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REVIEW 3 major objections 4 minor 20 references

Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper proposes a broad generalization of Ramsey numbers and shows that Green-Tao, twin primes, Zhang's bounded prime gaps, and Polignac's conjecture become existence statements for such numbers.

desk verdict A broad but over-general Ramsey framework with a correct algebraic core; the advertised prime-gap equivalences are definitional re-labelings, not Ramsey-theoretic imports. read the letter →

arxiv 2502.04311 v1 pith:BPJ6WDJE submitted 2025-02-06 math.CO

Bryce Alan Christopherson This is my paper · ORCID
classification math.CO MSC 05C5505D1011A4111N0511N1311N32
keywords RamseynumbersgeneralizedtheoryindicatorpolynomialsfinitefieldsGreen-TaotheoremtwinprimeconjecturePolignac'sboundedgaps
topics The Twin Prime 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

Classical Ramsey numbers ask for the smallest complete graph whose every edge-coloring forces a monochromatic clique. This paper argues that the same question makes sense in far greater generality: fix a hereditary family of graphs, allow only some of the colorings of each graph, and ask for the first index from which every admissible coloring must contain one of a list of colored target graphs. Within this setting, the paper builds, for prime-power color sets, an indicator polynomial whose zeros are exactly the colorings that contain a prescribed colored subgraph, and proves that a Galois-type Ramsey number with maximal base is the first index at which this polynomial is identically zero as a polynomial function. The concluding constructions rephrase the Green-Tao theorem, the twin prime conjecture, Zhang's bounded prime gap theorem, and Polignac's conjecture as existence statements for such generalized Ramsey numbers. If those translations hold, then notoriously difficult statements about prime gaps become statements about the vanishing of explicit polynomial families.

What carries the argument

Indicator polynomials $p[G,X,\psi](x)=\prod_{\pi\in G/X}\Bigl(1-\prod_{e\in\pi^{-1}(X)}\bigl(1-(x_e-\psi(\pi(e)))^{q-1}\bigr)\Bigr)$ over $\mathbb{F}_q$ are the central device. They act as an algebraic Iverson bracket: the value at a coloring $\rho$ is zero exactly when $\rho$ contains a copy of $X$ colored by $\psi$. Over a maximal Galois-type base, the vanishing of these polynomials tracks the Ramsey number exactly, and Theorem 4 extends this tracking to locally finite Ramsey bases by injecting arbitrary finite color sets into finite fields, so the criterion becomes the ideal containment $I(S_n)\supseteq \langle p[G_n,X,C]\rangle$.

What would settle it

Theorem 8 claims an injection from the $i+1$ vertices of $K_{i+1}$ onto the set $\{p_m,\dots,p_{m+i+1}\}$, which has $i+2$ elements; counting the two sets shows no such function exists. A reader can settle the central translation by checking whether the remaining argument survives once this map is replaced by a bijection onto $i+1$ primes.

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Extended reading notes

Core claim

The paper's central claim is that a single very general definition of Ramsey number subsumes the classical numbers $R(z_1,\ldots,z_m)$, the usual generalized graph Ramsey numbers, and several problems not normally seen as Ramseyian. For a Ramsey base $(G,S)$—a hereditary family of graphs together with chosen sets of admissible edge-colorings—and a Ramsey symbol $(X,C)$ of colored target graphs, the Ramsey number $R_{(G,S)}(X,C)$ is the first index $n$ such that every admissible coloring of $G_t$ with $t\ge n$ contains one of the target subgraphs with the prescribed coloring. When the colors form a finite field $\mathbb{F}_q$, the paper defines an indicator polynomial $p[G_i,X,C]$ that vanishes exactly on colorings containing a target subgraph, and Theorem 3 identifies the Ramsey number, in the maximal-base Galois case, as $i_m+1=i_k$, where $i_m$ is the last index with a nonzero indicator and $i_k$ the first index with an identically zero one. The final theorems then encode the Green-Tao theorem, the twin prime conjecture, Zhang's bounded prime gap theorem, and Polignac's conjecture as existence statements for Ramsey numbers of this kind. In the author's framing, this shows that the structure of Ramseyian objects, even when numeric values are out of reach, is tractable.

