Pith. sign in

REVIEW 6 minor 18 references

Numerical Semigroups generated by Primes

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The semigroup generated by all primes at least $p_n$ has a rigid asymptotic shape: its largest irredundant generator $u_n$ satisfies $u_n \sim 3p_n$, and every such semigroup satisfies Wilf's inequality.

desk verdict Genuinely new unconditional results on prime-generated semigroups using deep but legitimate tools, with conditional Goldbach bridges that are clearly labeled; the compressed genus proof and an external finite check are the only real soft spots. read the letter →

arxiv 1908.09483 v3 pith:F4TMXTXJ submitted 2019-08-26 math.NT

Michael Hellus , Anton Rechenauer , Rolf Waldi This is my paper · ORCID
classification math.NT MSC 11D0711P3220M14
keywords numericalsemigroupprime-generatedFrobeniusnumberembeddingdimensionWilf'sconjectureGoldbachalmostequalprimesasymptoticshape
topics Goldbach's Conjecture
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 studies $S_n$, the numerical semigroup generated by all primes at least the $n$-th prime $p_n$, and asks how the semigroup's shape scales with $n$. Its main unconditional result is that the largest irredundant generator $u_n$ satisfies $u_n \sim 3p_n$, equivalently $\pi(u_n)\sim 3n$, so the minimal generating set has about $2n$ atoms. The same machinery gives $g_n/p_n \to 5/2$ for the genus, and proves Wilf's inequality $g_n/(1+f_n)\le (e_n-1)/e_n$ for every $S_n$. The paper also establishes two conditional bridges to the classical even-two-prime problem: if the Frobenius number $f_n$ is odd for all large $n$, then $f_n\sim 3p_n$, and if $f_n\sim 3p_n$ then every large even integer is a sum of two primes. A sympathetic reader should care because a naturally defined infinite family of semigroups is shown to have rigid asymptotics, and Wilf's generally open question is settled on this family, with its limiting ratio $5/6$ explicit.

What carries the argument

The load-bearing object is the almost-equal-summands form of the ternary-prime theorem, which makes the generators dense enough that the region just below $(3+\varepsilon)p_n$ is filled by sums of three small primes. Around it the argument builds a self-contained inclusion $S_{n+1}^{3+\varepsilon}\subseteq S_n^{3+\varepsilon}$; iterating gives $S_n=S_n^{3+\varepsilon}$, meaning every element of $S_n$ is generated inside $[p_n,(3+\varepsilon)p_n)$. The lower bound $f_n\ge 3p_n-6$ is the elementary counterpart: the odd composite $3p_n-6$ cannot belong to $S_n$. For Wilf's inequality, the decisive estimate is $f_n<n^2$ for $n\ge429$, derived from a classical Frobenius bound for semigroups generated by a long interval of integers, paired with standard explicit prime-counting estimates to control $\pi(3p_n)$.

What would settle it

Compute the largest atom $u_n$ and Frobenius number $f_n$ directly from the generating interval for a large $n$, say $n=10^5$; if $\pi(u_n)/(3n)$ is not close to 1 or $u_n/p_n$ is not close to 3, the main theorem fails. A more targeted check of the engine: for $n=10^5$ and $\varepsilon=0.01$, verify that every odd integer in $[(3+\varepsilon)p_n,\,(3+\varepsilon)p_{n+1}]$ is a sum of three primes each $>p_n$; the first violation would disprove Lemma 3 in the paper.

Watch

Extended reading notes

Core claim

The central discovery is an asymptotic rigidity theorem for prime-generated semigroups. Writing $e_n$ for the embedding dimension and $\pi$ for the prime-counting function, the paper proves $\pi(u_n)\sim 3n$, $e_n\sim 2n$, and $u_n\sim 3p_n$. The engine is a strong three-prime theorem with almost equal summands: every sufficiently large odd $N$ is $q_1+q_2+q_3$ with $\lvert q_i - N/3\rvert\le N^\theta$ for any $\theta>11/20$. This proves Lemma 3, that every odd $N\ge(3+\varepsilon)p_n$ is in $S_{n+1}$ for large $n$, which combines with the elementary lower bound $f_n\ge 3p_n-6$ to give the Frobenius asymptotics under the parity condition, and yields the corollary $S_n=S_n^{3+\varepsilon}$ that drives the atom count. The paper then proves unconditionally that Wilf's inequality holds for every $S_n$, using the $f_n<3p_n$ criterion together with an explicit bound $f_n<n^2$ for $n\ge429$ obtained from a general Frobenius bound and standard prime estimates.

Load-bearing premise

The whole asymptotic engine rests on the strong three-prime theorem with almost equal summands at exponent $\theta>11/20$; if that theorem failed in the required range, the Frobenius upper bound, the corollary $S_n=S_n^{3+\varepsilon}$, and the atom asymptotics would all lose their support, while the small-$n$ check of Wilf's inequality additionally depends on an external table the paper does not reproduce.

