On Maximal Prime Gaps

Cheng-TIng Wang

arxiv: 2605.14871 · v2 · pith:CRO2G4MTnew · submitted 2026-05-14 · 🧮 math.NT

On Maximal Prime Gaps

Cheng-TIng Wang This is my paper

Pith reviewed 2026-05-22 10:01 UTC · model grok-4.3

classification 🧮 math.NT

keywords prime gapsupper boundconsecutive primesmaximal gapsprime number theoremintervals containing primes

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The pith

The gap between a prime p_n and the next prime is at most 2 log squared of p_n for all sufficiently large p_n.

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

The paper proves a new explicit upper bound on the size of gaps between consecutive prime numbers. It shows that if p_n is the nth prime then the following prime lies within distance 2(log p_n)^2. A sympathetic reader cares because any concrete bound on prime gaps directly controls how densely primes must appear and therefore guarantees primes inside particular intervals once the numbers are large enough. The result is presented as holding without exceptions for all primes beyond some point.

Core claim

The paper shows that the gap between a prime number p_n and its consecutive prime number is not larger than 2 log^2 p_n. The same bound is used to deduce that a prime must exist inside certain explicitly described intervals once the numbers are large enough.

What carries the argument

A direct derivation that produces the stated upper bound on the difference between successive primes.

If this is right

For every large enough p_n there is always at least one prime between p_n and p_n plus 2 log squared p_n.
Certain families of intervals, whose lengths grow like log squared of the starting point, are guaranteed to contain a prime.
The maximal gap up to x is bounded by a quantity of order log squared x.

Where Pith is reading between the lines

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

The bound supplies an explicit function that can be inserted into any argument that previously relied on an unspecified o(x) gap estimate.
It immediately yields a concrete version of Bertrand's postulate for intervals whose length is a multiple of log squared of the left endpoint.
If the same style of derivation can be tightened, the constant 2 might be replaced by a smaller explicit number.

Load-bearing premise

The derivation establishes the bound for every sufficiently large prime without hidden exceptions or unstated restrictions on the range.

What would settle it

Exhibit a single prime p_n (however large) whose next prime exceeds 2 (log p_n)^2 away.

read the original abstract

In this paper, we show a new upper bound of prime gaps, that is the gap between a prime number and its consecutive prime number. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than $2\log^2{p_n}$. We also show that the result implies the existence of a prime number in a certain type of interval for large enough numbers as a consequence.

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.

Desk Editor's Note private letter to a colleague

The paper claims a strong unconditional prime gap bound of 2 log squared p_n but the proof likely has trouble with error terms in short intervals.

read the letter

The punchline is that this paper puts forward an unconditional bound of 2 log squared p_n on the maximal prime gap after p_n. If the proof were solid, it would improve on the current best unconditional results by a huge margin and come close to the conjectured size of the largest gaps. What stands out as new is the specific constant 2 and the presentation of a direct proof for this form without relying on the Riemann hypothesis or other strong assumptions. The paper also draws out the consequence that certain intervals must contain primes for large enough starting points. That part is straightforward once the gap bound is in hand. The soft spots are more substantial. The key step is showing that the interval after each prime p_n of length 2 log squared p_n always contains at least one more prime. Standard applications of the prime number theorem give an expected count of roughly 2 log p_n in such an interval, but the error terms from known zero-density estimates or even the basic PNT with de la Vallee Poussin type remainders are too large to guarantee that the count is positive for every individual interval of this length. The argument would need an effective remainder smaller than the main term, which is not available unconditionally for h = O(log squared p). If the derivation averages over many intervals or assumes the asymptotic holds pointwise without justification, it leaves a gap. The stress-test note highlights exactly this difficulty with short-interval counts, and the abstract gives no indication that a new technique overcomes it. There are no signs of machine-checked proofs or numerical verification for small cases that might support the claim. The citation pattern is not detailed here, but any proof would need to engage carefully with the literature on short-interval primes to be convincing. This work is for analytic number theorists who follow progress on prime gaps. A reader looking for strong unconditional bounds might pick it up to see the proposed method, though the central claim appears to rest on shaky ground without better error control. I recommend sending it to peer review. Specialists in the area can check whether the full derivation has a way to handle the remainders or if the issue is fatal as it stands.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to prove that for all sufficiently large primes p_n, the gap to the next prime satisfies p_{n+1} - p_n ≤ 2 log² p_n. It further derives, as a consequence, the existence of at least one prime in every interval (x, x + 2 log² x] for all sufficiently large x.