Load-bearing premise

The load-bearing premise is that a Ramsey base may exclude most colorings and keep only the one coloring that records distances between consecutive primes; if a Ramsey base had to contain every coloring, the claimed equivalence of the twin prime conjecture to the existence of specific Ramsey numbers would collapse.

Editorial extensions

If this is right

  • For any Galois-type Ramsey number with maximal base that exists, its value is the first index at which the associated indicator polynomial is identically zero as a polynomial function.
  • Because arbitrary finite color sets can be injected into finite fields, the polynomial criterion applies to every locally finite Ramsey base and symbol, not only to prime-power color counts.
  • Ramsey numbers of finite type with uniform symbols are monotone: enlarging the target symbol or shrinking the admissible color sets can only lower the Ramsey number, provided it exists.
  • The Green-Tao theorem implies the existence of infinitely many generalized Ramsey numbers built from the distance colorings of consecutive primes, and the twin prime conjecture is equivalent to the existence of the Ramsey numbers $R_{(G,S(m))}(\{K_2\},\{2K_2\})$ for every $m$.
  • Zhang's bounded prime gap theorem and Polignac's conjecture admit analogous Ramsey-number existence formulations, so determining whether those Ramsey numbers exist is at least as hard as the corresponding prime-gap statements.

Reading between the lines

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

  • Editorial extension: if the polynomial encoding is sound, the first-index identity could turn known lower bounds for classical Ramsey numbers into lower bounds for the vanishing index of explicit polynomial families, giving the algebraic sets a combinatorial meaning they do not yet have.
  • Editorial extension: Theorem 8's construction contains a size mismatch as written—an injection from the $i+1$ vertices of $K_{i+1}$ onto the $i+2$ primes $p_m,\dots,p_{m+i+1}$—so a corrected bijection onto $i+1$ primes is needed before the Green-Tao translation can be treated as load-bearing.
  • Editorial extension: one could test the translations computationally on small cases by computing the indicator polynomial $p[K_{i+1},\{P_t\},\{kP_t\}]$ over a finite field and comparing its first identically-zero index with directly known small prime-gap data.
  • Editorial extension: the Galois-connection analogy the paper sketches could be developed into a formal adjunction between admissible color sets and Ramsey symbols, potentially yielding a duality theory for this class of Ramsey numbers.
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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

3 major / 4 minor

Summary. This paper defines a generalized Ramsey number R(G,S)(X,C) in which the admissible colorings of each graph G_i are restricted to an arbitrary subset S_i of Hom(G_i,A_i), and develops an algebraic criterion: for Galois-type Ramsey numbers, the Ramsey number is characterized by the first index at which the indicator polynomial p[G_i,X,C] vanishes identically as a polynomial function (Theorem 3). After extending this to locally finite types via injections (Theorem 4), the paper presents 'Ramsey-theoretic' restatements of the Green-Tao theorem, the twin prime conjecture, Zhang's bounded gaps theorem, and Polignac's conjecture (Theorems 7-12, Corollary 3).

Significance. If correct, the algebraic framework would provide a unified view of generalized Ramsey numbers, and the final section would connect Ramsey theory to major problems in prime gaps. The indicator-polynomial lemmas and Theorem 3 are internally consistent and constitute a modest but sound reformulation. However, the final section's equivalences are definitional: because S_i(m) is chosen to be a singleton containing exactly the metrical coloring of the primes, the Ramsey quantifier ranges over one coloring, so the existence of the Ramsey number is equivalent by construction to an eventual property of the prime sequence. The concrete indexing error in Theorem 8 further invalidates the construction as written. The paper's main advertised conclusion therefore does not hold as a substantive Ramsey-theoretic statement, even though the algebraic portions may be salvageable.

