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REVIEW 3 major objections 4 minor 20 references

On Maximal Prime Gaps

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper claims a new explicit upper bound on prime gaps: for every n, p_{n+1} − p_n < (51/16) log² p_n, a bound that would immediately imply Oppermann's conjecture and a strengthened Andrica conjecture.

desk verdict An elementary division error in Lemma 2.2 invalidates the main bound, so the paper does not establish what it claims. read the letter →

arxiv 2605.14871 v5 pith:CRO2G4MT submitted 2026-05-14 math.NT

Cheng-TIng Wang This is my paper · ORCID
classification math.NT MSC 11A4111N05
keywords primegapsOppermann'sconjectureAndrica'sCramér-typeboundsexplicitestimatesforprimesweightedprime-gapsumslog-squaredelementarynumbertheory
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 sets out to prove that consecutive primes are never too far apart: the gap g_n = p_{n+1} − p_n is always less than (51/16) times the square of the natural logarithm of p_n. This is an explicit constant-factor bound of the kind conjectured by Cramér, and it would settle Oppermann's century-old conjecture that a prime lies between a(a−1) and a² and between a² and a(a+1). It would also prove a strengthened form of Andrica's inequality: sqrt(p_{n+1}) − sqrt(p_n) < 1/2 for all n ≥ 31. The proof is elementary: it rewrites each gap in terms of weighted sums of averaged gaps, shows those sums double only slowly, then converts that doubling control into a log-squared estimate.

What carries the argument

The central objects are the running mean A_k = (p_{k+1} − 2)/k of the first k prime gaps and the weighted sums B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1. Theorem 2.1 gives exact identities such as g_n = n(B_n − T_n) and n − 1 = (B_n − T_n)/(T_n − B_{n−1}). The crucial mechanism is Lemma 2.2, which uses lower bounds for A_k and integral estimates to prove B_n < 2B_{n−1} for n ≥ 22; this doubling bound limits how fast the weighted gap sums can grow and is what eventually produces the constant 51/16.

What would settle it

Check the termwise inequality in Lemma 2.2 that leads to equation (2.9). For k = 7, A_7 = 17/7, so A_7/(k−1) = 17/42 ≈ 0.4048, while log(6)/6 + 2/7 ≈ 0.5843: the claimed step 'A_k/(k−1) > log(k−1)/(k−1) + 2/k' does not hold term by term. Whether the sum bound in (2.9) still holds by another route is the question that decides Lemma 2.2 and hence Theorem 2.5.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2.5: g_n < (51/16) log² p_n for every n ≥ 1. The engine is a set of summation identities (Theorem 2.1) expressing each gap g_n through B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1, where A_k is the average of the first k gaps. Lemma 2.2 asserts that B_n < 2B_{n−1} for every n ≥ 22; Lemma 2.3 converts that into g_n < (51/16) B_{n+1}; and explicit classical bounds on the nth prime, n log n < p_n < n(log n + log log n), let the author bound B_{n+1} by a logarithm-squared expression. The cases n ≤ 21 are checked directly.

Load-bearing premise

The entire chain rests on Lemma 2.2's assertion that the weighted sums B_n grow by less than a factor of two for every n ≥ 22; if the numerical inequality behind that doubling statement is not valid, the proof cannot exclude B_n ≥ 2B_{n−1}, and the log-squared bound does not follow.

Editorial extensions

If this is right

  • If correct, Oppermann's conjecture follows: for every integer a ≥ 2 there is at least one prime in (a(a−1), a²) and one in (a², a(a+1)).
  • If correct, a strengthened Andrica inequality follows: sqrt(p_{n+1}) − sqrt(p_n) < 1/2 for all n ≥ 31, whereas the classical conjecture only needs < 1.
  • The maximal prime gap among the first n primes is at most (51/16) log² p_n, an explicit constant-factor Cramér-type bound.
  • Any prime gap of size g must occur at a prime below e^(√(16g/51)), giving an explicit numerical relation between gap size and location.

