b)$ for $0\le p\le S$. Because the strips are disjoint, every representable number up to $S$ belongs to exactly one arithmetic progression $xa+kb$ modulo $a$; because of the reflection, the number of missing primes in the upper eighth of the interval is bounded by counting primes in progressions of the form $p\equiv -b-by\pmod a$ with $0\le by< S/8$. The quantitative estimates come from a large-sieve bound on primes in arithmetic progressions, explicit bounds for primes and for primes in progressions, and a uniform prime number theorem in arithmetic progressions for the asymptotic part.

What would settle it

Recompute $\pi(x)-\pi(7x/8)$ for $40000\le x<10^{10}$ and directly count primes of the form $ax+by$ for all coprime $3\le a<b$ in the finite ranges of Cases 4 and 6; if any of the asserted bounds fails, or if any coprime pair has $\pi(a,b)\le 0.005\,s(a,b)/\log s(a,b)$, Theorem 1.2 is false.

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Extended reading notes

Core claim

The central claim is that the primes represented by the binary form $ax+by$ below the Frobenius number are abundant, not merely infinite. The paper capitalizes on two structural facts: the sets $S_k=\{xa+kb: x\ge0,\ xa+kb\le S\}$ for $k=0,1,\ldots$ are disjoint, and for $0\le p\le S$ one has $p\notin T(a,b)$ exactly when $S-p\in T(a,b)$. Counting primes in the upper interval $7S/8<p\le S$ that fail to be representable then becomes a problem of counting primes in certain arithmetic progressions modulo $a$, which the authors estimate with the large sieve and explicit prime-counting bounds. The result is an explicit positive constant $0.005$ in Theorem 1.2, and, by summing the residue classes with the prime number theorem in arithmetic progressions, the asymptotic coefficient $1/2-1/(2(a-1))$ in Theorem 1.1.

Load-bearing premise

The proof's coverage of the small parameter ranges rests on finite computations — the constants $0.121243$, $0.024$ and $0.006$ — that are asserted with the words 'By a calculation' and shown without tables or code.

Editorial extensions

If this is right

For $a=3$, the count of representable primes below $S$ is asymptotically one quarter of all primes below $S$, so about 25% of primes up to $ab-a-b$ are of the form $3x+by$ for odd $b$ not divisible by 3.
The positivity conjecture for $\pi(a,b)$ follows for all coprime $3\le a<b$ as a direct corollary of Theorem 1.2, independent of any earlier proof.
The uniform lower bound $0.005\,S/\log S$ shows that near the Frobenius number the proportion of representable primes is bounded away from zero over all coprime pairs, not just along sequences.
For fixed $a$, the asymptotic coefficient $1/2-1/(2(a-1))$ increases toward $1/2$ as $a$ grows, consistent with the heuristic that, for large $a$, representable primes occupy about half of the residue classes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same disjoint-strip decomposition and the reflection identity could be applied to other binary linear forms, giving explicit lower bounds for prime values when both coefficients vary as well as when one is fixed.
If the asserted finite computations were replaced by published tables or code, the constant $0.005$ could be raised substantially; the paper's own conjectures suggest the true proportion lies between roughly $13/66$ and $1/2$ of all primes below the Frobenius number.
The equality cases proposed in the conjectures, $a=3,b=166$ and $a=3,b=5$, provide concrete benchmarks for any computational test of the sharp constants.
A natural next step not addressed in the paper is the case where $a$ grows slowly with $b$; the residue-class machinery used here is a candidate route toward such a two-variable asymptotic.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a real result, not a cosmetic repackaging. Theorem 1.1 gives a fixed-a asymptotic with coefficient 1/2 − 1/(2(a−1)), and Theorem 1.2 gives the first explicit linear lower bound c·s(a,b)/log s(a,b) for π(a,b). That goes beyond the previously confirmed positivity and the a→∞ asymptotic of Ding–Zhai–Zhao and Dai–Ding–Wang. The proofs are mostly clean analytic number theory: a disjointness lemma for residue classes, the large sieve, Siegel–Walfisz, and explicit prime bounds. They do not rely on the prior confirmations, which is nice. Lemma 2.7 (disjointness of the Sk(a,b)) is elegant and correct. The case split in Theorem 1.2 is careful, and the analytic estimates from Case 1, Case 2, and Case 5 all have healthy margins.