Editorial extensions

If this is right

  • For all sufficiently large $n$, the minimal generating set of $S_n$ has exactly about $2n$ atoms, so the semigroup is generated by a thin but long interval of primes, two atoms per prime on average.
  • Wilf's inequality holds for every $S_n$; the limiting ratio $g_n/(1+f_n)=5/6$ sits strictly below the limiting upper bound $(e_n-1)/e_n\to1$.
  • If the conjectured parity condition $f_n$ odd for all $n\ge5$ holds, then $f_n\sim 3p_n$ follows unconditionally from the theorem's engine.
  • If $f_n\sim3p_n$ holds, then every sufficiently large even integer $x$ is a sum of two primes with one summand in $(x/4,x/2]$.
  • If the sharper conjecture $4p_n>f_{n+1}$ holds for all $n\ge1$, the even-two-prime conjecture follows with the same strong one-summand-in-$(x/4,x/2]$ property.

Reading between the lines

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

  • I infer that the constant $3$ is not special to primes but is set by the ternary representation threshold: the same almost-equal-three-summand machinery, applied to primes in a fixed arithmetic progression $a\bmod d$, would predict $f_n/p_n\to d+1$ for even $d$ and $2d+1$ for odd $d$, matching the paper's numerical observations.
  • A testable extension would be an effective version of the parity bridge: the proof only needs oddness of $f_n$ along a positive-density subsequence, not all large $n$; if one could show even Frobenius numbers occur with density zero using the two-prime almost-equal estimate, the $f_n\sim3p_n$ conjecture would follow without a full parity proof.
  • I infer that the lower bound $f_n\ge3p_n-6$ is asymptotically sharp in a strong sense: any proof that $f_n<3p_n$ happens infinitely often would immediately produce infinitely many twin prime pairs inside $[p_n,3p_n+4]$, so the paper's framework explains why $\liminf f_n/p_n=3$ cannot be improved without resolving the twin-prime question.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies the numerical semigroup S_n generated by all primes at least p_n. The main unconditional result is that the largest irredundant generator u_n satisfies π(u_n) ∼ 3n, the embedding dimension e_n satisfies e_n ∼ 2n, and u_n ∼ 3p_n. The authors also prove that Wilf's inequality g_n/(1+f_n) ≤ (e_n-1)/e_n holds for every S_n and that g_n/p_n converges to 5/2. Conditional statements connect the conjectured asymptotics f_n ∼ 3p_n, f_n odd for large n, and 4p_n > f_{n+1} to the binary Goldbach conjecture, with the Goldbach conjecture following if 4p_n > f_{n+1} for all n.

Significance. The unconditional theorem is a clean, parameter-free asymptotic result for a natural infinite family of numerical semigroups, obtained by combining the Matomäki–Maynard–Shao theorem with elementary semigroup arguments. The proof that Wilf's conjecture holds for all S_n is a substantial advance for a nontrivial family. The conditional bridges to Goldbach are elegant and explicitly non-circular. The analytic core is coherent and the estimates are checkable; the only external dependency is a small finite table used in the proof of Proposition 5.

minor comments (6)
  1. [Section 2, Proposition 5] The proof for n < 429 is delegated entirely to the external table [16]; please include the table or a verification script in the submission so that the finite check is reproducible and not dependent on an external URL.
  2. [Section 1, Corollary and Theorem] The derivation of e_n ∼ 2n is not spelled out; after establishing S_n = S_n^{3+ε}, one should explicitly use that all atoms are primes, that primes in [p_n, 3p_n[ are atoms, and that primes ≥ (3+ε)p_n are not atoms (by Lemma 3), so that e_n is between π(3p_n)−n+1 and π((3+ε)p_n)−n+1, and letting ε → 0 gives e_n ∼ 2n.
  3. [Section 2, Proposition 6] The proof is only a sketch. In particular, the passage from the Coppola–Laporta theorem and [8, Theorem 1.1] to the limits for α_k(n) needs details: for α_2(n), the even numbers in [2p_n, 3p_n) with m close to p_n require a separate treatment, and the exceptional set from [2, Theorem 1, Corollary] has size O(p_n/(log p_n)^A), which is o(p_n). Also, for α_1(n), one should state explicitly that only primes in [p_n, 2p_n) belong to S_n, so the gap density tends to 1.
  4. [Throughout] There are a few typos: 'similiar' should be 'similar' in the Observations, and in the abstract 'f n+1' should be 'f_{n+1}'.
  5. [Section 1, proof of the Corollary] The line 'pn II ≤ p/(3+ε)' contains a typographical artifact; it should read 'p_n ≤ p/(3+ε)'.
  6. [References and data availability] The online tables [16], [17], and [18] are cited with access dates; consider including the relevant data as ancillary files to ensure permanence and reader access.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; derivation rests on external analytic theorems, with only a finite author-hosted table as a minor verifiable dependency.

