Almost primes between all squares

Adrian W. Dudek; Daniel R. Johnston

REVIEW 3 major objections 4 minor 18 references

Almost primes between all squares

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that for every $n \ge 1$, the interval between $n^2$ and $(n+1)^2$ contains an integer with at most four prime factors.

desk verdict The result is new and the sieve work is solid; the half-interval reading of Sorenson–Webster needs to be checked before full confidence. read the letter →

arxiv 2501.18048 v3 pith:P4RLP7G2 submitted 2025-01-29 math.NT

Adrian W. Dudek , Daniel R. Johnston This is my paper · ORCID

classification math.NT MSC 11N3611N05

keywords almostprimesLegendre'sconjecturesievemethodslinearweightedexplicitresultsprimegaps

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

This paper proves that between every pair of consecutive squares, $n^2$ and $(n+1)^2$, there is always an integer built from at most four primes. That is the first theorem of this kind to hold for every $n \ge 1$; earlier work located such almost primes only for sufficiently large $n$. The proof combines a recent computational verification of Legendre's conjecture up to a large bound with an explicit version of Kuhn's weighted sieve. As a byproduct, the paper gives a fully explicit weighted-sieve framework for generic sifting sets. If the argument is correct, it provides an unconditional step toward Legendre's conjecture: the obstruction to finding a prime in every square gap is, in this sense, at worst four prime factors.

What carries the argument

The key machinery is an explicit version of Kuhn's weighted sieve, stated for generic sifting sets. The weight $w(a) = 1 - \frac12 \sum_{z \le q < y,\, q^\ell \parallel a} \ell$ lets the count of integers with at most $k_2$ prime factors be bounded below by $S(\mathcal{A},\mathcal{P},z) - \frac12 \sum_{z \le q<y} S(\mathcal{A}_q,\mathcal{P},z)$ minus explicit error terms, keeping elements that have at most one small prime divisor in $[z,y)$. Lower and upper bounds for the sifting functions come from an explicit linear sieve, and explicit Mertens-type estimates control the product and remainder terms. The specific choice $z = X^{1/8}$, $y = X^{1/4}$, with $k_1=8$, $k_2=4$ and $\alpha = 0.07$, makes the positive main term dominate all errors once $N > 1.98\cdot10^{28}$.

What would settle it

Re-run or inspect the verification behind [17] and check, for every integer $n$ with $n^2 \le 4.97\cdot10^{27}$, whether both $(n^2,n(n+1))$ and $(n(n+1),(n+1)^2)$ contain a prime; if any such interval lacks a prime in one half, the small-$n$ lemma loses its force and the theorem would need another argument for that range.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: for every positive integer $n$, the interval $(n^2,(n+1)^2)$ contains an integer $a$ with $\Omega(a) \le 4$, where $\Omega(a)$ counts prime factors with multiplicity. The proof splits at $N = 1.98\cdot10^{28}$. For smaller square endpoints, a cited computation gives primes in each half of the square interval, and the observation that $4p$ lies between $n^2$ and $(n+1)^2$ when $p$ lies in a suitably rescaled interval lifts the coverage up to the cutoff. For larger $N$, the authors sieve the set $\mathcal{A} = \mathbb{Z} \cap (N, N+2\sqrt{N})$ with $z = X^{1/8}$, $y = X^{1/4}$, where $X = \lfloor N + 2\sqrt{N}\rfloor$, and apply an explicit weighted-sieve inequality to show that the count $r_4(\mathcal{A})$ of elements with at most four prime factors is positive. This establishes the first unconditional result of this form valid for all $n$ rather than only for sufficiently large $n$.

Load-bearing premise

The load-bearing premise is that the cited computation [17] verifies a prime in each half of the square interval up to about $4.97\cdot10^{27}$, not just a prime somewhere in the whole interval, since that half-interval strength is what lets the small cases be extended to $1.98\cdot10^{28}$.