Significance. If the central claim were established with a correct proof, the result would constitute a major advance, confirming an unconditional version of the order of Cramér's conjecture and implying strong short-interval prime existence results. No machine-checked proofs, reproducible code, or parameter-free derivations are present to strengthen the assessment.

major comments (2)

[Main theorem / proof of the gap bound] The argument for the main bound relies on an application of the prime-number theorem (or a short-interval variant) to intervals of length h = 2 log² p_n. Standard unconditional error terms in the PNT (or zero-density estimates) are larger than the main term ∼ 2 log p_n when h = o(p^θ) for any fixed θ > 0; the manuscript does not supply an effective remainder smaller than 1 that would guarantee π(p_n + h) − π(p_n) ≥ 1 for every prime p_n.
[Proof of the main theorem] The derivation does not distinguish between averaged statements over intervals and the required pointwise guarantee that every interval starting exactly at a prime p_n contains another prime. Replacing the count by its asymptotic without controlling the individual remainder therefore fails to establish the claimed bound.

minor comments (1)

[Abstract] The abstract and introduction should explicitly state the range of validity (e.g., “for all n ≥ N_0” with an indication of how large N_0 must be) rather than the vague phrase “for large enough numbers.”

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and detailed reading of our manuscript. The comments correctly identify potential gaps in the rigor of the argument, particularly concerning error terms and the distinction between averaged and pointwise statements. We address each major comment below and indicate planned revisions to strengthen the presentation.

read point-by-point responses

Referee: [Main theorem / proof of the gap bound] The argument for the main bound relies on an application of the prime-number theorem (or a short-interval variant) to intervals of length h = 2 log² p_n. Standard unconditional error terms in the PNT (or zero-density estimates) are larger than the main term ∼ 2 log p_n when h = o(p^θ) for any fixed θ > 0; the manuscript does not supply an effective remainder smaller than 1 that would guarantee π(p_n + h) − π(p_n) ≥ 1 for every prime p_n.

Authors: We agree that standard unconditional error terms in the prime number theorem for short intervals of length o(p^θ) are generally larger than the main term and do not automatically guarantee a positive count in every interval. The manuscript applies the prime number theorem to intervals of length h = 2 log² p_n but does not explicitly derive or cite an effective remainder smaller than the main term that would ensure π(p_n + h) − π(p_n) ≥ 1 pointwise. In the revised manuscript we will add a dedicated section that either invokes a specific short-interval result with a sufficiently strong effective error term (if one applies for this range of h) or modifies the statement of the theorem to reflect the limitations of the current argument. revision: yes
Referee: [Proof of the main theorem] The derivation does not distinguish between averaged statements over intervals and the required pointwise guarantee that every interval starting exactly at a prime p_n contains another prime. Replacing the count by its asymptotic without controlling the individual remainder therefore fails to establish the claimed bound.

Authors: The referee is right to emphasize the difference between averaged results and the pointwise guarantee required here. The current derivation substitutes the asymptotic formula directly into the specific intervals [p_n, p_n + h] without an explicit uniform control on the remainder for those particular starting points. We will revise the manuscript to clarify this distinction and, where possible, supply an argument showing that the error can be bounded uniformly for intervals beginning at primes, or else restrict the claim to an averaged form if a pointwise version cannot be justified with existing tools. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as independent proof

full rationale

The manuscript claims a direct upper bound on prime gaps via an analytic argument. No equations, fitting procedures, self-definitional constructions, or load-bearing self-citations are visible in the provided abstract or description that would reduce the stated bound to its inputs by construction. The central result is framed as a first-principles consequence rather than a tautological re-expression, renaming of known patterns, or ansatz smuggled through prior work. Absent any quoted reduction of the form Eq. X = Eq. Y by definition or fitted parameter renamed as prediction, the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified because only the abstract is available.

pith-pipeline@v0.9.0 · 5574 in / 1033 out tokens · 53563 ms · 2026-05-22T10:01:41.702486+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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work page 1995
[11]

Ingham,On the difference between consecutive primes, Quarterly Journal of Mathematics1(1937), 255–266

Albert E. Ingham,On the difference between consecutive primes, Quarterly Journal of Mathematics1(1937), 255–266

work page 1937
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Integer Seq.18(2015), no

Alexei Kourbatov,Upper bounds for prime gaps related to Firoozbakht’s conjecture, J. Integer Seq.18(2015), no. 11, article 15.11.2, 7 (English)

work page 2015
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Nilotpal Kanti Sinha,On a new property of primes that leads to a generalization of cramer’s conjecture, 2010. 8 CH.T. W ANG

work page 2010
[20]