Reading between the lines

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

  • The method leaves the constant 51/16 dependent on the explicit bounds for the nth prime; tightening those bounds would lower the constant directly, so the paper implicitly offers a parameterized route toward Cramér-type constants without pursuing it.
  • The same summation-identity machinery could be applied to other sparse sequences wherever a Bertrand-type bound is available; the paper does not explore this extension.
  • Because 51/16 is far larger than the heuristic lower limit near 1.12 for the limsup of g_n/log² p_n, the proved bound does not discriminate between competing conjectures about the true limiting ratio.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims an unconditional bound for every prime gap: g_n = p_{n+1} - p_n < (51/16) log^2 p_n for all n (the arXiv metadata abstract states 13/3 instead of 51/16). The proof introduces averages A_k, B_n, and T_n, proves elementary identities in Theorem 2.1, then uses a chain of lemmas: Lemma 2.2 asserts B_n < 2B_{n-1} for n >= 22, Lemma 2.3 converts this into g_n < (51/16) B_{n+1}, Lemma 2.4 bounds A_n by 2 log(n-1), and Theorem 2.5 assembles the final gap bound. Consequences claimed include Oppermann's conjecture (Theorem 3.1) and a strengthened Andrica bound (Theorem 3.2).

Significance. If the proof were correct, the result would be a landmark: it would improve all known unconditional upper bounds on prime gaps (currently of size about x^{0.525}) to O(log^2 x), prove Oppermann's conjecture, and establish a strong Andrica-type inequality. Unfortunately, the central derivation is not valid. The main algebraic step in Lemma 2.2 is simply incorrect, and Lemma 2.3 contains a further unjustified inequality. The paper does contain some correct elementary identities and makes standard use of Rosser-type bounds, but these do not support the claimed theorem. The finite 'direct calculation' ranges are not documented with code or tables, which is an additional reproducibility concern for a result of this magnitude.

major comments (3)
  1. [Lemma 2.2, Eq. (2.9)] The step from (2.8) to (2.9) is algebraically false. From A_k > log(k-1) + 2/k one may divide by k-1 to obtain A_k/(k-1) > log(k-1)/(k-1) + 2/[k(k-1)], not log(k-1)/(k-1) + 2/k. The displayed sum in (2.9) with the extra sum_{k=7}^{n-1} 2/k is therefore not justified. This erroneous term is exactly what produces the log(n-1) term in (2.10) and drives the contradiction for all n >= 22. With the correct convergent sum sum 2/[k(k-1)], the lower bound loses the logarithmic growth and the claimed contradiction does not follow. Since Lemma 2.2 is the engine for Lemma 2.3 and Theorem 2.5, the main theorem is unsupported.
  2. [Lemma 2.3, Eq. (2.14)] Even accepting Lemma 2.2, the proof of Lemma 2.3 contains an invalid inequality. From the order relations T_n < B_n < B_{n+1} < 2B_n the paper asserts 16T_n - 4√(B_{n+1}T_n) ≤ [B_{n+1}+T_n - √Δ]/(2·1/16) ≤ B_n. This is not a consequence of those order relations; for example, take T_n = 1/2 and B_{n+1} = 1, which are compatible with the order relations. Then 16T_n - 4√(B_{n+1}T_n) ≈ 5.17, while [B_{n+1}+T_n - √Δ]/[2·1/16] ≈ 0.34, so the first inequality fails. No additional argument is supplied. This inequality is load-bearing for the contradiction that yields g_n < (51/16)B_{n+1}.
  3. [Theorems 2.5, 3.1, 3.2] The proof depends on several claimed finite verifications: 1 ≤ n ≤ 21 in Theorem 2.5, 2 ≤ a ≤ 488 in Theorem 3.1, and 31 ≤ n ≤ 21141 in Theorem 3.2. These are asserted as 'direct calculation' with no code, tables, or reproducible method. Since the theorem is meant to be an unconditional result about all prime gaps, these ranges are part of the proof. This is secondary to the algebraic errors above, but it should be addressed in any revision.
minor comments (4)
  1. [Abstract] The arXiv metadata abstract states the bound as (13/3) log^2 p_n, while the body and the abstract in the full text state (51/16) log^2 p_n. These must be reconciled.
  2. [After Eq. (2.11)] The sentence 'But (2.11) implies that 21 ≤ n' appears to have the inequality direction backwards: the derivation requires that (2.11) be impossible for n ≥ 22, i.e. it should imply n ≤ 21 (or fail for all n ≥ 22), not n ≥ 21.
  3. [Eq. (2.8) surrounding text] The text says that (2.8) implies A_k > log(k-1)+2/k 'for all k ≤ 7' and A_k > log(k-1) 'for all k ≤ 6'. The inequalities and the use of the sums later indicate these should be k ≥ 7 and k ≥ 6 (or similar); as written the directions are inconsistent.
  4. [General] There are several typographical issues, including 'Betrand's postulate' in Section 1 and the hyphenation of Erdős's name. These do not affect the mathematics but should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bound is derived from external prime estimates and finite computation; no fitted input is renamed as a prediction and no self-citation chain is load-bearing.

