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The number of primes not in a numerical semigroup

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Nonrepresentable primes are at least 4% of primes below ab-a-b.

desk verdict 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. read the letter →

arxiv 2506.03625 v2 pith:M67RERYK submitted 2025-06-04 math.NT

Yong-Gao Chen , Hui Zhu This is my paper
classification math.NT MSC 11D0711N1311Y35
keywords numericalsemigroupFrobeniusproblemprimesinarithmeticprogressionsdistributionofSiegel-Walfisztheoremnonrepresentablelowerboundpositivedensity
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

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.

What carries the argument

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).

What would settle it

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.

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Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Theorem 1.2, Case 2] 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.
  2. [Theorem 1.2, Case 3] 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.
  3. [Theorem 1.2, Case 4] 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.
  4. [Theorem 1.5] 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.
minor comments (5)
  1. [Abstract] The phrase "pose following conjecture" should read "pose the following conjecture".
  2. [Proof of Theorem 1.1] 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.
  3. [Proof of Theorem 1.1, upper bound] 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.
  4. [Lemma 2.1 application] 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.
  5. [General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stated lower bounds are derived from independent prime-counting estimates and explicit finite verifications, not from assuming the conclusion.

full rationale

The derivation of Theorem 1.1 represents pi*(a,b) exactly as a count of primes in residue classes bv mod a with v=1,...,a-1, then applies lower and upper bounds from Corollary 2.3 and Lemma 2.4, which are independent external prime estimates. The bound (1/2 + 1/(2(a-1))) S/log S is reached by summing v over reduced residue classes and letting b tend to infinity; the 'best possible' assertion is an upper bound with the same leading constant, again from the same external estimates, so it is not a fit to the target quantity. Theorem 1.2 introduces a parameter delta and proves a general inequality Delta(delta,a,S)pi(S) < pi*(a,b) using Lemma 2.1, the Siegel-Walfisz theorem, and Rosser-Schoenfeld bounds; the constants 0.0445, 0.0401, 0.0425, and 0.05334 are obtained by evaluating this inequality, not by normalizing to pi*(a,b) data. The remaining cases marked 'a calculation shows' are finite verifications of explicit inequalities or enumerations over bounded ranges of (a,b); although the computations are not reported in the text and are therefore a reproducibility concern, they do not assume the theorem and are not equivalent to the conclusion. No load-bearing self-citation occurs, and no equation is defined in terms of the result it is supposed to prove. Accordingly, the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim does not introduce new objects, forces, or fitted physical constants. The only hand-chosen numbers are case-splitting constants and computation cutoffs in the proofs; they are selected to satisfy explicit inequalities, not tuned to make the theorem true. The mathematical content rests on standard prime-number theorems plus unverified finite computations.

free parameters (2)
  • Case-splitting constants delta in Theorem 1.2 = 0.1; 0.0904; 0.095; 0.05
    Hand-chosen in separate ranges of a to keep the lower bound Delta above 0.04. They are proof constants, not fitted to numeric data, and do not enter the final theorem statement.
  • Cutoff values for S in Theorem 1.5 = 18595; 60180
    Hand-picked split points for the a=9 and a=10 verification intervals; no data fitting, just convenience thresholds for the computer check.
assumptions (5)
  • standard math Siegel-Walfisz theorem (Lemma 2.2): primes in arithmetic progressions are equidistributed up to logarithmic moduli.
    Used to establish Corollary 2.3, which supplies the lower and upper estimates for pi(bv; a, bv) for large a and b in Theorem 1.1.
  • standard math Bennett-Martin-O'Bryant-Rechnitzer explicit bounds for primes in arithmetic progressions (Lemma 2.4).
    Provides explicit x/(phi(m) log x) inequalities for moduli a<=1200 and x>=50a^2, used for the a<=1200 case in Theorem 1.1.
  • standard math Montgomery-Vaughan large sieve bound (Lemma 2.1).
    Bounds the number of representable primes up to delta*S in each residue class, a key ingredient in the proof of Theorem 1.2.
  • standard math Rosser-Schoenfeld elementary estimates for pi(x) (Lemma 2.5).
    Used to convert S/logS estimates into pi(S) estimates and to compare pi(S) with pi(delta*S) in the later cases.
  • ad hoc to paper The reported finite computations are correct.
    The paper repeatedly asserts 'a calculation shows' for finite ranges (a<=180, b<=180; a=9, b<=2325; a=10, b<=6687) without code or output; the universal claims depend on these checks being accurate.

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Pith. "Pith review of The number of primes not in a numerical semigroup." pith.science (2026). https://pith.science/paper/M67RERYK

@misc{pith2026250603625,
  author       = {Pith},
  title        = {Pith review of: The number of primes not in a numerical semigroup},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M67RERYK}},
  note         = {Machine review of arXiv:2506.03625}
}
abstract

For two coprime positive integers $a$ and $b$,let $\pi^* (a, b)$ be the number of primes that cannot be represented as $au+bv$, where $u$ and $v$ are nonnegative integers. It is clear that $\pi^* (a, b)\le \pi (ab-a-b)$, where $\pi (x)$ denotes the number of primes not exceeding $x$. In this paper, we prove that $\pi^* (a, b)\ge 0.04\pi (ab-a-b)$ and pose following conjecture: $\pi^* (a, b)\ge \frac 12 \pi (ab-a-b)$. This conjecture is confirmed for $1\le a\le 10$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    For two coin values a1<a2 with gcd 1, every sufficiently large amount is payable with coprime coin counts, and the threshold G satisfies a1a2 <= G << a1a2(log a1a2)^2; general k is also finite.

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