A tyro's approach to the tilted sieve: beyond the Erd\H{o}s--Rankin bound

Tristan Freiberg

REVIEW

A tyro's approach to the tilted sieve: beyond the Erd\H{o}s--Rankin bound

Tristan Freiberg

Number Theory · Submitted 23 Sep 2026 · 40 references

Pith's verdict: A genuinely new quantitative bound that beats a 70-year-old benchmark, honestly credited to its AI source, with a proof more self-contained than the 'tyro' title suggests — but the t-uniform concentration in Lemma 5.5 is where a referee must look.

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arXiv ID2609.28253 (version 1)

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CitationTristan Freiberg. "A tyro's approach to the tilted sieve: beyond the Erd\H{o}s--Rankin bound." arXiv:2609.28253 (2026). https://pith.science/paper/PF3K7ZDP

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Editor's note

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Three things you should know. First, Theorem 1.1 is real and new in its quantitative form: Y(x) >> x log x/((log_2 x)(log_3 x)) appears nowhere else in the cited literature, and it does beat Rankin's 1938 scale by the unbounded factor (log_2 x)/(log_3 x)^2. Second, the proof is considerably more careful than the "tyro's approach" framing implies: it is self-contained, every parameter is fixed before the boundary cases are counted, and the probabilistic machinery is standard second-moment analysis. Third, the paper is scrupulous about attribution — the power tilt and inverse-probability group selection are GPT-5.6 Sol's [16]; this paper's contribution is the adaptation to a shorter interval where whole residue classes serve as groups and primes are covered individually. That is a real but modest step: it isolates what the tilt alone achieves, without Maynard weights or hypergraph covering.

I want to correct the reader's take on one point. The flagged "weakest assumption" — the crude divisor bound (5.3) in Lemma 5.1 — is not where the risk sits. I rechecked the sizes: with U approximately log x/((log_2 x)(log_3 x)) and w = (log x)^100, the Lemma 5.1 error exponent is on the order of (log x)^(-97), about seven orders below the (log x)^(-90) target. The bound is crude but has enormous margin; it is non-load-bearing.

The genuinely load-bearing, least-checked object is Lemma 5.5: the claim that E*G^tau = 1 + O((log x)^(-90)) with full uniformity in t and in n. Every subsequent step — the second moments in Lemma 6.1, Proposition 6.3, and Theorem 1.1 itself — hangs on that uniformity. I traced the rest of the chain and it is internally consistent: the (5.7)-(5.8) cancellation, the CRT counting in Lemma 5.4, the G > x truncation, and the diagonal/off-diagonal split. But Lemma 5.5 is intricate, asserted rather than machine-checked, and the paper's own Appendix C.3 concedes that model agreement is not evidence of correctness. A hidden non-uniformity there would propagate straight into (6.4). I could not find one, but nothing in the text rules it out at the explicitly constant level.

The honesty about provenance cuts both ways. It makes the paper unusually trustworthy about what the AI did and what the author verified; it also means a referee's burden is to check the written proof, not lean on prior credibility.

Recommendation: send it out. It improves a 70-year-old benchmark with a self-contained argument and a genuinely useful historical appendix. A referee should concentrate on Lemma 5.5's uniformity claims. If those hold — and the surrounding machinery suggests they do — this is a clean, citable paper, valuable more for isolating the tilt's contribution than for the bound itself, which [15] and [16] both exceed.

Referee report

Significance

The paper has genuine value if the proof is correct. Methodologically, it isolates the contribution of the tilted sieve: the tilt alone, without Maynard sieve weights or a hypergraph covering theorem, already exceeds the classical Erdős–Rankin scale by an unbounded factor. The proof is largely self-contained (PNT, Mertens, elementary probability) and internally consistent: I re-derived the central chain — the Euler-product tail in Lemma 5.5, the three contributions in Proposition 6.3, and the final balance in §7.3 (200e^γ c < 1/4 versus #ℛ′ ~ x/(4 log x)) — and found the margins comfortable at every step. The parameter choices are genuine rather than fitted: τ in (7.3) is derived from the tail condition M/Q ≍ (log_2 x)(log_3 x), and the constants close consistently (H·Q ≈ 200e^γ c x/((log_2 x)^2 (log_3 x)^2) at the chosen t, and the expected prime survivors A·H/log H ≈ 200e^γ c x/log x). Strengths: complete ordinary proofs (no machine-checked proofs, but no gaps found), three detailed appendices reproducing the classical constructions, and an unusually transparent AI-provenance disclosure (Appendix C).