Editorial extensions

If this is right

For every integer $n \ge 1$, the interval between consecutive squares contains an integer with at most four prime factors, so the almost-prime analogue of Legendre's conjecture with $k=4$ holds unconditionally.
Because the weighted-sieve inequalities are proved for generic sifting sets, other interval problems can reuse the explicit constants without re-deriving the sieve bounds.
With adjusted parameters, the authors note, the same method should give at most three prime factors between consecutive cubes and at most two between consecutive fourth powers for all $n$.
The effective range begins at $N > 1.98\cdot10^{28}$, unlike the earlier prime-between-cubes result, which only starts at a doubly exponential scale.

Reading between the lines

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

In our reading, the small-range computation is the part most worth checking first: the argument needs a prime in each half of the square interval, which is stronger than the cited title's claim of verifying Legendre's conjecture, and a failure there would not be repaired by the sieve portion.
A natural next step is to optimize the sieve parameters $k_1$, $k_2$, and $\alpha$; the paper's own rough estimate suggests that reaching $k=3$ for all $n$ would require the starting point to move to roughly $10^{50}$, so progress is more likely to come from extending the computational small-$n$ coverage.
Any future computation that certifies half-interval primes to a larger bound would directly extend the range of Theorem 1.2 and could reduce the number of prime factors in the small-$n$ regime.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper proves that for every n ≥ 1, the interval (n^2, (n+1)^2) contains an integer with at most four prime factors. That is genuine progress: earlier explicit results were either asymptotic or allowed more prime factors, and this is the first all-n statement of its kind. The main technical contribution, an explicit version of Kuhn's weighted sieve for generic sifting sets, is cleanly stated and should be reusable. Constants are tracked carefully, and the self-cited linear sieve is a general tool rather than a conclusion-shaped assumption, so circularity is low. The appendix's explicit Mertens bound is also a useful small piece.

The soft spot is exactly where the stress-test points. Lemma 2.1 assumes that the Sorenson–Webster computation gives a prime in each half of (n^2, (n+1)^2). The title of [17] only advertises Legendre's conjecture, which is weaker. If [17] only certifies the full interval, the proof loses coverage between 4.97·10^27 and the sieve cutoff, and the 4p argument falls apart. The authors need to state which theorem in [17] supplies the half-interval primes, or provide the data directly. I don't think this is a dealbreaker — the stronger claim is plausible and likely extractable from the algorithm — but it is load-bearing and cannot be waved away.

The stress-test's secondary worry about the final numerical margin does not survive arithmetic. At N = 1.98·10^28, the positive term in (4.20) is about 5.8·10^11 and the negative term about 5.7·10^11, so the margin is small but real, and it widens as N grows.

This paper is for anyone working on explicit sieve methods or Legendre-adjacent problems. It deserves a serious referee. My recommendation: send it out, and have the referee verify the Sorenson–Webster half-interval claim before sign-off.

Referee Report

3 major / 4 minor

Summary. The paper proves that for every integer n ≥ 1, the interval (n^2, (n+1)^2) contains an integer with at most four prime factors (Theorem 1.2). The proof splits at n^2 ≈ 1.98·10^28: for smaller n, a computation of Sorenson and Webster is invoked; for larger n, an explicit version of the linear sieve due to Bordignon, Johnston, and Starichkova is combined with a new explicit version of Kuhn's weighted sieve. The authors derive explicit constant estimates for the sieve input and use them to show that the weighted sieve count is positive.

