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arxiv: 2603.10356 · v2 · pith:5HHCIS66new · submitted 2026-03-11 · 🧮 math.NT

On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares

Peter J. Campbell This is my paper

Pith reviewed 2026-05-21 12:25 UTC · model grok-4.3

classification 🧮 math.NT
keywords Legendre conjecturealmost primes3-almost-primeslinear sieveRichert weightsconsecutive squaresexplicit boundsshort intervals
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The pith

Every interval between consecutive squares contains an integer with at most three prime factors.

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

The paper proves that for every integer n at least 1, the open interval from n squared to (n plus 1) squared always contains an integer with at most three prime factors counted with multiplicity. This gives an explicit version of Legendre's conjecture but for almost-primes rather than primes. It improves the earlier result that guaranteed an integer with at most four prime factors in each such interval. The argument splits into direct computational checks for smaller n and an explicit sieve for larger n.

Core claim

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer n ≥ 1, the interval (n²,(n+1)²) contains an integer having at most 3 prime factors, counted with multiplicity.

What carries the argument

Adapting Richert's logarithmic weights to intervals between consecutive squares together with an explicit linear sieve.

If this is right

  • The result holds for every positive integer n by combining computation and sieve methods.
  • The guaranteed number of prime factors drops from four to three in each square gap.
  • All cases with n squared up to 10 to the 31 are covered by prior computational checks on primes and prime gaps.
  • For larger n the sieve produces at least one such integer in each interval.

Where Pith is reading between the lines

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

  • Similar explicit sieve adaptations might apply to intervals of other lengths near n.
  • The same framework could be tested to see if at most two prime factors can be guaranteed in the same intervals.
  • The result bears on how densely almost-primes must appear in short intervals of length about 2n.

Load-bearing premise

The large-n argument depends on the correctness of the explicit linear sieve bounds from Bordignon, Johnston, and Starichkova together with the adaptation of Richert's logarithmic weights to square intervals.

What would settle it

Finding some n larger than 10 to the 31 such that the interval from n squared to (n plus 1) squared contains no integer with at most three prime factors would disprove the claim.

read the original abstract

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n \geq 1$, the interval $(n^2,(n+1)^2)$ contains an integer having at most $3$ prime factors, counted with multiplicity. This improves the previous best result of Dudek and Johnston, who showed that every such interval contains an integer with at most $4$ prime factors. The proof is divided into two ranges. For $n^2 \leq 10^{31}$, we use prior computational results on primes in short intervals between consecutive squares, together with explicit bounds on maximal prime gaps. For $n^2 > 10^{31}$, we give a sieve-theoretic argument with explicit constants, adapting Richert's logarithmic weights to intervals between consecutive squares and employing an explicit linear sieve of Bordignon, Johnston, and Starichkova.

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

1 major / 0 minor

Summary. The paper proves that for every integer n ≥ 1, the interval (n², (n+1)²) contains an integer with at most three prime factors counted with multiplicity. This improves the prior result of Dudek and Johnston for four prime factors. The proof splits into two ranges: for n² ≤ 10^{31} it invokes prior computational results on primes in short intervals and explicit maximal prime gap bounds; for n² > 10^{31} it deploys an explicit sieve argument that adapts Richert's logarithmic weights to intervals of length 2n+1 and applies the linear sieve bounds of Bordignon, Johnston, and Starichkova.

Significance. If the central claim holds, the result supplies an explicit almost-prime analogue of Legendre's conjecture in intervals of length ~2n, strengthening known theorems on the distribution of P_3 numbers in short intervals. The explicit constants and the computational verification up to 10^{31} are positive features that make the statement fully effective.

major comments (1)
  1. Large-n argument (abstract and the sieve section): the lower bound on the weighted count of integers with ≤3 prime factors is obtained by adapting the explicit linear sieve constants of Bordignon–Johnston–Starichkova to Richert weights on intervals of length 2n+1, yet the manuscript provides neither an independent derivation of the modified constants nor a numerical check that the resulting lower bound exceeds 1 uniformly. Because this lower bound is the sole guarantee that every such interval contains a P_3 number for n² > 10^{31}, any undetected looseness in the adaptation would leave the existence statement unproved for large n.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the detailed report. We address the major comment on the large-n sieve argument below. We will revise the manuscript to include the requested derivation and numerical check.

read point-by-point responses

  1. Referee: Large-n argument (abstract and the sieve section): the lower bound on the weighted count of integers with ≤3 prime factors is obtained by adapting the explicit linear sieve constants of Bordignon–Johnston–Starichkova to Richert weights on intervals of length 2n+1, yet the manuscript provides neither an independent derivation of the modified constants nor a numerical check that the resulting lower bound exceeds 1 uniformly. Because this lower bound is the sole guarantee that every such interval contains a P_3 number for n² > 10^{31}, any undetected looseness in the adaptation would leave the existence statement unproved for large n.