Major objections (2)

[§5.3–5.4, Lemma 5.4 (second average), Lemma 5.5] The estimate 𝔼*G^τ = 1 + O((log x)^{-90}) in (5.11) is the hinge on which Lemmas 6.1, Proposition 6.3 and Theorem 1.1 rest, and it is claimed uniformly in t ∈ [log_3 x, 3 log_3 x] and in n. The proof leaves the t-uniformity implicit in exactly two places: in the tail bound (5.14), the estimate Σ_{p>w}(p^τ−1)/p² ≪ 1/w is obtained from Σ_{m>w} m^{−2+τ} ≪ w^{−1+τ}, which is uniform in t only after applying (4.15) (τ log w = o(1)); and in the G > x contribution (5.13), the bound G^τ ≤ exp(U τ log H) uses τ log H = O(log_3 x) at the endpoint. Since the eventual choice (7.3) sets t = 2 log_3 x + 2 log_4 x near the upper edge of (4.12), and since Appendix C.3 explicitly disclaims machine verification, I ask that the proof state these two endpoint checks explicitly rather than leave the reader to assemble them from (4.11), (4.15) and (4.16). I have verified both bounds — the tail is O(U²/w) = O((log x)^{−98}) and G^τ = x^{o(1)}, so the margins are comfortable — and thus this is a request for explicitness at the load-bearing point, not a detected error.
[§5.4, Lemma 5.5] The uniformity in n claimed for the second auxiliary average in Lemma 5.5 rests entirely on the counting in the second part of Lemma 5.4, where divisibility d | G_n(q,r) is turned into CRT classes q ≡ −n h_i^{-1} mod p_i, r ≡ −n ℓ_i^{-1} mod p_i with offsets 0 < |h_i|, |ℓ_i| < U. The counting bound (5.10) is uniform in n only because the offsets are bounded independently of n and because the members of E_{q,n}∖{n} are pairwise coprime to n (Lemma 5.2); the manuscript never says this explicitly. Given that Lemma 5.5's second statement is used in Lemma 6.1 to average the conditional correlations of (5.8) over n ∈ 𝒞, I recommend adding one sentence to Lemma 5.4's proof identifying precisely where the bound is independent of n and why it remains valid for every fixed candidate.

Minor objections (5)

[Abstract] The typesetting of the abstract and several early lines is broken (e.g., "WegiveanelementaryexpositionofthetiltedsieveintroducedbyGPT-5.6Sol[ 16]"), with spaces missing between words; the displayed abstract should be reset so that the text is readable.
[§5.1] The crude divisor bound (5.3) in Lemma 5.1 is real but non-load-bearing: substituting U ≍ log x/((log_2 x)(log_3 x)) and w = (log x)^{100} gives the Lemma 5.1 error exponent O(k² log H/(w log w)) of order (log x)^{-97}, roughly seven orders of magnitude below the (log x)^{-90} target used in (5.6) and (5.11). A one-sentence remark quantifying this margin would preempt a reader's natural concern about the crudeness of (5.3).
[§7.3] In the proof of Theorem 1.1, the admissible range of the fixed constant c is given only as "sufficiently small that 200e^γ c < 1/4"; I suggest stating the explicit bound c < 1/(800e^γ) so that the arithmetic closing the proof (𝔼#ℳ < #ℛ′ ~ x/(4 log x)) is immediately checkable.
[Appendix B] The historical footnote asserting a gap in Ricci's survivor count, attributed to GPT-6 Astra, is a substantive historical claim; given the paper's own caution in Appendix C.3 that AI agreement is not evidence of correctness, I recommend the author state explicitly how this identification was verified against the original source (e.g., a page-level check of [33, pp. 203–204]) before publication.
[§5.4] The statement of Lemma 5.5 separates the bound (5.11) from the uniformity clause ("The bound is uniform in the fixed reserve prime q in the first case and in the fixed candidate n in the second"); merging the uniformity clause into the displayed statement would make it harder for a reader to miss this essential range of validity.

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The paper

The paper claims

One residue class per prime $p \le x$ can cover an interval of length $\gg x\log x/((\log_2 x)\log_3 x)$, beating the Erdős–Rankin scale by an unbounded factor.