The soft spot is the layer of finite computation that carries the final constant. Lemma 2.4 asserts a lower bound for π(x)−π(7x/8) on two intervals, 40000≤x<10^5 and 10^5≤x<10^10, with no code or tables. Cases 4 and 6 in Section 4 rely on similar asserted numerical checks in small S ranges. This is not cosmetic: in Case 3 the margin is only 0.121243 − 0.116219 = 0.005024, so a relative error of about 0.5% in either constant, or a slightly larger true maximum in the upper bound 0.116219, would drop the final bound below 0.005. I cannot verify these constants from the manuscript, and neither can any reader. That makes Theorem 1.2, as written, a conditional result: if the computations are correct, the theorem is true; if they are off, the bound may fail.

None of this looks like a fatal flaw. The ranges are finite and modest (primes up to 10^10 can be counted exactly with a sieve), and the analytic machinery is standard. The right fix is for the authors to supply the code, or at least a table of the extremal values that justify each 'by a calculation' claim, perhaps as ancillary files. A referee should be able to demand this without finding any deeper problem.

Who is this for? Specialists in analytic number theory working on primes in arithmetic progressions or in numerical semigroups. It is a solid, focused contribution, and it deserves a serious referee. I would recommend sending it to peer review, with a request to make the finite computations reproducible. If the authors provide the data, I would be happy to cite Theorem 1.1 and the explicit lower bound; as it stands, the unverified constants give me pause.

Referee Report

3 major / 5 minor

Summary. The manuscript studies π(a,b), the number of primes representable as ax+by with x,y≥0 and not exceeding the Frobenius number S=ab−a−b, for coprime integers 3≤a<b. Theorem 1.1 shows that for fixed a, π(a,b)∼(1/2−1/(2(a−1)))π(S) as b→∞. Theorem 1.2 establishes the uniform lower bound π(a,b)>0.005 S/log S. The proof decomposes primes in T(a,b) by residue classes kb mod a, using the disjointness of the sets S_k(a,b) (Lemma 2.7), large-sieve and Siegel–Walfisz estimates to bound missing primes (Lemma 2.9), and an explicit lower bound for π(S)−π(7S/8) (Lemma 2.4). Small-parameter cases are handled by asserted finite computations in Section 4. The paper concludes with two numerical conjectures.

Significance. Theorem 1.2 is a genuine improvement in explicitness: it gives an absolute constant and the order S/log S, rather than the previously available C(ε)S/(log S)^{2+ε} for sufficiently large parameters. Theorem 1.1 gives a clean asymptotic for fixed a and does not depend on the recent works [2] and [4]. The analytic structure is coherent and uses standard external theorems; there is no circularity. The main weakness is that a load-bearing layer of finite numerical assertions is not reproducible from the manuscript, and in Section 4, Case 3, the final margin is only 0.005024, so these assertions are not cosmetic.

major comments (3)

[§2, Lemma 2.4; §4, Cases 1–3] The proof of Lemma 2.4 is analytic for x≥10^10, but for 10^5≤x<10^10 and 40000≤x<10^5 it says only 'By a calculation' and provides no code, tables, or rounding conventions. The resulting constant 0.121243 is used in Cases 1–3 of Theorem 1.2, and in Case 3 the margin is 0.121243−0.116219=0.005024, only 0.000024 above the target 0.005. Thus a small numerical error in either bound would invalidate Theorem 1.2. The bounds '<0.11272' in Case 2 and '<0.116219' in Case 3 are likewise asserted rather than derived. Please supply a complete and independently checkable verification for all these assertions, such as code with output tables or interval-arithmetic certificates.
[§4, Case 4] For 21≤a≤200 and S<40000, the proof reduces to the asserted lower bounds '#{...}>0.024 S/logS' for 202≤b≤40000/(a−1)+1 and '#{...}>0.219 S/logS' for a<b≤201. These finite enumerations are the only support for Theorem 1.2 in this range, and no code, tables, or reproducible constants are provided. Require the authors to supply the computations or replace them by a rigorous estimate; without one of these, the theorem is unverified for this entire parameter range.
[§4, Case 6] For 3≤a≤20 and a<b<50a², the lower bound '≥0.006 S/logS' is asserted as a finite computation over 0≤x≤150. This is the only argument covering all small a with b in this range, so it is essential to Theorem 1.2. Please provide reproducible code/tables or a rigorous verification; the current 'By Lemma 2.7 ... = ...' text does not show how the numerical value 0.006 is obtained.

