Effective short intervals containing primes

Matt Visser (Victoria University of Wellington)

REVIEW 3 major objections 4 minor 11 references

Effective short intervals containing primes

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For every x >= 1, a prime lies in [x, x + x^(1 - 1/106)].

desk verdict A useful but uneven assembly of explicit short-interval prime bounds; the headline claim for n≥106 depends on an unproved 'inspection' strengthening that needs to be substantiated before the result can be trusted. read the letter →

arxiv 2508.18786 v3 pith:7G2GXKP6 submitted 2025-08-26 math.NT

Matt Visser (Victoria University of Wellington) This is my paper · ORCID

classification math.NT MSC 11N05

keywords primesinshortintervalseffectiveboundsprimegapsbetweenpowerssub-linearexplicitestimatesuniversalgapbound

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

The paper's aim is to turn a century-old ineffective existence theorem for primes in short intervals into a fully explicit, universal statement. It combines three independent effective inputs: results on primes between 90th powers and between cubes, verified maximal prime gaps, and explicit high-precision linear interval data. The strongest claim is that for every n >= 106 and every real x >= 1, the interval [x, x + x^(1 - 1/n)] contains at least one prime. If correct, this means the next prime after any x is no farther than x^(0.9906), and the classical 1930/1933 sub-linear interval theorems hold for all x rather than only for 'sufficiently large' x. The paper also gives explicit astronomical thresholds for smaller exponents, such as n >= 4 with x >= exp(exp(33)).

What carries the argument

The load-bearing object is the family of sub-linear intervals [x, x + x^(1 - 1/n)], with n as a dial controlling interval length. The argument's mechanism is a crossover identity: an effective prime-between-powers result [i^m, (i+1)^m] is equivalent, by the binomial expansion of (i+1)^m, to an interval of length m x^(1 - 1/m) ending at x; comparing that length with x^(1 - 1/n) gives an explicit threshold x = [m^m]^(n/(n-m)). For global coverage at n = 106, the paper stitches three regimes: maximal prime gaps below 2 x 10^19, tabulated effective linear intervals [(1 - 1/Delta)x, x] up to (Delta_max - 1)^106, and the 90th-power interval above that threshold.

What would settle it

Find one real x >= 1 with no prime in [x, x + x^(1 - 1/106)]; since the theorem is universal, that single counterexample would settle it. Short of that, show that the proofs behind references [6] and [7] only establish prime-between-powers for integer bases, not all real x, which would break the 'by inspection' steps (2.5) and (2.10).

Watch

Extended reading notes

Core claim

The paper's central claim is that old existence theorems for primes in short intervals can be made fully explicit by splicing together three independent effective inputs. Using a cited 90th-power theorem, interpreted by inspection as [x, x + 90 x^(1 - 1/90)] for all real x >= 1, plus the cited cube theorem with its improved threshold, plus the known table of maximal prime gaps and explicit linear interval data, the paper derives three effective statements: for n >= 4 and x >= exp(exp(33)) the interval [x, x + x^(1 - 1/n)] contains a prime; for n >= 91 and x >= [90^90]^(n/(n-90)) it contains a prime; and for n >= 106 the statement holds for every x >= 1. The last statement yields the universa

Load-bearing premise

The load-bearing premise is the paper's 'by inspection' claim that the cited 90th-power theorem and cube theorem actually prove the stronger all-real-x interval statements (2.5) and (2.10); if that reading is wrong, the n >= 106 all-x result would not be established by this paper.

Editorial extensions

If this is right

Every starting point x >= 1 is guaranteed to contain a prime within distance x^(1 - 1/106), giving an explicit and checkable stopping rule for prime searches.
The classical 1930/1933 interval exponents hold unconditionally for all real x, not just asymptotically.
For any n >= 106 the same universal guarantee holds, so one can trade a slightly longer allowable gap for exactly the same coverage.
For n between 4 and 106, explicit but astronomical thresholds are supplied; for example n = 4 requires x >= exp(exp(33)).
The n = 105 case leaves a bounded middle range undecided, and only finitely many additional explicit linear-interval pairs would be needed to close it.