The reading

The paper claims that one residue class modulo each prime $p \le x$ can cover an interval of length $\gg x\log x/((\log_2 x)\log_3 x)$, exceeding the classical Erdős–Rankin scale by the unbounded factor $(\log_2 x)/(\log_3 x)^2$. This is an improvement over Rankin's bound from 1938, obtained with the prime number theorem and Mertens' reciprocal-prime formula as the only external inputs. The engine is the tilted preliminary sieve: residue classes are biased toward zero so that the survival probability of a squarefree composite candidate takes the clean form $A n^{-\tau}$, and inverse-probability weights then cover whole groups of composite survivors together, with no Maynard sieve weights or hypergraph covering theorem required. The result is weaker than the strongest known bounds, but it isolates what the tilt alone contributes. The paper is explicit that the theorem and proof emerged from an AI-assisted dialogue and were verified by the author.

What carries the argument

The machine is the power tilt, a biased distribution on residue classes for each preliminary prime $w < p \le v$: the probability of choosing a nonzero class is adjusted so that the survival ratio of a multiple to a nonmultiple of $p$ is $\lambda_p = p^{-\tau}$ with $\tau = t/\log x$. For squarefree $w$-rough integers this yields the multiplicative survival law $\mathbb{P}(n \in \mathcal{N}) = A n^{-\tau}$, with $A = \prod_{w<p\le v} B_p$ of Mertens scale, and for candidate sets $\mathcal{D}$ a joint survival estimate $\Pi(\mathcal{D}) = A^k \operatorname{rad}(\prod n)^{-\tau} \exp(O(k^2\log H/(w\log w)))$ whose error is crushed by the large choice $w = (\log x)^{100}$. Two further components carry the argument: correlation estimates showing that the shared-prime factor $G^{\tau}$ between two candidate groups has mean $1 + O((\log x)^{-90})$ under either relevant average, and inverse-probability weighting $X_{q,a} = \mathbf{1}\{E_{q,a} \subseteq \mathcal{N}\}/\Pi(E_{q,a})$ combined with a second-moment method, showing that almost all composite survivors are covered by groups modulo the reserve primes $\mathcal{R}$ while primes and exceptions are mopped up individually.

Extended reading notes

Core claim

The central claim is Theorem 1.1: for all sufficiently large $x$, $Y(x) \gg x\log x/((\log_2 x)\log_3 x)$, where $Y(x)$ is the length of the longest interval coverable by choosing one residue class modulo each prime $p \le x$. By the Chinese remainder theorem such a covering produces a run of consecutive composite numbers, so the bound transfers to prime gaps: $G(X) \gg (\log X)(\log_2 X)/((\log_3 X)\log_4 X)$. The decisive step is a bias in the preliminary residue choices, the power tilt $\lambda_p = p^{-\tau}$ with $\tau = t/\log x$, which gives every squarefree $w$-rough composite candidate the survival probability $A n^{-\tau}$ with $A$ of Mertens scale. Joint survival of candidate groups is controlled by the radical of their product, with an error made negligible by the huge choice $w = (\log x)^{100}$; correlations between distinct groups contribute only the factor $G^{\tau}$ from shared primes, which averages to $1 + O((\log x)^{-90})$. Tuning $t = 2\log_3 x + 2\log_4 x$ makes the expected number of uncovered composites $o(x/\log x)$, while the prime survivors stay below the supply of final reserve primes, so the covering is completed by mopping up. The gain over Rankin, the factor $(\log_2 x)/(\log_3 x)^2$, already appears before any method for covering primes in groups is invoked.

Load-bearing premise

The argument hinges on the joint-survival estimate and its crude uniform bound on how many large prime factors the candidates in a group can share; if the most structured groups defeat that bound, the second-moment control fails and the covering runs short of final reserve primes.

Editorial extensions

If this is right

The covering bound transfers to the largest gap between consecutive primes: $G(X) \gg (\log X)(\log_2 X)/((\log_3 X)\log_4 X)$ for all sufficiently large $X$.
Rankin's scale is exceeded by the unbounded factor $(\log_2 x)/(\log_3 x)^2$, so the improvement grows without limit even though the bound stays below the strongest known covering scales.
Composite survivors are reclassified from an exceptional set to be removed into a resource: one residue class modulo a reserve prime covers an entire group of them at once.
The improvement appears before any covering of primes in groups, isolating the tilt's contribution from Maynard sieve weights and hypergraph covering machinery.
The argument is self-contained, using only the prime number theorem and Mertens' reciprocal-prime formula as external analytic inputs.