Significance. If the proof is correct, this is the first result of this kind valid for all n ≥ 1, giving an explicit analogue of Legendre's conjecture for almost primes. The paper also provides a self-contained explicit version of Kuhn's weighted sieve for generic sifting sets, which may be useful for future applications. The constants are tracked carefully and the auxiliary computations (e.g., Mertens-type estimates) are documented, making the argument reproducible in principle.

major comments (3)

[Section 4, Eq. (4.9)] The bound e^{2h(s)} ≤ e^2 · 3e^{-s}/s used in (4.9) is false for s ≥ 3. Since h(s) = 3e^{-s}/s for s ≥ 3, one has e^{2h(s)} = e^{6e^{-s}/s}; at s = 3.3 this is about 1.069, whereas e^2 · 3e^{-s}/s is about 0.248. With the correct value, the maximum of f(s) − εC2(ε)e^{2h(s)} on [3,4] is below 0.808, so the constant 8.8C(s) − 7.113 in (4.20) becomes negative and the claimed positivity of r4(A) is not obtained. This is a load-bearing numerical error and must be corrected.
[Lemma 2.1] The lemma asserts that the computation in [17] yields a prime in each half-interval (n^2, n(n+1)) and (n(n+1), (n+1)^2) for all n with n^2 ≤ 4.97·10^27. The cited title only advertises verification of Legendre's conjecture, which guarantees a prime somewhere in the full interval (n^2, (n+1)^2), not necessarily in each half. The half-interval statement is load-bearing for the 4p extension up to n^2 ≤ 1.98·10^28. The authors should verify from [17] (or supply the computation) that the stronger half-interval assertion is indeed available.
[Proposition 3.5, Eq. (3.13)] The factor (1 + k2/(2(log X)^2)) in the statement of Proposition 3.5 and in equation (3.13) is inconsistent with the derivation from Lemma 3.3. Since log z = log X / k1, the correction term from Lemma 3.3 is k1^2/(2(log X)^2), not k2/(2(log X)^2). The numerical evaluation in (4.15) uses 32 in the numerator, corresponding to k1^2/2 with k1 = 8, so the displayed formulas should read 1 + k1^2/(2(log X)^2).

minor comments (4)

[Lemma 3.1] The first sentence states that no element of A is divisible by a prime in P; in the application P is the set of all primes, which would make the hypothesis false. The intended condition is that no element of A is divisible by a prime in the complement of P, as in Lemma 2.2.
[Lemma 4.1] The lower bound '> 107' for y should read '> 10^7' (or a more precise value). With the corrected value, the choice c2 = 0.07 > 1.1/log(10^7) is valid.
[Proposition A.2, Case 2] In the displayed bound for the product, '2√z' should be '2/√z'; the subsequent numerical estimate uses the correct expression.
[Lemma 2.2 / Proposition A.2] The condition (2.4) is required for all 1 < u < z, but Proposition A.2 proves the estimate only for u ≥ 3. The remaining range u < 3 is easy to handle but should be explicitly mentioned.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the large-n sieve argument is generic and the small-n input is external; self-citation is independent support.

full rationale

The proof splits at N = 1.98e28. Lemma 2.1 covers the small range by citing the Sorenson–Webster computation [17]; whether or not that computation has the half-interval strength the lemma asserts, it is an external computational input rather than a quantity fitted or predicted within this paper, so any mismatch with the cited title is a correctness concern, not circularity. For the large range, the paper proves a generic explicit Kuhn weighted sieve (Lemma 3.1) and applies it to the interval set A(N). The sifting bounds call on Lemma 2.2 from [2], a paper co-authored by the present second author, but that lemma is a general explicit linear sieve with hypotheses (finite set A, multiplicative density g, remainder term R) that do not include the target statement; it is not constructed to force a bound on Omega(a). The constants Q = 2 and epsilon = 1.97e-3 are derived in Proposition A.2 from explicit Mertens-type estimates and a finite direct computation, and the parameters alpha = 0.07 and s = 3.3 are chosen after the analytic bounds are proved, with a positive numerical margin in (4.20). No equation reduces to its own input, no fitted parameter is renamed as a prediction, and the self-citation is independent support rather than a circular chain. Therefore the derivation is not circular.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The proof introduces no new physical or mathematical entities. The free parameters are choices of sieve exponents and numerical parameters that satisfy explicit inequalities; they are not fitted to the target result. The load-bearing external inputs are the explicit linear sieve theorem from [2] and the computational verification from [17].