Reading between the lines

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

A direct check of the two 'by inspection' strengthenings in the cited proofs is a natural next step; if either fails, the universal claim would still likely hold for all x above some finite threshold, but not exactly as stated.
Any future reduction of the 90th-power exponent would automatically lower the all-x n through the crossover formula x = [m^m]^(n/(n-m)).
The n = 105 middle gap is a concrete computational target: a few more (x_Delta, Delta) pairs would either close it or reveal a genuine boundary.
The same splice-and-cover recipe could be applied to other explicit interval data, such as primes in arithmetic progressions, whenever analogous tables exist.

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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

This paper takes known effective prime-between-powers results, a table of maximal prime gaps, and Cully-Hugill–Lee linear intervals, and assembles them into explicit statements about primes in [x, x+x^(1−1/n)]. The specific package is new: the n≥106, x≥1 claim and the n≥4, x≥exp(exp(33)) claim are not in the cited literature. As a compilation, it is clean and honest, and the paper explicitly notes where the argument stops short (n=105 has an unresolved gap). No new analytic method is introduced, but that is not a flaw if the assembly is correct.

The soft spots are real. First, (3.1) is false as stated: for n=2 and x=8, the interval [8, 8+√8] ≈ [8,10.83] contains no prime. The statement may hold for n≥106, but the universal quantifier over n≥2 is overbroad and should be corrected. Second, and more seriously, the derivation of (2.5) from Cully-Hugill and Johnston's Theorem 1.4 is asserted by 'inspection' without any detail. The cited theorem only guarantees a prime between consecutive 90th powers, i.e., in [i^90,(i+1)^90] for integer i. The stronger claim that for every real x≥1 the interval [x, x+90x^(89/90)] contains a prime is not a formal consequence of that statement by simple algebra; it depends on where inside the power interval the prime actually sits. The same issue affects (2.10) from Dudek's cube theorem. If this strengthening is not actually established, the coverage for large x in (5.2) collapses: using only the integer-power theorem with k=ceil(x^(1/90)) gives a threshold around x≈10^1345 for n=106, which exceeds the endpoint of the linear-interval coverage, so the final all-x≥1 claim would be unproved. The author needs to provide the missing details or quote a theorem that directly gives the real-x statement.

I should note that the reader's concern about (2.18) does not survive close reading: for n in (3,n1], the [27]^{n/(n−3)} term dominates exp(exp(33)), so the condition is not actually omitted. That is a minor point in the author's favor.

Who is this for? Readers who need explicit prime-gap bounds for computations or for removing 'sufficiently large' conditions will find it useful, once the gaps are fixed. It deserves a serious referee, but the referee should insist on a full justification of (2.5) (or a revised statement that does not depend on it) and a correction of (3.1). My own inclination is skeptical until the 'inspection' claim is made reproducible.

Referee Report

3 major / 4 minor

Summary. The paper combines several known effective results—primes between 90th powers (Cully-Hugill–Johnston), primes between cubes (Dudek/Cully-Hugill), the table of maximal prime gaps, and explicit linear intervals due to Cully-Hugill–Lee—to produce explicit ranges in which the interval [x, x + x^{1-1/n}] is guaranteed to contain a prime. The main claims are: for all n ≥ 4 and x ≥ exp(exp(33)); for all n ≥ 91 and x ≥ [90^90]^{n/(n-90)}; and, for all n ≥ 106 and every x ≥ 1, the interval contains a prime. The final all-x statement, if established, gives a fully explicit prime-gap bound of size x^{1-1/106}.