Reading between the lines

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

The tilt parameter $t$ and the split of reserve primes between group covering and mopping up leave a tradeoff the paper explores along one curve; joint optimization of $t$ with the reserve split might push this elementary scale further without adding prime-group machinery.
The joint-survival estimate is the fragile link: a sharper replacement for the crude divisor-count inequality of Lemma 5.1 would tighten the error terms and could allow the interval length to grow, likely extending the same scheme toward the stronger scales.
Because the tilt's contribution is isolated, one could recombine it with Maynard weights or a hypergraph step and test numerically whether the composite-group step survives at the longer intervals used by the stronger constructions.
The choice $w = (\log x)^{100}$ is deliberately extravagant; testing whether much smaller exponents preserve the correlation bounds would reveal how much slack the construction carries.

Original abstract

We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \cite{GPT2026}, showing that one residue class modulo each prime $p \le x$ can cover an interval of length \begin{equation*} \gg \frac{x\log x}{(\log_{2} x)\log_{3} x}. \end{equation*} This improves the classical Erd\H{o}s--Rankin bound by a factor of $(\log_{2} x)/(\log_{3} x)^{2}$. Although weaker than the strongest known bounds, it shows what the tilt and its associated covering of composite survivors achieve without Maynard sieve weights or a hypergraph covering theorem. The proof uses the prime number theorem, Mertens' reciprocal-prime formula, and elementary probability. The appendices provide a historical survey and self-contained proofs of the classical Erd\H{o}s--Rankin bound.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-24: 1 request, 4,414 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.96 · human participants 0.02 · controlled comparison 0.16 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.98 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \cite{GPT2026}, showing that one residue class modulo each prime $p \le x$ can cover an interval of length \begin{equation*} \gg \frac{x\log x}{(\log_{2} x)\log_{3} x}. \end{equation*} This improves the classical Erd\H{o}s--Rankin bound by a factor of $(\log_{2} x)/(\log_{3} x)^{2}$. Although weaker than the strongest known bounds, it shows what the tilt and its associated covering of composite survivors achieve without Maynard sieve weights or a hypergraph covering theorem. The proof uses the prime number theorem, Mertens' reciprocal-prime formula, and elementary probability. The appendices provide a historical survey and self-contained proofs of the classical Erd\H{o}s--Rankin bound.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \cite{GPT2026}, showing that one residue class modulo each prime $p \le x$ can cover an interval of length \begin{equation*} \gg \frac{x\log x}{(\log_{2} x)\log_{3} x}.	contribution 0.55	0.66	0.79	asserted 1.00
s1	\end{equation*} This improves the classical Erd\H{o}s--Rankin bound by a factor of $(\log_{2} x)/(\log_{3} x)^{2}$.	result 0.79	0.74	0.52	asserted 1.00
s2	Although weaker than the strongest known bounds, it shows what the tilt and its associated covering of composite survivors achieve without Maynard sieve weights or a hypergraph covering theorem.	limitation 0.66	0.61	0.09	asserted 1.00
s3	The proof uses the prime number theorem, Mertens' reciprocal-prime formula, and elementary probability.	method 1.00	0.71	0.02	asserted 1.00
s4	The appendices provide a historical survey and self-contained proofs of the classical Erd\H{o}s--Rankin bound.	contribution 0.76	0.56	0.02	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (40)

[16]

GPT-5.6Sol. “Atiltedresidue-classconstructionforlongprime-freeintervals.” Preprint,48pp.,25August2026; archived atweb.archive.org. Lean 4 formalization by B. Alexeev atgithub.com/plby/lean-proofs

View on Pith
[15]

Ford, K., B. Green, S. Konyagin, J. Maynard and T. Tao. “Long gaps between primes.”J. Amer. Math. Soc. 31(1):65–105, 2018.doi:10.1090/jams/876

View on Pith doi:10.1090/jams/876 2018
[1]

Backlund, R. J. “Über die Differenzen zwischen den Zahlen, die zu den𝑛 ersten Primzahlen teilerfremd sind.” Ann. Acad. Sci. Fenn. Ser. A 32(2):1–9, 1929.JFM:55.0687.01

View on Pith 1929
[2]

Banks, W., K. Ford and T. Tao. “Large prime gaps and probabilistic models.”Invent. Math. 233(3):1471–1518,

View on Pith
[3]

Brauer, A. and H. Zeitz. “Über eine zahlentheoretische Behauptung von Legendre.”Sitzungsber. Berliner Math. Ges. 29:116–125, 1930.JFM:56.0156.02

View on Pith 1930
[4]

Brun, V. “Le crible d’Eratosthène et le théorème de Goldbach.”Videnskapsselskapets Skrifter. I. Mat.-naturv. Klasse, 1920, no. 3, pp. 1–36.https://archive.org/details/lecriblederatost00brun

View on Pith 1920
[5]