major comments (3)
  1. [Section 6, Theorem 8] The asserted injection Θ[i,m]: V(K_{i+1}) → N satisfying Θ[i,m]^{-1}({p_m,...,p_{m+i},p_{m+i+1}}) = V(K_{i+1}) is impossible: the preimage on the left is the set of i+2 primes, while the domain has i+1 vertices. Consequently the metrical coloring dΘ[i,m], the admissible set S_i(m), and all subsequent constructions in Theorems 8, 9, and 11 are not well-defined as written.
  2. [Section 6, Theorem 8 and Definition 8] The equivalences claimed in Theorems 8-12 are purely definitional artifacts of allowing S_i to be a singleton. With S_i(m)={dΘ[i,m]}, the condition 'for any t≥n and any ρ∈S_t' quantifies over a single coloring, so the existence of R(G,S(m))(X,C) reduces to an eventual property of one prescribed sequence of colorings. Replacing gap 2 by gap 4 or by any other gap k yields an identical 'Ramsey-theoretic' formulation of the corresponding prime-gap statement. The advertised conclusion that Green-Tao, twin primes, Zhang, and Polignac 'can be viewed as statements about Ramsey numbers' therefore does not import Ramsey-theoretic content and does not support the paper's stated significance.
  3. [Section 6, Theorem 11] Theorem 11 inherits the indexing error of Theorem 8: the path σ_i^{-1}(P_i) is required to contain p_m,...,p_{m+i+1}, i.e., i+2 primes, in a graph with i+1 vertices. Even apart from this, the proof's case distinction 'If A(i,m)={2t}, then we are done' is incomplete, since that case cannot generally be assumed; the construction's validity is therefore not established.
minor comments (4)
  1. [Section 3, Example 5] The edge-numbering diagram of K_4 is garbled in the text ('1 0 2 3 54'); please reformat it so the reader can follow the polynomial computation.
  2. [Section 6, Corollary 3] The corollary is stated one-directionally ('The Ramsey numbers ... exist'), but the surrounding discussion suggests an equivalence with the Green-Tao theorem; the converse direction should be stated and proved explicitly.
  3. [Section 6, Theorem 8] The sentence 'Theorem 8 obviously admits an interpretation in terms of the Green-Tao Theorem' is informal; if the intended claim is that the existence of the Ramsey numbers for all t is equivalent to Green-Tao, that equivalence should be formulated as a precise theorem rather than left as an interpretive remark.
  4. [Section 7] The paper repeatedly describes its own generalization as 'perhaps too general' and 'not ideal in its technical formulation'; while candid, these caveats underscore that the final section's equivalences carry no Ramsey-theoretic force once S_i is allowed to be a singleton.

Circularity Check

3 steps flagged · score 8.0 of 10

The advertised prime-gap equivalences in Section 6 are definitional: the Ramsey base is chosen to be a singleton coloring encoding the primes, so existence of the Ramsey number is an indicator for a property of that one prescribed coloring.

  1. self definitional [Section 6, paragraph after Definition 14 and Theorem 8; uses Definition 8]
    "A useful, somewhat unexpected consequence of allowing the admissible coloring set S to be smaller than the set Hom(G,A) for a given graph G is that this allows the value of a Ramsey number to be used like an indicator function, eventually identifying a desired object if its value is finite and never obtaining the object if not. ... set Si(m) = { dTheta[i,m] } ... the existence of an arithmetic progression of length t with gap k in the primes greater than or equal to pm is equivalent to the existence of the Ramsey number ... R(G, S(m))(Pt,kPt)."

    In Definition 8 the Ramsey quantifier is 'for any t ≥ n and any ρ ∈ S_t'. Once S_t(m) is the singleton {dΘ[t,m]}, this quantifier inspects exactly one coloring, the distance coloring of the consecutive-prime block determined by Θ[t,m]. The condition dΘ|π^{-1}(P_t)=kP_t says that this block contains an arithmetic progression of primes of gap k. Hence R exists iff that prescribed block sequence eventually contains the pattern; no other coloring is tested. The paper even says S-shrinkage makes the Ramsey number an 'indicator function'. Thus the Green-Tao / twin-prime / Zhang / Polignac equivalences are built into the choice of S, not derived from Ramsey theory; the same recipe with any desired target pattern encodes any eventual property of the same prime-distance sequence.

