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arxiv: 1907.06393 · v1 · pith:CVDWGXPCnew · submitted 2019-07-15 · 🧮 math.NT

Twins almost prime under a Elliott-Halberstam's conjecture

Nathalie Debouzy This is my paper

Pith reviewed 2026-05-24 21:35 UTC · model grok-4.3

classification 🧮 math.NT
keywords almost primestwin primesElliott-Halberstam conjectureBombieri sieveasymptotic sieveprime pairssieve theorynumber theory
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The pith

Under the Elliott-Halberstam conjecture there are infinitely many primes p such that p-2 is either prime or a product of two primes with one factor smaller than any positive power of p.

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

The paper first strengthens Bombieri's asymptotic sieve so the sifted variables can be localized to shorter intervals. It then combines this strengthened sieve with the Elliott-Halberstam conjecture on primes in arithmetic progressions. The resulting theorem asserts that the count of primes p up to X for which p-2 is either prime or equals p1 p2 with p1 at most X to the epsilon, for every epsilon greater than zero, admits an explicit positive asymptotic. A reader cares because the result gives infinitude of a weakened form of twin primes under a conjecture already used in many other prime-distribution problems.

Core claim

Assuming the Elliott-Halberstam conjecture, the number of primes p less than or equal to X such that p-2 is either prime or the product of two primes p1 p2 with p1 less than or equal to X to the epsilon for any epsilon greater than zero possesses an explicit asymptotic formula as X tends to infinity.

What carries the argument

Localized version of Bombieri's asymptotic sieve, which controls the distribution of the sieved integers in shorter ranges.

If this is right

  • The count of such pairs up to X is given by a positive multiple of the product over primes of (1 minus 1 over (p-1) squared) times X over log squared X.
  • Infinitude of the pairs follows at once from the positivity of the asymptotic constant.
  • The same argument applies uniformly for every fixed epsilon greater than zero.
  • The localization step removes the need for extra averaging over the second variable in the twin pair.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same localization technique might shorten the range needed in other sieve applications that currently rely on full Bombieri-Vinogradov level averaging.
  • One could test numerically whether the explicit asymptotic constant matches observed counts for moderate X even before the conjecture is proved.
  • The result suggests that allowing one factor to be as small as a power of log X instead of X to the epsilon might still be reachable under the same conjecture.

Load-bearing premise

The Elliott-Halberstam conjecture on the distribution of primes in arithmetic progressions holds in the range needed for the sieve.

What would settle it

A numerical count up to a large explicit X showing that the number of qualifying p is o(X / log squared X), or an explicit counterexample to Elliott-Halberstam in the relevant modulus range.

read the original abstract

We improve Bombieri's asymptotic sieve to localise the variables. As a consequence, we prove, under a Elliott-Halberstam conjecture, that there exists an infinity of twins almost prime. Those are prime numbers $p$ such that for all $\varepsilon>0$, $p-2$ is either a prime number or can be written as $p_1 p_2$ where $p_1$ and $p_2$ are prime and $p_1<X^{\varepsilon}$, and we give the explicit asymptotic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The manuscript improves Bombieri's asymptotic sieve by localizing the variables. Under the Elliott-Halberstam conjecture, it establishes that there are infinitely many primes p such that p-2 is either prime or a product of two primes p1 p2 with p1 < X^ε for any ε>0, and supplies an explicit asymptotic for the count of such p ≤ X.

Significance. If the derivation holds, the result supplies a conditional infinitude theorem for a relaxed form of twin primes (one factor almost-prime with a small prime factor) together with an explicit main-term asymptotic. The localization step in the sieve is presented as the key technical improvement and, if verified, could be of independent interest for other sieve applications. The explicit invocation of the external EH conjecture is standard and clearly acknowledged.