Constantinescu, P. “Large gaps between primes.” First-year project, London School of Geometry and Number Theory, 2018, 28 pp.petruconstantinescu.github.io/large_gaps.pdf

View on Pith 2018
[6]

Cramér, H. “On the order of magnitude of the difference between consecutive prime numbers.”Acta Arith. 2(1):23–46, 1936.doi:10.4064/aa-2-1-23-46

View on Pith doi:10.4064/aa-2-1-23-46 1936
[7]

de Bruijn, N. G. “The asymptotic behaviour of a function occurring in the theory of primes.”J. Indian Math. Soc. (N.S.) 15:25–32, 1951

View on Pith 1951
[8]

de Bruijn, N. G. “On the number of positive integers ⩽ 𝑥 and free of prime factors > 𝑦 .” Proc. Kon. Ned. Akad. Wetensch. Ser. A 54(1):50–60, 1951.doi:10.1016/S1385-7258(51)50008-2

View on Pith doi:10.1016/s1385-7258(51)50008-2 1951
[9]

Dirichlet, P. G. L. “Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält.” 1837. Reprinted in R. Dedekind (ed.),Vorlesungen über Zahlentheorie, pp. 342–359, 1879; reissued by Cambridge University Press, Cambridge, 2013,doi:...

View on Pith doi:10.1017/cbo9781139237321.012 2013
[10]

Dupré, A.Examen d’une proposition de Legendre relative à la théorie des nombres. Mallet-Bachelier, Paris, 1859. gallica.bnf.fr/ark:/12148/bpt6k939379s

View on Pith
[11]

Erdős Problems.“Proof claims.”Discussion thread for Erdős Problem #4, maintained by T. F. Bloom, accessed 13 September 2026; archived atweb.archive.org

View on Pith 2026
[12]

Erdős, P. “On the difference of consecutive primes.”Quart. J. Math. Oxford Ser. os-6(1):124–128, 1935. doi:10.1093/qmath/os-6.1.124

View on Pith doi:10.1093/qmath/os-6.1.124 1935
[13]

Erdős, P. “Some problems on number theory.” InProceedings of the Seventeenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1986),Congr. Numer.54:225–244,

View on Pith 1986
[14]

Ford, K., B. Green, S. Konyagin and T. Tao. “Large gaps between consecutive prime numbers.”Ann. of Math. (2) 183(3):935–974, 2016.doi:10.4007/annals.2016.183.3.4

View on Pith doi:10.4007/annals.2016.183.3.4 2016
[17]

Granville, A. “Harald Cramér and the distribution of prime numbers.”Scand. Actuar. J. 1995(1):12–28, 1995. doi:10.1080/03461238.1995.10413946

View on Pith arXiv 1995
[18]

Heilbronn, H. and E. Landau. “Bemerkungen zur vorstehenden Arbeit von Herrn Bochner.”Math. Z. 37:10–16,

View on Pith
[19]

Legendre, A.-M.Théorie des nombres. 3rd ed., vol. 2, Firmin Didot frères, Paris, 1830, pp. 71–79.doi:10.3931/e- rara-59737

View on Pith doi:10.3931/e-
[20]

Maier, H. “Primes in short intervals.”Michigan Math. J. 32(2):221–225, 1985.doi:10.1307/mmj/1029003189

View on Pith arXiv 1985
[21]

Maier, H. and C. Pomerance. “Unusually large gaps between consecutive primes.”Trans. Amer. Math. Soc. 322(1):201–237, 1990.doi:10.1090/S0002-9947-1990-0972703-X

View on Pith doi:10.1090/s0002-9947-1990-0972703-x 1990
[22]

Maier, H. and C. L. Stewart. “On intervals with few prime numbers.”J. Reine Angew. Math.608:183–199, 2007. doi:10.1515/CRELLE.2007.057

View on Pith doi:10.1515/crelle.2007.057 2007
[23]

Maynard, J. “Large gaps between primes.” Ann. of Math. (2) 183(3):915–933, 2016. doi:10.4007/annals.2016.183.3.3

View on Pith doi:10.4007/annals.2016.183.3.3 2016
[24]

Montgomery, H. L. and R. C. Vaughan. Multiplicative number theory I: Classical theory . Cam- bridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2006. doi:10.1017/CBO9780511618314 34 T. FREIBERG

View on Pith doi:10.1017/cbo9780511618314 2006
[25]

Nyman, B. and T. R. Nicely. “New prime gaps between1015 and 5 × 10 16.” J. Integer Seq. 6(3):Article 03.3.1, 6 pp., 2003.cs.uwaterloo.ca/journals/JIS/VOL6/Nicely/nicely2.html