Significance. If the main theorem is correct, it is a clean and useful explicit contribution: it makes classical sub-linear prime-in-short-interval results fully effective for all x ≥ 1, at the cost of a large exponent (n ≥ 106). The method is elementary and transparently combines several strong external results. The paper also gives explicit numerical thresholds in Tables 1–3, which are easily checked. However, the central derivation rests on an unproved strengthening of a cited theorem, and one stated auxiliary claim is false as written. The significance is therefore conditional on closing those gaps.

major comments (3)

[§2.1, Eq. (2.5)] Equation (2.5) is load-bearing, but the justification 'Inspection of the proof of [6, Theorem 1.4] shows' is not supplied. The cited theorem, as stated in (2.4), guarantees a prime in (i^90, (i+1)^90) for each integer i. This does not formally imply the interval statement [x, x+90x^{89/90}] for every real x ≥ 1, because the prime guaranteed by the power-interval theorem may lie below x. The derivation of (2.8), and eventually (5.2), depends on (2.5). The author must either reproduce the argument from [6] that yields the stronger statement, or quote an explicitly stated stronger theorem from [6] with a precise location. The same issue affects (2.10) from Dudek's cube theorem, although the final n ≥ 106 claim depends primarily on (2.5).
[§3, Eq. (3.1)] Equation (3.1) is false as stated: it claims for all n ≥ 2 and all x < 2×10^19 that [x, x+x^{1-1/n}] contains a prime. For n = 2 and x = 8, the interval is [8, 8+√8] ≈ [8, 10.828], which contains no prime. The later use for n ≥ 106 can be repaired because x^{1-1/106} is much larger than √x for the x-values in question, and the maximal-gap table appears to suffice there; but the universal assertion must be corrected, e.g. by restricting the claimed n-range. In addition, 'easily verified by inspecting a table' is not a proof; the author should state the exact verified property (which gaps are bounded by p_i^{1-1/n}) and cite the exhaustive computation or include enough data to make the verification reproducible.
[§5, Eq. (5.1)–(5.2)] The final all-x coverage depends on three external inputs: the maximal-gap verification for x < 2×10^19, the linear-interval data from [11] as reproduced in Table 2, and the unproved strengthening (2.5). Even if (2.5) is accepted, the paper should make explicit how the maximal-gap bound and the (x_Δ, Δ) pairs are combined for n = 106, and in particular why the union of intervals [x_Δ, (Δ−1)^n] indeed covers [4×10^18, (Δ_max−1)^n]. The overlap is plausible from Tables 1 and 3, but the verification is asserted rather than demonstrated. Given the central claim's dependence on this numerical covering, a short verification script or a more detailed table would materially strengthen the paper.

minor comments (4)

[Abstract] The abstract states x ≥ exp(3 exp(33)), while the body consistently uses exp(exp(33)). This discrepancy should be resolved in favor of the body's value unless the abstract's larger threshold is intended.
[§2.2] Equation (2.9) says i ≥ exp(exp(33)), but (2.10) says x ≥ exp(exp(33)); for consistency, state whether the endpoints are inclusive and whether the inequalities are strict or non-strict in all displayed results.
[References] References [8]–[10] rely on Wikipedia, Prime Pages, and software code. For a formal explicit-number-theory paper, it would be preferable to cite a peer-reviewed computation or provide the full table and a description of the verification method.
[Throughout] There are several minor typographical/formatting issues: 'Cully-Hugill+Johnston' should be 'Cully-Hugill and Johnston'; some displayed equations have inconsistent spacing and missing parentheses in the extracted text; the threshold in Table 1 for n=91 is written with a period in '1.762594084 · 1016005' and should be formatted consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a combination of external effective bounds and computational gap data; the 'inspection' strengthening is a verification gap, not a circular step.

full rationale

The paper's chain is straightforward: it imports effective results on primes between powers (Cully–Hugill/Johnston for 90th powers, Dudek/Cully-Hugill for cubes), converts them to short-interval statements, then bridges the remaining range using maximal-prime-gap tables and effective linear intervals from Cully-Hugill–Lee. No parameter is fitted to the target claim, no self-citation is load-bearing (the author appears in none of the references), and the final all-x≥1 statement is not assumed as an input. The only potentially problematic points are (2.5) and (2.10), where the paper asserts 'Inspection of the proof' yields a stronger real-interval statement than the stated integer-power theorem. That is an unsupported or at least unverified strengthening, and if wrong it would invalidate the conclusion; but it is not circular in the sense of defining X in terms of Y or renaming a fitted input as a prediction. It is a correctness/verification gap, not a self-referential derivation. The use of maximal prime gaps up to 2×10^19 is likewise computational evidence rather than a fully formal proof, but again it is external data, not circularity. Under the stated review rules, this paper merits a score of 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It rests on several external theorems and on the unproved assertion that those theorems can be strengthened into interval statements. The main risk is the 'inspection' claims.