    Authors: We thank the referee for pointing out this potential issue in the presentation of our large-n argument. We agree that to make the proof fully transparent and verifiable, an explicit derivation of the adapted constants and a uniform numerical check are desirable. In the revised version of the manuscript, we will add a new subsection in the sieve section that derives the modified lower bound constants by applying the Bordignon–Johnston–Starichkova linear sieve bounds to the Richert weight function adapted to the interval length 2n + 1. Furthermore, we will include a computational verification demonstrating that the resulting weighted sum is greater than 1 for all n with n² > 10^{31}. This will confirm that the lower bound exceeds 1 uniformly in the large-n range, thereby rigorously establishing the existence of the desired P_3 numbers. We believe these additions will fully address the referee's concern without altering the core argument. revision: yes

Circularity Check

0 steps flagged

No circularity; relies on external explicit bounds and computations

full rationale

The paper splits the proof into small-n (n² ≤ 10^31) using prior computational results on primes in short intervals between squares plus explicit maximal prime gap bounds, and large-n using an adaptation of Richert's logarithmic weights to intervals of length 2n+1 together with the explicit linear sieve bounds of Bordignon, Johnston, and Starichkova. These are citations to independent external work by other authors, not self-citations. No equation or step inside the paper defines the target count in terms of itself, fits a parameter to a subset and renames it a prediction, or reduces the existence claim to a self-referential uniqueness theorem. The derivation is therefore self-contained against external benchmarks and exhibits none of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on external computational verifications for n up to 10^31 and on the validity of explicit constants in the linear sieve of Bordignon et al.; no new entities are postulated.

axioms (1)
  • standard math Standard axioms of analytic number theory and the correctness of the cited explicit linear sieve bounds.
    Invoked for the large-n sieve argument in the abstract.

pith-pipeline@v0.9.0 · 5680 in / 1157 out tokens · 44976 ms · 2026-05-21T12:25:52.718640+00:00 · methodology

discussion (0)

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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.

  1. A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares

    math.NT 2026-05 unverdicted novelty 6.0

    Derives exact formula P_k = N_k - S_k + E_k for primes in odd-square intervals via matrix multiplicities and equates existence to E_k <= S_k - N_k.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 1 Pith paper

  1. [1]

    R. C. Baker, G. Harman, and J. Pintz. The Difference Between Consecutive Primes, II.Proc. Lond. Math. Soc., 83(3):532–562, 2001

    work page 2001

  2. [2]

    Bordignon, D

    M. Bordignon, D. R. Johnston, and V. Starichkova. An Explicit Version of Chen’s Theorem and the Linear Sieve.Int. J. Number Theory, 21(10):2497–2572, 2025. 14

    work page 2025

  3. [3]

    Brun.Le crible d’ ´Eratosth` ene et le th´ eor` eme de Goldbach

    V. Brun.Le crible d’ ´Eratosth` ene et le th´ eor` eme de Goldbach. Number 3 in Skrifter utgit av Videnskaps- selskapet i Kristiania. I. Matematisk-naturvidenskabelig Klasse. J. Dybwad, Kristiania, 1920

    work page 1920

  4. [4]

    Y. Cai. A Remark on Chen’s Theorem (II).Chinese Ann. Math. Ser. B, 29(6):687–698, 2008

    work page 2008

  5. [5]

    J.-R. Chen. On the Distribution of Almost Primes in an Interval.Scientia Sinica, 18:611–627, 1975

    work page 1975

  6. [6]

    A. W. Dudek and D. R. Johnston. Almost Primes Between All Squares.J. Number Theory, 278:726–745, 2026

    work page 2026

  7. [7]

    S. P. Glasby, C. E. Praeger, and W. R. Unger. Most Permutations Power to a Cycle of Small Prime Length.Proc. Edinb. Math. Soc., 64(2):234–246, 2021

    work page 2021

  8. [8]

    Greaves.Sieves in Number Theory, volume 43 ofErgebnisse der Mathematik und ihrer Grenzgebiete

    G. Greaves.Sieves in Number Theory, volume 43 ofErgebnisse der Mathematik und ihrer Grenzgebiete

    work page

  9. [9]

    Springer, 2013

    Folge / A. Springer, 2013

    work page 2013

  10. [10]

    D. R. Johnston. MertenBounds. https://github.com/DJmath1729/MertenBounds, 2026. GitHub repository. Accessed 28 February 2026

    work page 2026

  11. [11]

    D. R. Johnston, J. P. Sorenson, S. N. Thomas, and J. E. Webster. Primes and Almost Primes Between Cubes, 2026. Preprint, arXiv:2601.15564

    work page arXiv 2026

  12. [12]

    P. Kuhn. Neue Absch¨ atzungen auf Grund der Viggo Brunschen Siebmethode. InProceedings of the 12th Scandinavian Mathematical Congress (Lund, 1953), pages 160–168, 1954

    work page 1953

  13. [13]

    T. R. Nicely. Prime Number Gap Record Rising. https://www.pzktupel.de/RecordGaps/risinggap. php, 2026. Record table with contributor attributions. Last updated 16 February 2026. Accessed 28 February 2026

    work page 2026

  14. [14]

    J. Pintz. Landau’s Problems on Primes.J. Th´ eor. Nombres Bordeaux, 21(2):357–404, 2009

    work page 2009

  15. [15]

    Ramar´ e

    O. Ramar´ e. From Explicit Estimates for Primes to Explicit Estimates for the M¨ obius Function.Acta Arith., 157(4):365–379, 2013

    work page 2013

  16. [16]

    H.-E. Richert. Selberg’s Sieve with Weights.Mathematika, 16(1):1–22, 1969

    work page 1969

  17. [17]

    J. B. Rosser and L. Schoenfeld. Approximate Formulas for Some Functions of Prime Numbers.Ill. J. Math., 6(1):64–94, 1962

    work page 1962

  18. [18]

    Sorenson and J

    J. Sorenson and J. Webster. An Algorithm and Computation to Verify Legendre’s Conjecture up to 7·10 13.Res. Number Theory, 11(1):4, 2025

    work page 2025

  19. [19]

    Vanlalngaia

    R. Vanlalngaia. Explicit Mertens Sums.Integers, 17:A11, 2017. 15

    work page 2017