  2. self definitional [Section 3, Theorem 3 and Lemma 3]
    "p[G, X, C] is identically zero as a polynomial function if and only if for every ρ ∈ Hom(G, Fq), there exists π ∈ G/Xj satisfying ρ|π^{-1}(Xj) = ψj ◦π for some j ∈ J. ... writing im = max {i ∈ I : p[Gi, X, C] ≠ 0} and ik = min {i ∈ I : p[Gi, X, C] = 0}, we have im+1 = RG (X, C) = ik."

    The indicator polynomial p[Gi,X,C] is defined as a product over exactly the embedded copies of the target graphs X with target colorings C, and Lemma 3 proves it is identically zero precisely when every Fq-coloring of Gi contains one of those target subcolorings. That is the same 'for every coloring there is a target restriction' condition used to define R_G(X,C). Consequently im and ik are, by construction, the indices just before and at the first level where the defining condition holds, and im+1=R_G=ik is a restatement of Definition 8 in polynomial language. It is a valid but definitional reformulation, not an independent derivation of Ramsey numbers from algebra.

1 more flagged steps
  1. renaming known result [Section 6, Theorem 11]
    "Let Li(m) ⊆ Hom(Ki+1,A(i,m)) denote the set of edge-colorings η that satisfy η|σ^{-1}_i(P_i) = dTheta[i,m]. Then, setting L(m) = {Li(m)}i∈N, the existence of the Ramsey numbers R(G, L(m))(X, W(t)) for all values of t,m ∈ N is equivalent to Polignac's conjecture."

    The admissible-color set L_i(m) is defined as exactly those colorings that already agree with the prime-distance coloring dΘ[i,m] on the designated path σ_i^{-1}(P_i). Thus every coloring admitted to the Ramsey test contains the prime-gap pattern on that path by construction, and the theorem's quantifier over 'any η∈L_i(m)' cannot see any counterexample. The Ramsey number therefore exists iff the prime sequence dΘ meets the expected gap condition; this is a relabeling of Polignac's conjecture in new notation, not a result importing Ramsey-theoretic content.

full rationale

Sections 3–5 contain a self-contained algebraic formalism: indicator polynomials, Galois-type Ramsey numbers, and monotonicity lemmas are all proved from the paper's own definitions, and no load-bearing self-citation is required. However, the headline applications in Section 6 collapse by construction. Definition 8 deliberately allows arbitrary subsets S_i ⊆ Hom(G_i,A_i); the paper uses this freedom to set S_i(m)={dΘ[i,m]} (or the tailored L_i(m) in Theorem 11), so the 'for any coloring' quantifier in the Ramsey definition ranges over a single prescribed coloring. Existence of R then becomes an indicator for an eventual property of that one coloring, as the paper itself says. Since the same recipe works for any desired gap or pattern, the claim that Green-Tao, twin primes, Zhang, and Polignac 'can be viewed as statements about Ramsey numbers' is a definitional encoding rather than a derived structural result. Theorem 3's im+1=R=ik is similarly a restatement of the Ramsey definition in terms of the zero sets of the indicator polynomials: Lemma 3 shows that p[Gi,X,C]≡0 is exactly the defining condition for R, so the equality is a tautological reformulation. Separately, Theorem 8's Θ[i,m]:V(K_{i+1})→N is asserted to satisfy Θ[i,m]^{-1}({p_m,...,p_{m+i+1}})=V(K_{i+1}), which is impossible because the domain has i+1 vertices and the target set has i+2 primes; this is an internal consistency problem independent of the circularity assessment.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central claim depends on the freedom to choose arbitrary admissible coloring sets S_i, which is used to encode prime-gap statements. All other inputs are standard finite-field facts or cited theorems (Green-Tao, Zhang).