View on Pith 2003
[26]

OpenAI. “Improved long gaps between primes.”Preprint, 8 pp., 2026; archived atweb.archive.org. Lean 4 formalization atgithub.com/openai/LongGapsBetweenPrimes. The paper attributes the proof to GPT-6 Astra

View on Pith 2026
[27]

Pintz, J. “Very large gaps between consecutive primes.” J. Number Theory 63(2):286–301, 1997. doi:10.1006/jnth.1997.2081

View on Pith arXiv 1997
[28]

Pippenger, N. and J. Spencer. “Asymptotic behavior of the chromatic index for hypergraphs.”Journal of Combinatorial Theory, Series A 51(1):24–42, 1989.doi:10.1016/0097-3165(89)90074-5

View on Pith doi:10.1016/0097-3165(89)90074-5 1989
[29]

Pomerance, C. “Two methods in elementary analytic number theory.” InNumber theory and applications , R. A. Mollin (ed.), pp. 135–161. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 265. Kluwer Academic Publishers, Dordrecht, 1989

View on Pith 1989
[30]

Ramaré, O. “Rankin’s trick and Brun’s sieve.” InExcursions in multiplicative number theory , pp. 251–258. Birkhäuser Advanced Texts Basler Lehrbücher, Birkhäuser, Cham, 2022.doi:10.1007/978-3-030-73169-4_25

View on Pith doi:10.1007/978-3-030-73169-4_25 2022
[31]

Rankin, R. A. “The difference between consecutive prime numbers.”J. London Math. Soc. s1-13(4):242–247,

View on Pith
[32]

Rankin, R. A. “The difference between consecutive prime numbers V.”Proc. Edinburgh Math. Soc. (2) 13(4):331–332, 1963.doi:10.1017/S0013091500025633

View on Pith doi:10.1017/s0013091500025633 1963
[33]

Ricci, G. “Ricerche aritmetiche sui polinomi, II. (Intorno a una proposizione non vera di Legendre).” Rend. Circ. Mat. Palermo 58:190–208, 1934.doi:10.1007/BF03019710

View on Pith doi:10.1007/bf03019710 1934
[34]

Schönhage, A. “Eine Bemerkung zur Konstruktion grosser Primzahllücken.”Arch. Math. 14:29–30, 1963. doi:10.1007/BF01234916

View on Pith doi:10.1007/bf01234916 1963
[35]

Westzynthius, E. “Über die Verteilung der Zahlen, die zu den𝑛 ersten Primzahlen teilerfremd sind.” Comment. Phys.-Math. 5(25):1–37, 1931.Zbl:0003.24601

View on Pith arXiv 1931
[36]

Zeitz, H.Elementare Betrachtung über eine zahlentheoretische Behauptung von Legendre . Privatdruck, Berlin, 1930. A TYRO’S APPROACH TO THE TILTED SIEVE 35 Appendix A. The classical Erdős–Rankin construction We give a detailed proof of the classical Erdős–Rankin bound, using an elementary smooth- number estimate and a greedy choice of residue classes. We f...

View on Pith 1930
[1933]

doi:10.1007/BF01474553

View on Pith doi:10.1007/bf01474553
[1938]

doi:10.1112/jlms/s1-13.4.242

View on Pith doi:10.1112/jlms/s1-13.4.242
[1986]

users.renyi.hu/~p_erdos/1986-16.pdf

View on Pith 1986
[2023]

doi:10.1007/s00222-023-01199-0

View on Pith doi:10.1007/s00222-023-01199-0

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[1] [16]

GPT-5.6Sol. “Atiltedresidue-classconstructionforlongprime-freeintervals.” Preprint,48pp.,25August2026; archived atweb.archive.org. Lean 4 formalization by B. Alexeev atgithub.com/plby/lean-proofs

View on Pith

[2] [15]

Ford, K., B. Green, S. Konyagin, J. Maynard and T. Tao. “Long gaps between primes.”J. Amer. Math. Soc. 31(1):65–105, 2018.doi:10.1090/jams/876

View on Pith doi:10.1090/jams/876 2018

[3] [1]

Backlund, R. J. “Über die Differenzen zwischen den Zahlen, die zu den𝑛 ersten Primzahlen teilerfremd sind.” Ann. Acad. Sci. Fenn. Ser. A 32(2):1–9, 1929.JFM:55.0687.01

View on Pith 1929

[4] [2]