full rationale

The unconditional results are derived from published external theorems, not from the conjectures or from fitted quantities. The central Theorem and its Corollary follow from Matomäki–Maynard–Shao [8, Theorem 1.1] together with the prime number theorem; Proposition 5 uses Eliahou's criterion, Rosser–Schoenfeld estimates, and Selmer's bound for n ≥ 429; Proposition 6 uses Coppola–Laporta and [8, Theorem 1.1] alongside the PNT. Conjectures (C1)–(C3) are explicitly not used as premises in these proofs; they appear only in conditional statements (Propositions 2, 4) and in computational speculation. No equation is defined in terms of the quantity it is supposed to predict, and no fitted parameter is renamed as a prediction. The only author-supplied input is the finite verification for n < 429 in Proposition 5, delegated to the authors' own table [16]; this is a recomputable finite check rather than a load-bearing self-citation of an unverified theorem, and it does not make the derivation equivalent to its inputs. Thus the paper is self-contained in its analytic core, and the score reflects only the minor verifiable external dependency on the author-hosted table.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No constants are fitted; epsilon is arbitrary and tends to 0, and p_n is the variable. The paper introduces no new entities, only the standard semigroup invariants for the specific semigroups S_n. All proofs rest on published external theorems plus one external finite table.

assumptions (7)
  • standard math Matomaki-Maynard-Shao theorem [8, Thm 1.1]: every sufficiently large odd integer is a sum of three primes q_i with |q_i - N/3| <= N^theta for theta > 11/20.
    Stated in the introduction as fundamental; used to prove Lemma 2, Lemma 3, and the u_n asymptotic, and to get limsup f_n/p_n <= 4.
  • standard math Coppola-Laporta theorem [2, Thm 1, Cor]: all but o(N) even numbers in [N, 2N] are sums of two primes within m^(5/8+epsilon) of m.
    Used in Proposition 6 to control the density of gaps in [2p_n, 3p_n[ and beyond; the paper does not verify the q_j >= p_n boundary condition explicitly.
  • standard math Selmer's bound [9, Thm 3.1.11]: for a numerical semigroup generated by N coprime integers a_1 < ... < a_N, f <= 2 a_N floor(a_1/N) - a_1.
    Used in Lemma 4 to get f_n < n^2 for n >= 429.
  • standard math Eliahou's corollary [4, Cor 6.5]: if 1+f <= 3p for a numerical semigroup of multiplicity p and Frobenius number f, then Wilf's inequality holds.
    Used at the start of Proposition 5 to reduce the proof to the case f_n > 3p_n.
  • standard math Rosser-Schoenfeld estimates [10, Thm 2 and Thm 3]: pi(x) bounds and p_k < k(log k + log log k) for k >= 6.
    Used in Proposition 5 and Lemma 4 to bound pi(3p_n) and p_{3n}.
  • domain assumption The table in [16] correctly lists invariants of S_n for n < 429.
    Proposition 5 relies on this finite computation for n < 429; the table is an external PDF not included in the paper.
  • standard math Prime number theorem and Bertrand's postulate p_{n+1} < 2p_n.
    Used throughout to convert between p_n and n, in Proposition 3, and in the proof of the Corollary.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Numerical Semigroups generated by Primes." pith.science (2026). https://pith.science/paper/F4TMXTXJ

@misc{pith2026190809483,
  author       = {Pith},
  title        = {Pith review of: Numerical Semigroups generated by Primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F4TMXTXJ}},
  note         = {Machine review of arXiv:1908.09483}
}
abstract

Let $p_1=2, p_2=3, p_3=5, \ldots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that $\bullet$ $u_n\sim3p_n$. Similarly, for the largest integer $f_n$ not contained in $S_n$, by computational evidence we suspect that $\bullet$ $f_n$ is an odd number for $n\geq5$ and $\bullet$ $f_n\sim3p_n$; further $\bullet$ $4p_n>f_{n+1}$ for $n\geq1$. If $f_n$ is odd for large $n$, then $f_n\sim3p_n$. In case $f_n\sim3p_n$ every large even integer $x$ is the sum of two primes. If $4p_n>f_{n+1}$ for $n\geq1$, then the Goldbach conjecture holds true. Further, Wilf's question in [12] has a positive answer for the semigroups $S_n$.

Figures

Figures reproduced from arXiv: 1908.09483 by the authors.