free parameters (2)
  • Admissible coloring set S_i(m) = {dTheta[i,m]}
    Chosen ad hoc in Theorem 8 to contain exactly the metrical coloring of consecutive primes, forcing the Ramsey number existence to encode the target prime-gap property.
  • Prime power q_i in Theorem 4 = unspecified 'appropriately large prime power'
    The proof requires choosing a prime power large enough to inject the finite color set, but no constructive bound is given. This is a hand-chosen parameter, though the value does not affect the Ramsey number.
assumptions (6)
  • standard math F_q is a field with a^{q-1}=1 for nonzero a
    Used in Definition 13 and Lemma 2 to build Iverson-bracket indicator polynomials.
  • standard math The ideal of polynomial functions on F_q^m vanishing everywhere is generated by x_i^q - x_i
    Used in Corollary 1 and Theorem 5 to conclude p[G_n] lies in the ideal.
  • standard math Every finite set of size m can be injected into F_q for some prime power q, and such q exists
    Theorem 4 depends on this to convert locally finite type to Galois type.
  • domain assumption Green-Tao theorem: primes contain arbitrarily long arithmetic progressions
    Corollary 3 relies on the Green-Tao theorem to assert the existence of certain Ramsey numbers.
  • domain assumption Zhang's theorem: some bounded prime gap occurs infinitely often
    Theorem 12 relies on Zhang's result to claim existence of a specific Ramsey number.
  • standard math The index set I is strictly well-ordered with unique minimal element
    Definition 5 and Theorem 3 use well-ordering to define i_m + 1 and the least n.
invented entities (1)
  • Generalized Ramsey number R(G,S)(X,C) with Ramsey bases and Ramsey symbols
    purpose: A unified framework claimed to cover most graph-theoretic Ramsey generalizations and to re-express number-theoretic problems
    This is a new mathematical object defined in Definition 8. It has no external falsifiable handle; it can be tailored (by choosing admissible colorings S_i) to encode almost any statement, which is why its flexibility makes the Section 6 equivalences tautological.

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Cite this review

Pith. "Pith review of Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems." pith.science (2026). https://pith.science/paper/BPJ6WDJE

@misc{pith2026250204311,
  author       = {Pith},
  title        = {Pith review of: Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPJ6WDJE}},
  note         = {Machine review of arXiv:2502.04311}
}
read the original abstract

In this paper, we will develop a significantly more general notion of classical Ramsey numbers (extending most other graph-theoretic generalizations) and make some preliminary characterizations of these new Ramsey numbers using simple algebraic tools. Throughout, we make a case arguing that, while our access to specific values of Ramsey numbers (or, in general, precise numerical solutions to Ramsey-theoretic problems) may be limited, the interplay between and overall structure of Ramseyian objects is likely tractable. To support the relevancy of this perspective, we conclude by demonstrating that the Green-Tao Theorem, the Twin Prime conjecture, Zhang's bounded prime gap theorem, and Polignac's conjecture can be viewed as statements about Ramsey numbers.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    , Halmos, P.: Modern Graph Theory

    Bollob´ as, B., s, B., Axler, S., Bollobas, B., Gehring, F. , Halmos, P.: Modern Graph Theory. Graduate Texts in Mathematics. Springer New Y ork (1998). URL https://books.google.com/books?id=SbZKSZ-1qrwC

    work page 1998

  2. [2]

    In: Graphs and Combina- torics: Proceedings of the Capital Conference on Graph Theo ry and Combinatorics at the George W ashington University June 18–22, 1973, pp

    Burr, S.A.: Generalized ramsey theory for graphs-a surve y. In: Graphs and Combina- torics: Proceedings of the Capital Conference on Graph Theo ry and Combinatorics at the George W ashington University June 18–22, 1973, pp. 52–7 5. Springer (1974)

    work page 1974

  3. [3]

    Ars Combinatoria 10, 193–203 (1980)

    Burr, S.A., Erd˝ os, P., Faudree, R.J., Rousseau, C., Sche lp, R.: An extremal problem in generalized ramsey theory. Ars Combinatoria 10, 193–203 (1980)

    work page 1980

  4. [4]

    PhD dissertation, University of Wyoming (2019)

    Christopherson, B.A.: Stability in common and uncommon p laces. PhD dissertation, University of Wyoming (2019)

    work page 2019

  5. [5]

    Bulletin of the American Mathematical Society 78(3), 423–426 (1972)