assumptions (6)

domain assumption Cully-Hugill and Johnston's theorem on primes between 90th powers.
Relied on in Section 2.1 to establish (2.4) and the stronger (2.5).
ad hoc to paper The assertion that (2.5) follows 'by inspection' from the proof of [6, Theorem 1.4].
This is a stronger claim than the stated theorem and is not demonstrated in this paper.
domain assumption Dudek's theorem on primes between cubes for i ≥ exp(exp(33)).
Used in Section 2.2 to establish (2.9) and (2.10).
ad hoc to paper The assertion that (2.10) follows 'by inspection' from Dudek's proof.
Stronger than the stated theorem; not verified in this paper.
domain assumption The correctness of maximal prime gap tables up to 2e19.
Used in Section 3 to assert (3.1). The statement as written is false for n=2, so the use is problematic.
domain assumption The linear interval estimates of Cully-Hugill and Lee (Table 2).
Used in Section 4 to cover intermediate ranges.

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Cite this review

Pith. "Pith review of Effective short intervals containing primes." pith.science (2026). https://pith.science/paper/7G2GXKP6

@misc{pith2026250818786,
  author       = {Pith},
  title        = {Pith review of: Effective short intervals containing primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7G2GXKP6}},
  note         = {Machine review of arXiv:2508.18786}
}

abstract

95 years ago Hoheisel proved the existence of primes in the sub-linear interval \[ \left[x, x+x^{1-{1\over 33000}}\right] \qquad \hbox{for $x$ sufficiently large}. \] This was improved by Heilbronn, proving existence of primes in the sub-linear interval \[ \left[x, x+x^{1-{1\over 250}}\right] \qquad \hbox{for $x$ sufficiently large}. \] More recently Baker, Harman, Pintz proved existence of primes in the sub-linear interval \[ \left[x, x+ x^{1-{19\over 40}}\right] \qquad \hbox{for $x$ sufficiently large}. \] In the present article I will, to the extent possible, make some of these statements effective. Specifically, among other things, I shall show that \[ \forall n \geq 4, \qquad\forall x \geq \exp(3\exp(33)), \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]; \] \[ \forall n \geq 91, \qquad\forall x \geq [90^{90}]^{n/(n-90)} , \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] Furthermore \[ \forall n \geq 106, \qquad\forall x \geq 1, \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] In particular this last observation makes both the Hoheisel and Heilbronn results fully explicit and effective. This (relatively) specific observation can be extended and generalized in various manners.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

[6]

On the error term in the explicit formula of Riemann–von Mangoldt. II

Michaela Cully-Hugill and Daniel R. Johnston, “On the error term in the explicit formula of Riemann–von Mangoldt. II.”, Funct. Approx. Comment. Math. Advance Publication 1 - 20, 2025. doi:10.7169/facm/241110-18-11 [arXiv: 2402.04272 [math.NT]]

arXiv 2025
[11]

Explicit Interval Estimates for Prime Numbers

Michaela Cully-Hugill and Ethan S. Lee, “Explicit Interval Estimates for Prime Numbers”, Math. Comp. 91 (2022), 1955–1970. doi:10.1090/mcom/3719 [arXiv: 2103.05986 [math.NT]] – 9 –

work page Pith review arXiv 2022
[1]

Primzahlprobleme in der Analysis

Guido Hoheisel, “Primzahlprobleme in der Analysis”, Sitzungsberichte Berliner Akad. d. Wiss., (1930) 580-588

work page 1930
[2]

Uber den Primzahlsatz von Herrn Hoheisel

Hans Heilbronn, “Uber den Primzahlsatz von Herrn Hoheisel”, Math. Z. 36 (1933), 394–423. doi:10.1007/BF01188631

work page doi:10.1007/bf01188631 1933
[3]