Banks, W., K. Ford and T. Tao. “Large prime gaps and probabilistic models.”Invent. Math. 233(3):1471–1518,

View on Pith

[5] [3]

Brauer, A. and H. Zeitz. “Über eine zahlentheoretische Behauptung von Legendre.”Sitzungsber. Berliner Math. Ges. 29:116–125, 1930.JFM:56.0156.02

View on Pith 1930

[6] [4]

Brun, V. “Le crible d’Eratosthène et le théorème de Goldbach.”Videnskapsselskapets Skrifter. I. Mat.-naturv. Klasse, 1920, no. 3, pp. 1–36.https://archive.org/details/lecriblederatost00brun

View on Pith 1920

[7] [5]

Constantinescu, P. “Large gaps between primes.” First-year project, London School of Geometry and Number Theory, 2018, 28 pp.petruconstantinescu.github.io/large_gaps.pdf

View on Pith 2018

[8] [6]

Cramér, H. “On the order of magnitude of the difference between consecutive prime numbers.”Acta Arith. 2(1):23–46, 1936.doi:10.4064/aa-2-1-23-46

View on Pith doi:10.4064/aa-2-1-23-46 1936

[9] [7]

de Bruijn, N. G. “The asymptotic behaviour of a function occurring in the theory of primes.”J. Indian Math. Soc. (N.S.) 15:25–32, 1951

View on Pith 1951

[10] [8]

de Bruijn, N. G. “On the number of positive integers ⩽ 𝑥 and free of prime factors > 𝑦 .” Proc. Kon. Ned. Akad. Wetensch. Ser. A 54(1):50–60, 1951.doi:10.1016/S1385-7258(51)50008-2

View on Pith doi:10.1016/s1385-7258(51)50008-2 1951

[11] [9]

Dirichlet, P. G. L. “Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält.” 1837. Reprinted in R. Dedekind (ed.),Vorlesungen über Zahlentheorie, pp. 342–359, 1879; reissued by Cambridge University Press, Cambridge, 2013,doi:...

View on Pith doi:10.1017/cbo9781139237321.012 2013

[12] [10]

Dupré, A.Examen d’une proposition de Legendre relative à la théorie des nombres. Mallet-Bachelier, Paris, 1859. gallica.bnf.fr/ark:/12148/bpt6k939379s

View on Pith

[13] [11]

Erdős Problems.“Proof claims.”Discussion thread for Erdős Problem #4, maintained by T. F. Bloom, accessed 13 September 2026; archived atweb.archive.org

View on Pith 2026

[14] [12]

Erdős, P. “On the difference of consecutive primes.”Quart. J. Math. Oxford Ser. os-6(1):124–128, 1935. doi:10.1093/qmath/os-6.1.124

View on Pith doi:10.1093/qmath/os-6.1.124 1935

[15] [13]

Erdős, P. “Some problems on number theory.” InProceedings of the Seventeenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1986),Congr. Numer.54:225–244,

View on Pith 1986

[16] [14]

Ford, K., B. Green, S. Konyagin and T. Tao. “Large gaps between consecutive prime numbers.”Ann. of Math. (2) 183(3):935–974, 2016.doi:10.4007/annals.2016.183.3.4

View on Pith doi:10.4007/annals.2016.183.3.4 2016

[17] [17]

Granville, A. “Harald Cramér and the distribution of prime numbers.”Scand. Actuar. J. 1995(1):12–28, 1995. doi:10.1080/03461238.1995.10413946

View on Pith arXiv 1995

[18] [18]

Heilbronn, H. and E. Landau. “Bemerkungen zur vorstehenden Arbeit von Herrn Bochner.”Math. Z. 37:10–16,

View on Pith

[19] [19]

Legendre, A.-M.Théorie des nombres. 3rd ed., vol. 2, Firmin Didot frères, Paris, 1830, pp. 71–79.doi:10.3931/e- rara-59737

View on Pith doi:10.3931/e-

[20] [20]

Maier, H. “Primes in short intervals.”Michigan Math. J. 32(2):221–225, 1985.doi:10.1307/mmj/1029003189

View on Pith arXiv 1985

[21] [21]

Maier, H. and C. Pomerance. “Unusually large gaps between consecutive primes.”Trans. Amer. Math. Soc. 322(1):201–237, 1990.doi:10.1090/S0002-9947-1990-0972703-X

View on Pith doi:10.1090/s0002-9947-1990-0972703-x 1990

[22] [22]

Maier, H. and C. L. Stewart. “On intervals with few prime numbers.”J. Reine Angew. Math.608:183–199, 2007. doi:10.1515/CRELLE.2007.057