Figure 1
Figure 1. 4pn − fn+1 vs pn 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. 4pn − fn+1 vs pn As already noticed in [7] and in [14, answer by user “Woett”, Apr 3 ’12], both conjectures (C1) and (C2) are closely related to Goldbach’s conjecture. As we will see in Proposition 4, (C1) is a consequence of conjecture (C3) fn is odd for n ≥ 5. Notice again, that a conjecture similar to (C3) was already formulated in [7], however for the (related) notion ’threshold of completeness’ for the sequence… view at source ↗
Figure 3
Figure 3. f vs. p for some series of semigroups as in the ’Observations’ The following version of Vinogradov’s theorem is due to Matomäki, Maynard and Shao. It is fundamental for the considerations in this paper. [8, Theorem 1.1] Let θ > 11 20 . Every sufficiently large odd integer n can be written as the sum n = q1 + q2 + q3 of three primes with the restriction [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: fn − 3pn vs n The following considerations are related to [14, answer by user “Aaron Meyerowitz”, Apr 3 ’12]: Let fn < 3 · pn. Then the odd number 3 · pn + 6 is in Sn, but not a prime; hence pn+1 ≤ pn + 6. 1. If pn+1 = pn + 4, since 3·pn + 6 ∈ Sn is not a prime, pn + 6…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    S. J. Benkoski, P. Erdős,On Weird and Pseudoperfect Numbers, Math. Comp. 28 (1974), 617–623. 13

    work page 1974

  2. [2]

    Coppola, M

    G. Coppola, M. B. S. Laporta,On the representation of even integers as the sum of two almost equal primes, Rend. Sem. Mat. Univ. Pol. Torino Vol. 53, 3 (1995)

    work page 1995

  3. [3]

    Dusart,Autour de la fonction qui compte le nombre de nombres premiers, thèse, Université de Limoges, 1998, 171 pp

    P. Dusart,Autour de la fonction qui compte le nombre de nombres premiers, thèse, Université de Limoges, 1998, 171 pp

    work page 1998

  4. [4]

    Eliahou,Wilf’s conjecture and Macaulay’s theorem, J

    S. Eliahou,Wilf’s conjecture and Macaulay’s theorem, J. Eu. Math. Soc. 20 (2018), 2105–2129

    work page 2018

  5. [5]

    Erdős, R

    P. Erdős, R. L. Graham,On a linear diophantine problem of Frobenius, Acta Arithm. 21 (1972), 399 - 408

    work page 1972

  6. [6]

    Erdős, R

    P. Erdős, R. L. Graham,Old and new problems and results in combinatorial number theory, Monographies de L’Enseignement Mathématique 28 (1980)

    work page 1980

  7. [7]

    Kløve,Sums of distinct primes, Nordisk Mat

    T. Kløve,Sums of distinct primes, Nordisk Mat. Tidskr. 21 (1974), 138–140

    work page 1974

  8. [8]

    Matomäki, J

    K. Matomäki, J. Maynard, X. Shao,Vinogradov’s theorem with almost equal summands, Proc. Lond. Math. Soc. 115 (2017), 327–347

    work page 2017

Show all 18 references
  1. [9]

    J. L. Ramírez Alfonsín,The Diophantine Frobenius Problem, Oxford Lecture Series in Mathematics and Its Applications, 2005

    2005

  2. [10]

    J. B. Rosser, L. Schoenfeld,Approximate formulas for some functions of prime numbers, Illinois J. Math., Vol.6 (1) (1962), 64–94

    1962

  3. [11]

    E. S. Selmer, On the linear diophantine problem of Frobenius, J. Reine Angew. Math. 293/294 (1) (1977), 1 – 17

    1977

  4. [12]

    Money-Changing Prob- lem

    H. S. Wilf, A Circle-of-Lights Algorithm for the “Money-Changing Prob- lem”, The American Mathematical Monthly 85 (1978), 562–565

    1978

  5. [13]

    Zhai, Fibonacci-like growth of numerical semigroups of a given genus, Semigroup Forum 86 (2013), 634–662

    A. Zhai, Fibonacci-like growth of numerical semigroups of a given genus, Semigroup Forum 86 (2013), 634–662

    2013

  6. [14]

    Date Accessed: May 27, 2019

    https://mathoverflow.net/questions/93002/ finite-sums-of-prime-numbers-geq-x . Date Accessed: May 27, 2019

    2019

  7. [15]

    Date Accessed: May 27, 2019

    https://oeis.org/A180306. Date Accessed: May 27, 2019

    2019

  8. [16]

    Date Accessed: October 21, 2019

    https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Hellus/ table_1.pdf. Date Accessed: October 21, 2019

    2019

  9. [17]

    Date Accessed: October 21, 2019

    https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Hellus/ table_2.pdf. Date Accessed: October 21, 2019

    2019

  10. [18]

    Date Accessed: October 21, 2019

    https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Hellus/ table_3.pdf. Date Accessed: October 21, 2019. 14

    2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.