minor comments (5)

[Abstract and §1] The abstract says π(a,b) counts primes 'less than or equal to s(a,b)' while the introduction says 'less than s(a,b)'; since s(a,b)∉T(a,b) the two statements coincide, but the wording should be uniform.
[§4, Cases 4 and 6] Inequalities such as 0≤y≤a/2−1 use a non-integer upper bound when a is odd; write floor(a/2)−1 or otherwise clarify that y ranges over integers.
[§2, Lemma 2.4 proof] In the displayed chain for x≥10^10, the second error term at 7x/8 is missing its denominator in the typeset version; please correct the display.
[§1, Theorem 1.1] In the displayed 'In particular' line, the formula 'o(π(s(3,b))' is missing a closing parenthesis; it should be 'o(π(s(3,b)))'.
[References] Reference [2] should use the standard arXiv identifier format, e.g. arXiv:2411.09446, for consistency with [4].

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found.

full rationale

The derivation chain of Theorems 1.1 and 1.2 rests on external results (the large sieve, the Siegel–Walfisz theorem, explicit prime-counting bounds by Bennett–Martin–O’Bryant–Rechnitzer and by Rosser–Schoenfeld) and on elementary disjointness/counting lemmas. No parameter is fitted to the target quantity, no prior theorem by the present authors is invoked, and the conjectures stated in the introduction are explicitly not used in the proofs. The computational assertions in Lemma 2.4 and Cases 4 and 6 are unsupported reproducibility-wise, but they are finite verifications, not definitions or fitted inputs; an unverified numerical check is a correctness risk, not a circular step. Neither Theorem 1.1 nor Theorem 1.2 assumes the inequality or asymptotic it proves.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or new entities. The paper's contribution rests on standard analytic number theory plus finite numerical checks that are asserted but not made reproducible.

assumptions (4)

standard math Siegel-Walfisz theorem (Lemma 2.5) holds uniformly for fixed modulus a
Used in Theorem 1.1 to replace π(S;a,kb) by li(S)/φ(a) with acceptable error.
standard math Montgomery-Vaughan large sieve bound (Lemma 2.1) with the stated constant 2
Used in Lemma 2.9 to bound primes in arithmetic progressions in short intervals.
standard math Explicit bounds for primes in arithmetic progressions and for π(x) from Bennett-Martin-O'Bryant-Rechnitzer (Lemmas 2.3 and 2.6)
Supplies the numerical constants 0.0008375 and the 5/(2 log x) correction term.
ad hoc to paper Finite computations asserted by 'By a calculation' in Lemma 2.4 and Section 4 Cases 4 and 6 are correct
These unshown lower bounds (0.121243, 0.024, 0.006) are load-bearing for the small-range part of Theorem 1.2.

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Cite this review

Pith. "Pith review of Primes of the form $ax+by$." pith.science (2026). https://pith.science/paper/MMVIRB2T

@misc{pith2026250603620,
  author       = {Pith},
  title        = {Pith review of: Primes of the form $ax+by$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MMVIRB2T}},
  note         = {Machine review of arXiv:2506.03620}
}

abstract

For two coprime positive integers $a,b$, let $T(a,b)=\{ ax+by : x,y\in \mathbb{Z}_{\ge 0} \} $ and let $s(a,b)=ab-a-b$. It is well known that all integers which are greater than $s(a,b)$ are in $T(a,b)$. Let $\pi (a, b)$ be the number of primes in $T(a,b)$ which are less than or equal to $s(a,b)$. It is easy to see that $\pi (2, 3)=0$ and $\pi (2, b)=1$ for all odd integers $b\ge 5$. In this paper, we prove that if $b>a\ge 3$ with $\gcd (a, b)=1$, then $\pi (a, b)>0.005 s(a,b)/\log s(a,b)$. We conjecture that $\frac{13}{66}\pi (s(a,b))\le \pi (a, b)\le \frac 12\pi (s(a,b))$ for all $b>a\ge 3$ with $\gcd (a, b)=1$.