free parameters (4)

k1 and k2 = k1 = 8, k2 = 4
Exponents in z = X^(1/k1), y = X^(1/k2); chosen by hand to make the sieve inequalities work, not fitted to data.
alpha = 0.07
Parameter in Proposition 3.5 satisfying 0 < alpha < 1/8; chosen by hand to balance the main terms and error terms.
s = 3.3
Sieve variable s = log D / log z chosen in [3,4] so that C(s) > 0.839 and the final coefficient is positive.
c1 and c2 = c1 = 0.01, c2 = 0.07
Hand-picked constants in Lemma 4.1 that dominate explicit estimates; they are chosen to satisfy inequalities, not fitted to the target theorem.

assumptions (3)

domain assumption Explicit linear sieve bounds from [2, Theorem 6] and the associated constants C1(epsilon), C2(epsilon).
Lemma 2.2 imports a recent explicit linear sieve theorem coauthored by the second author; if these bounds or table values are incorrect, the final inequalities collapse.
domain assumption Sorenson-Webster computation verifies primes in both half-intervals around squares up to about 7e13.
Lemma 2.1 relies on this computational input, but the cited title only states Legendre's conjecture; the stronger half-interval content is an assumption taken on trust.
standard math Standard explicit estimates for Mertens products, divisor sums, and prime counting from Rosser-Schoenfeld, Vanlalngaia, and Ramare.
These are cited external theorems with proofs elsewhere; they are used for bounding products, squarefree divisor sums, and prime counting functions.

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Cite this review

Pith. "Pith review of Almost primes between all squares." pith.science (2026). https://pith.science/paper/P4RLP7G2

@misc{pith2026250118048,
  author       = {Pith},
  title        = {Pith review of: Almost primes between all squares},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P4RLP7G2}},
  note         = {Machine review of arXiv:2501.18048}
}

abstract

We prove that for all $n\geq 1$ there exists a number between $n^2$ and $(n+1)^2$ with at most 4 prime factors. This is the first result of this kind that holds for every $n\geq 1$ rather than just sufficiently large $n$. Our approach relies on a recent computation by Sorenson and Webster, along with an explicit version of the linear sieve. As part of our proof, we also prove an explicit version of Kuhn's weighted sieve. This is done for generic sifting sets to enhance the future applicability of our methods.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [17]

An algorithm to verify Legendre’s conjecture up to 7 · 1013

J. Sorenson and J. Webster. “An algorithm to verify Legendre’s conjecture up to 7 · 1013”. Res. Number Theory 11.4 (2025)

work page 2025

[2] [1]

The difference between consecutive primes, II

R. C. Baker, G. Harman, and J. Pintz. “The difference between consecutive primes, II”. Proc. Amer. Math. Soc. 83.3 (2001), pp. 532–562

work page 2001

[3] [2]

An explicit version of Chen’s theorem and the linear sieve

M. Bordignon, D. R. Johnston, and V. Starichkova. “An explicit version of Chen’s theorem and the linear sieve”. Int. J. Number Theory (to appear) (2025)

work page 2025

[4] [3]

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”.Videnskaps. Skr., Mat. Natur. Kl. Kristiana 3 (1920)

work page 1920

[5] [4]

On Chen’s theorem (II)

Y. Cai. “On Chen’s theorem (II)”. J. Number Theory 128.5 (2008), pp. 1336–1357

work page 2008

[6] [5]

On the distribution of almost primes in an interval

J. R. Chen. “On the distribution of almost primes in an interval”. Sci. Sinica 18 (1975), pp. 611–627

work page 1975

[7] [6]

Primes between consecutive powers

M. Cully-Hugill. “Primes between consecutive powers”. J. Number Theory 247 (2023), pp. 100–117

work page 2023

[8] [7]

An explicit result for primes between cubes

A. W. Dudek. “An explicit result for primes between cubes”. Funct. Approx. Com- ment. Math. 55.2 (2016), pp. 177–197

work page 2016