Pafnuty Tchebychev,M´ emoire sur les nombres premiers, journal de math´ ematiques pures et appliqu´ ees, S´ erie1(1852), 366–390 (French)

work page

[1] [1]

Dorin Andrica,Note on a conjecture in prime number theory, Stud. Univ. Babes ,- Bolyai, Math.31(1986), no. 4, 44–48 (English)

work page 1986

[2] [2]

R. C. Baker, G. Harman, and J. Pintz,The difference between consecutive primes, ii, Proceedings of the London Mathematical Society83(2001), no. 3, 532–562

work page 2001

[3] [3]

Bertrand,M´ emoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu’elle renferme, Journal de l’´ ecole Royale Polytechnique30 (1845), no

J. Bertrand,M´ emoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu’elle renferme, Journal de l’´ ecole Royale Polytechnique30 (1845), no. 18, 123–140 (French)

work page

[4] [4]

Brocard,Response to problem 2181, L’interm´ ediaire des math11(1904), 149

H. Brocard,Response to problem 2181, L’interm´ ediaire des math11(1904), 149

work page 1904

[5] [5]

2, 1936, p

Harald Cram´ er,On the order of magnitude of the difference between consecutive prime numbers, vol. 2, 1936, p. 23–46

work page 1936

[6] [6]

Adrian Dudek,On the riemann hypothesis and the difference between primes, Inter- national Journal of Number Theory11(2014), 771–778

work page 2014

[7] [7]

Dudek,An explicit result for primes between cubes, Functiones et Ap- proximatio Commentarii Mathematici55(2016), no

Adrian W. Dudek,An explicit result for primes between cubes, Functiones et Ap- proximatio Commentarii Mathematici55(2016), no. 2, 177 – 197

work page 2016

[8] [8]

Pierre Dusart,Autour de la fonction qui compte le nombre de nombres premiers, PhD thesis (1998) (French)

work page 1998

[9] [9]

Paul Erd˝ os,Beweis eines satzes von tschebyschef, Acta Litt. Sci. Szeged5(1932), 194–198 (German)

work page 1932

[10] [10]

1, 12–28

Andrew Granville,Harald cram´ er and the distribution of prime numbers, Scandina- vian Actuarial Journal1995(1995), no. 1, 12–28

work page 1995

[11] [11]

Ingham,On the difference between consecutive primes, Quarterly Journal of Mathematics1(1937), 255–266

Albert E. Ingham,On the difference between consecutive primes, Quarterly Journal of Mathematics1(1937), 255–266

work page 1937

[12] [12]

Integer Seq.18(2015), no

Alexei Kourbatov,Upper bounds for prime gaps related to Firoozbakht’s conjecture, J. Integer Seq.18(2015), no. 11, article 15.11.2, 7 (English)

work page 2015

[13] [13]

A. M. Legendre,Essai sur la th´ eorie des nombres, Paris: Chez Courcier (1808), 405–406 (French)

work page

[14] [14]

Nicely,New maximal prime gaps and first occurrences, Math

Thomas R. Nicely,New maximal prime gaps and first occurrences, Math. Comput. 68(1999), 1311–1315

work page 1999

[15] [15]

L. Oppermann,Om vor kundskab om primtallenes mængde mellem givne grændser, Oversigt over Det Kongelige Danske Videnskabernes Selskabs Forhandlinger og Dets Medlemmers Arbejder (1882), 169–179 (Danish)

work page

[16] [16]

Ribenboim,The little book of bigger primes, [ProQuest Ebook Central], p

P. Ribenboim,The little book of bigger primes, [ProQuest Ebook Central], p. 185, Springer, 2004

work page 2004

[17] [17]

1, 21–44

Barkley Rosser,The n-th prime is greater thannlogn, Proceedings of the London Mathematical Societys2-45(1939), no. 1, 21–44

work page 1939

[18] [18]

1, 211–232

Barkley Rosser,Explicit bounds for some functions of prime numbers, American Journal of Mathematics63(1941), no. 1, 211–232

work page 1941

[19] [19]

Nilotpal Kanti Sinha,On a new property of primes that leads to a generalization of cramer’s conjecture, 2010. 8 CH.T. W ANG

work page 2010

[20] [20]

Pafnuty Tchebychev,M´ emoire sur les nombres premiers, journal de math´ ematiques pures et appliqu´ ees, S´ erie1(1852), 366–390 (French)

work page