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages · cited by 1 Pith paper

[2]

T. Dai, Y. Ding and H. Wang, Note on a conjecture of Ram ´ ırez Alfons ´ ın and Ska lba, arXiv2411.09446v3 (2024)

work page Pith review arXiv 2024
[4]

Y. Ding, W. Zhai and L. Zhao, On a conjecture of Ram ´ ırez Al- fons ´ ın and Ska lba II, to appear in J. Th´ eor. Nombres Bordeaux, arXiv:2309.09796v2 (2023)

work page Pith review arXiv 2023
[1]

M. A. Bennett, G. Martin and K. O’Bryant, A. Rechnitzer, Explicit bounds for primes in arithmetic progressions, Illinois J. Math. 62 (2018), 427–532. Primes of the formax+by13

work page 2018
[3]

Davenport, Multiplicative Number Theory, 2nd edn., GTM 74, Springer, New York, 1980

H. Davenport, Multiplicative Number Theory, 2nd edn., GTM 74, Springer, New York, 1980

work page 1980
[5]

Ding, On a conjecture of Ram ´ ırez Alfons ´ ın and Ska lba, J

Y. Ding, On a conjecture of Ram ´ ırez Alfons ´ ın and Ska lba, J. Number Theory 245 (2023), 292–302

work page 2023
[6]

H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119–134

work page 1973
[7]

J. L. Ram ´ ırez Alfons ´ ın and M. Ska lba, Primes in numerical semigroups, C. R. Math. Acad. Sci. Paris 358 (2020), 1001–1004

work page 2020
[8]

J. B. Rosser and L. Schoenfeld, Approximate formulas for some func- tions of prime numbers, Illinois J. Math. 6 (1962), 64–94

work page 1962

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [2]

T. Dai, Y. Ding and H. Wang, Note on a conjecture of Ram ´ ırez Alfons ´ ın and Ska lba, arXiv2411.09446v3 (2024)

work page Pith review arXiv 2024

[2] [4]

Y. Ding, W. Zhai and L. Zhao, On a conjecture of Ram ´ ırez Al- fons ´ ın and Ska lba II, to appear in J. Th´ eor. Nombres Bordeaux, arXiv:2309.09796v2 (2023)

work page Pith review arXiv 2023

[3] [1]

M. A. Bennett, G. Martin and K. O’Bryant, A. Rechnitzer, Explicit bounds for primes in arithmetic progressions, Illinois J. Math. 62 (2018), 427–532. Primes of the formax+by13

work page 2018

[4] [3]

Davenport, Multiplicative Number Theory, 2nd edn., GTM 74, Springer, New York, 1980

H. Davenport, Multiplicative Number Theory, 2nd edn., GTM 74, Springer, New York, 1980

work page 1980

[5] [5]

Ding, On a conjecture of Ram ´ ırez Alfons ´ ın and Ska lba, J

Y. Ding, On a conjecture of Ram ´ ırez Alfons ´ ın and Ska lba, J. Number Theory 245 (2023), 292–302

work page 2023

[6] [6]

H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119–134

work page 1973

[7] [7]

J. L. Ram ´ ırez Alfons ´ ın and M. Ska lba, Primes in numerical semigroups, C. R. Math. Acad. Sci. Paris 358 (2020), 1001–1004

work page 2020

[8] [8]

J. B. Rosser and L. Schoenfeld, Approximate formulas for some func- tions of prime numbers, Illinois J. Math. 6 (1962), 64–94

work page 1962