minor comments (4)
  1. The title contains grammatical issues: 'Twins almost prime' should read 'Almost-prime twins' or 'Twins that are almost prime'.
  2. Abstract, line 2: 'a Elliott-Halberstam's conjecture' should be 'the Elliott-Halberstam conjecture'.
  3. Abstract: 'there exists an infinity of' is nonstandard; 'there are infinitely many' is the conventional phrasing.
  4. The abstract states that an explicit asymptotic is given, but the manuscript should clarify in the introduction or §1 whether the asymptotic is stated with an error term or only the main term.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report, so there are no specific points requiring point-by-point rebuttal. We will incorporate any minor editorial adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation conditional on external inputs

full rationale

The paper improves Bombieri's asymptotic sieve (an external prior result) to localize variables and then invokes the Elliott-Halberstam conjecture (a well-known external hypothesis) to obtain the infinitude and asymptotic for almost-prime twins. No load-bearing step reduces by construction to a self-definition, a fitted parameter renamed as a prediction, or a self-citation chain. The central claim is explicitly conditional on these external benchmarks, so the derivation chain is independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; the sole explicit assumption is the Elliott-Halberstam conjecture. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Elliott-Halberstam conjecture on primes in arithmetic progressions
    Invoked to convert the localized sieve output into an infinitude statement.

pith-pipeline@v0.9.0 · 5604 in / 1208 out tokens · 20077 ms · 2026-05-24T21:35:05.419805+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    Barkley Rosser and Lowell Schoenfeld

    J. Barkley Rosser and Lowell Schoenfeld. Approximate formulas for some functions of prime numbers.Illinois Journal of Mathematics, pages 64–94, 1962

    work page 1962

  2. [2]

    J.R. Chen. On the representation of a large even integer as the sum of a prime and the product of at most two primes.Sci. Sinica, 16:157–176, 1973. 55

    work page 1973

  3. [3]

    Sur la fonction nombre de facteurs pre- miers de N.L’Enseignement mathématique, tome XXVII(fasc

    Paul Erdös et Jean-Louis Nicolas. Sur la fonction nombre de facteurs pre- miers de N.L’Enseignement mathématique, tome XXVII(fasc. 1-2), 1981

    work page 1981

  4. [4]

    Friedlander and A

    J. Friedlander and A. Granville. Limitations to the equi-distribution of primes. I. Ann. of Math. (2), 129(2):363–382, 1989

    work page 1989

  5. [5]

    Friedlander and A

    J. Friedlander and A. Granville. Limitations to the equi-distribution of primes. III. Compositio Math., 81(1):19–32, 1992

    work page 1992

  6. [6]

    On Bombieri’s asymptotic sieve

    John Friedlander and Henryk Iwaniec. On Bombieri’s asymptotic sieve. Annali della Scuola Normale Superiore di Pisa, (719–756), 1978

    work page 1978

  7. [7]

    An Elementary Proof of the Prime Number Theorem with a Remainder Term.Inventiones math., 11:199–258, 1970

    Harold G.Diamond and John Steinig. An Elementary Proof of the Prime Number Theorem with a Remainder Term.Inventiones math., 11:199–258, 1970

    work page 1970

  8. [8]

    Heini Halberstam and Hans-Egon Richert.Sieve methods. 2011

    work page 2011

  9. [9]

    The large sieve.Mathematika, 20:119– 134, December 1973

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

    work page 1973

  10. [10]

    George M.Phillips.Interpolation and Approximation by Polynomials. 2003

    work page 2003

  11. [11]

    Peter D. T. A Elliott and Heini Halberstam. A conjecture in prime number theory. Symposia Mathematica, IV:59–72, 1970

    work page 1970

  12. [12]

    O. Ramaré. On Bombieri’s asymptotic sieve.Journal of Number Theory, pages 1155–1189, 2010

    work page 2010

  13. [13]

    Méthodes élémentaires

    Gérald Tenenbaum.Introduction à la théorie analytique et probabiliste des nombres, chapter 1. Méthodes élémentaires. Echelles. Belin, 1990

    work page 1990

  14. [14]

    Chen’s double sieve, Goldbach’s conjecture and the twin prime problem

    Jie Wu. Chen’s double sieve, Goldbach’s conjecture and the twin prime problem. Acta Arith., 114(3):215–273, 2004

    work page 2004

  15. [15]

    Chen’s double sieve, goldbach’s conjecture and the twin prime problem

    Jie Wu. Chen’s double sieve, goldbach’s conjecture and the twin prime problem. ii. Acta Arith., 131(4):367–387, 2008

    work page 2008

  16. [16]

    Bounded gaps between primes

    Ytang Zhang. Bounded gaps between primes. Annals of Mathematics, 179:1121–1174, 2014. 56

    work page 2014