full rationale

The claimed upper bound g_n < (51/16) log^2 p_n is obtained from the averaging identities in Theorem 2.1, the external bounds of Bertrand and Rosser (p_n > n log n, p_n < n(log n + log log n)), and a finite check for small n. Theorem 2.1 consists of algebraic identities following from the definition of A_k and B_n; these are tautological but are used only as rearrangement tools, not as empirical predictions. Lemma 2.2 and Lemma 2.4 rely on standard, externally established inequalities and on finite computation, not on the target theorem or on any fitted parameter. There are no self-citations, no imported uniqueness theorems from the authors' prior work, and no fitted constants that are later relabeled as predictions. The apparent algebraic error identified by the reader—where dividing A_k > log(k-1) + 2/k by k-1 would produce +2/[k(k-1)] rather than +2/k—is a correctness/validity concern, not a circularity concern: even if that step fails, the proof does not assume g_n < (51/16) log^2 p_n in its inputs. Because the derivation chain is self-contained apart from external estimates and finite verification, there is no circularity to report.

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

The central claim rests on standard theorems (Bertrand, Rosser) and on a finite computation asserted without display. No free parameters are fit to data, and no new entities are postulated. The main weakness is the erroneous use of +2/k in Lemma 2.2, which is an ad hoc algebraic mistake rather than an external input.

assumptions (4)
  • standard math Bertrand's postulate: for every n>1, there is a prime between n and 2n, used to bound p_{n+1} < 2p_n.
    Invoked in Lemma 2.2 around equation (2.6) to estimate 2A_{n-1} - A_n.
  • standard math Rosser's bounds: n log n < p_n for n≥1 and p_n < n(log n + log log n) for n≥6.
    Used throughout, e.g., (2.8), Lemma 2.4, and (2.11), to convert gap sums into logarithms and to bound p_n above.
  • domain assumption The functions log x / x and log log x / x are decreasing for x≥6.
    Assumed in Lemma 2.4 to justify integral comparisons and monotonicity. (Note: log x is increasing, so this is a stated assumption about the specific ratios.)
  • ad hoc to paper The 'direct calculation' claims for n≤21 and a≤488 are correct.
    Theorem 2.5 relies on checking 1≤n≤21, and Theorem 3.1 on checking 2≤a≤488, but no computation, code, or table is provided.

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Cite this review

Pith. "Pith review of On Maximal Prime Gaps." pith.science (2026). https://pith.science/paper/CRO2G4MT

@misc{pith2026260514871,
  author       = {Pith},
  title        = {Pith review of: On Maximal Prime Gaps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CRO2G4MT}},
  note         = {Machine review of arXiv:2605.14871}
}
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 $\frac{13}{3}\log^2{p_n}$. 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.

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Reference graph

Works this paper leans on

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