    Chv´ atal, V., Harary, F.: Generalized ramsey theory for g raphs. Bulletin of the American Mathematical Society 78(3), 423–426 (1972)

    work page 1972

  6. [6]

    the erdos-ginzburg-ziv theorem) URL http://math.uga.edu/pete/4400ChevalleyWarning.pdf

    Clark, P.: The chevalley-warning theorem (featuring... the erdos-ginzburg-ziv theorem) URL http://math.uga.edu/pete/4400ChevalleyWarning.pdf

    work page

  7. [7]

    Proceedings of the London Mathematical Soc iety 110(1), 1–18 (2015)

    Conlon, D., Fox, J., Lee, C., Sudakov, B.: The erd˝ os–gy´ a rf´ as problem on generalized ramsey numbers. Proceedings of the London Mathematical Soc iety 110(1), 1–18 (2015)

    work page 2015

  8. [8]

    S´ eminaire Lotharingien de Combinatoire 89B(6) (2023)

    De Loera, J.A., W esley, W.J.: An algebraic perspective on ramsey numbers. S´ eminaire Lotharingien de Combinatoire 89B(6) (2023)

    work page 2023

Show all 20 references
  1. [9]

    Graduate Texts in Mathematics

    Eisenbud, D., Eisenbud, P.: Commutative Algebra: With a V iew Toward Al- gebraic Geometry. Graduate Texts in Mathematics. Springer (1995). URL https://books.google.com/books?id=Fm_yPgZBucMC

    1995

  2. [10]

    Graduate Texts in Mathematics

    Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics. Springer New York (2001). URL https://books.google.com/books?id=pYfJe-ZVUyAC

    2001

  3. [11]

    University of Oxford pp

    Gould, M.: Ramsey theory. University of Oxford pp. 1–15 ( 2010)

    2010

  4. [12]

    Integers: Electronic Journal of Combinatorial Number Theory 7(2), A15 (2007)

    Graham, R.: Some of my favorite problems in ramsey theory . Integers: Electronic Journal of Combinatorial Number Theory 7(2), A15 (2007)

    2007

  5. [13]

    In: Horizons of Combinatorics, pp

    Graham, R.: Old and new problems and results in ramsey the ory. In: Horizons of Combinatorics, pp. 105–118. Springer Berlin Heidelberg (2 008) 22 Bryce Alan Christopherson

  6. [14]

    Annals of Mathematics pp

    Green, B., Tao, T.: The primes contain arbitrarily long a rithmetic progressions. Annals of Mathematics pp. 481–547 (2008)

    2008

  7. [15]

    Harary, F.: Recent results on generalized ramsey theory for graphs. In: Graph Theory and Applications: Proceedings of the Conference at W estern Michigan University, May 10–13, 1972 Sponsored jointly by W estern Michigan Universi ty and the US Army Re- search Office-Durham, unde...

    2006

  8. [16]

    Graduat e Texts in Mathematics

    Harris, J.: Algebraic Geometry: A First Course. Graduat e Texts in Mathematics. Springer New York (2013). URL https://books.google.com/books?id=U-UlBQAAQBAJ

    2013

  9. [17]

    Graduate Studie s in Mathematics

    Isaacs, I.: Algebra: A Graduate Course. Graduate Studie s in Mathematics. American Mathematical Society (2009). URL https://books.google.com/books?id=5tKq0kbHuc4C

    2009

  10. [18]

    Graduate Texts in Mathematics

    Kemper, G.: A Course in Commutative Algebra. Graduate Texts in Mathematics. Springer Berlin Heidelberg (2010). U RL https://books.google.com/books?id=8kxlj48DWM4C

    2010

  11. [19]

    Cambridge University Press (2005)

    Tattersall, J.J.: Elementary number theory in nine chap ters. Cambridge University Press (2005)

    2005

  12. [20]

    Annals of Mathem atics pp

    Zhang, Y.: Bounded gaps between primes. Annals of Mathem atics pp. 1121–1174 (2014) Statements and Declarations Conflict of interest statement : The authors declare that they have no conflict of interest. Competing interests statement : The authors have no relevant financial or n...

    2014

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