View on Pith doi:10.1515/crelle.2007.057 2007

[23] [23]

Maynard, J. “Large gaps between primes.” Ann. of Math. (2) 183(3):915–933, 2016. doi:10.4007/annals.2016.183.3.3

View on Pith doi:10.4007/annals.2016.183.3.3 2016

[24] [24]

Montgomery, H. L. and R. C. Vaughan. Multiplicative number theory I: Classical theory . Cam- bridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2006. doi:10.1017/CBO9780511618314 34 T. FREIBERG

View on Pith doi:10.1017/cbo9780511618314 2006

[25] [25]

Nyman, B. and T. R. Nicely. “New prime gaps between1015 and 5 × 10 16.” J. Integer Seq. 6(3):Article 03.3.1, 6 pp., 2003.cs.uwaterloo.ca/journals/JIS/VOL6/Nicely/nicely2.html

View on Pith 2003

[26] [26]

OpenAI. “Improved long gaps between primes.”Preprint, 8 pp., 2026; archived atweb.archive.org. Lean 4 formalization atgithub.com/openai/LongGapsBetweenPrimes. The paper attributes the proof to GPT-6 Astra

View on Pith 2026

[27] [27]

Pintz, J. “Very large gaps between consecutive primes.” J. Number Theory 63(2):286–301, 1997. doi:10.1006/jnth.1997.2081

View on Pith arXiv 1997

[28] [28]

Pippenger, N. and J. Spencer. “Asymptotic behavior of the chromatic index for hypergraphs.”Journal of Combinatorial Theory, Series A 51(1):24–42, 1989.doi:10.1016/0097-3165(89)90074-5

View on Pith doi:10.1016/0097-3165(89)90074-5 1989

[29] [29]

Pomerance, C. “Two methods in elementary analytic number theory.” InNumber theory and applications , R. A. Mollin (ed.), pp. 135–161. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 265. Kluwer Academic Publishers, Dordrecht, 1989

View on Pith 1989

[30] [30]

Ramaré, O. “Rankin’s trick and Brun’s sieve.” InExcursions in multiplicative number theory , pp. 251–258. Birkhäuser Advanced Texts Basler Lehrbücher, Birkhäuser, Cham, 2022.doi:10.1007/978-3-030-73169-4_25

View on Pith doi:10.1007/978-3-030-73169-4_25 2022

[31] [31]

Rankin, R. A. “The difference between consecutive prime numbers.”J. London Math. Soc. s1-13(4):242–247,

View on Pith

[32] [32]

Rankin, R. A. “The difference between consecutive prime numbers V.”Proc. Edinburgh Math. Soc. (2) 13(4):331–332, 1963.doi:10.1017/S0013091500025633

View on Pith doi:10.1017/s0013091500025633 1963

[33] [33]

Ricci, G. “Ricerche aritmetiche sui polinomi, II. (Intorno a una proposizione non vera di Legendre).” Rend. Circ. Mat. Palermo 58:190–208, 1934.doi:10.1007/BF03019710

View on Pith doi:10.1007/bf03019710 1934

[34] [34]

Schönhage, A. “Eine Bemerkung zur Konstruktion grosser Primzahllücken.”Arch. Math. 14:29–30, 1963. doi:10.1007/BF01234916

View on Pith doi:10.1007/bf01234916 1963

[35] [35]

Westzynthius, E. “Über die Verteilung der Zahlen, die zu den𝑛 ersten Primzahlen teilerfremd sind.” Comment. Phys.-Math. 5(25):1–37, 1931.Zbl:0003.24601

View on Pith arXiv 1931

[36] [36]

Zeitz, H.Elementare Betrachtung über eine zahlentheoretische Behauptung von Legendre . Privatdruck, Berlin, 1930. A TYRO’S APPROACH TO THE TILTED SIEVE 35 Appendix A. The classical Erdős–Rankin construction We give a detailed proof of the classical Erdős–Rankin bound, using an elementary smooth- number estimate and a greedy choice of residue classes. We f...

View on Pith 1930

[37] [1933]

doi:10.1007/BF01474553

View on Pith doi:10.1007/bf01474553

[38] [1938]

doi:10.1112/jlms/s1-13.4.242

View on Pith doi:10.1112/jlms/s1-13.4.242

[39] [1986]

users.renyi.hu/~p_erdos/1986-16.pdf

View on Pith 1986

[40] [2023]

doi:10.1007/s00222-023-01199-0

View on Pith doi:10.1007/s00222-023-01199-0