The Difference Between Consecutive Primes, II

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

work page 2001
[4]

Primes between consecutive powers

Michaela Cully-Hugill, “Primes between consecutive powers”, Journal of Number Theory, 247 (2023) 100–117. doi:10.1016/j.jnt.2022.12.002 [arXiv: 2107.14468 [math.NT]]

work page Pith review arXiv 2023
[5]

On the error term in the explicit formula of Riemann--von Mangoldt

Michaela Cully-Hugill and Daniel R. Johnston, “On the error term in the explicit formula of Riemann–von Mangoldt”, International Journal of Number Theory Vol. 19, No. 6 (2023) 1205?1228 doi:10.1142/S1793042123500598 [arXiv: 2111.10001 [math.NT]]

work page Pith review arXiv 2023
[7]

An Explicit Result for Primes Between Cubes

Adrian Dudek, “An Explicit Result for Primes Between Cubes”, Funct. Approx. Comment. Math. 55 # 2(2016) 177–197. doi:10.7169/facm/2016.55.2.3 [arXiv: 1401.4233 [math.NT]]

work page Pith review arXiv 2016

Show all 11 references

[8]

Prime gap

Wikipedia, “Prime gap”, https://en.wikipedia.org/wiki/Prime gap (accessed on 19 August 2025)

2025
[9]

Prime pages, Table of known maximal gaps, https://t5k.org/notes/GapsTable.html (accessed on 19 August 2025)

2025
[10]

Kim Walisch, 2025, primecount software, https://github.com/kimwalisch/primecount

2025

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [6]

On the error term in the explicit formula of Riemann–von Mangoldt. II

Michaela Cully-Hugill and Daniel R. Johnston, “On the error term in the explicit formula of Riemann–von Mangoldt. II.”, Funct. Approx. Comment. Math. Advance Publication 1 - 20, 2025. doi:10.7169/facm/241110-18-11 [arXiv: 2402.04272 [math.NT]]

arXiv 2025

[2] [11]

Explicit Interval Estimates for Prime Numbers

Michaela Cully-Hugill and Ethan S. Lee, “Explicit Interval Estimates for Prime Numbers”, Math. Comp. 91 (2022), 1955–1970. doi:10.1090/mcom/3719 [arXiv: 2103.05986 [math.NT]] – 9 –

work page Pith review arXiv 2022

[3] [1]

Primzahlprobleme in der Analysis

Guido Hoheisel, “Primzahlprobleme in der Analysis”, Sitzungsberichte Berliner Akad. d. Wiss., (1930) 580-588

work page 1930

[4] [2]

Uber den Primzahlsatz von Herrn Hoheisel

Hans Heilbronn, “Uber den Primzahlsatz von Herrn Hoheisel”, Math. Z. 36 (1933), 394–423. doi:10.1007/BF01188631

work page doi:10.1007/bf01188631 1933

[5] [3]

The Difference Between Consecutive Primes, II

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

work page 2001

[6] [4]

Primes between consecutive powers

Michaela Cully-Hugill, “Primes between consecutive powers”, Journal of Number Theory, 247 (2023) 100–117. doi:10.1016/j.jnt.2022.12.002 [arXiv: 2107.14468 [math.NT]]

work page Pith review arXiv 2023

[7] [5]

On the error term in the explicit formula of Riemann--von Mangoldt

Michaela Cully-Hugill and Daniel R. Johnston, “On the error term in the explicit formula of Riemann–von Mangoldt”, International Journal of Number Theory Vol. 19, No. 6 (2023) 1205?1228 doi:10.1142/S1793042123500598 [arXiv: 2111.10001 [math.NT]]

work page Pith review arXiv 2023

[8] [7]

An Explicit Result for Primes Between Cubes

Adrian Dudek, “An Explicit Result for Primes Between Cubes”, Funct. Approx. Comment. Math. 55 # 2(2016) 177–197. doi:10.7169/facm/2016.55.2.3 [arXiv: 1401.4233 [math.NT]]

work page Pith review arXiv 2016