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An improvement on the largest prime factors of consecutive integers

T0 review · 4 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The author proves that for large x, more than 28% of integers n below x have P^+(n) < P^+(n+1), and that a positive density satisfies the stronger condition P^+(n) < P^+(n+1) < x^{41/107+ε}.

desk verdict Plausible improvement of the Erdős–Turán lower density to 0.280, but the final constant depends on unreported numerical work and at least one explicitly guessed lemma; worth a serious referee, not acceptance as written. read the letter →

arxiv 2607.16032 v1 pith:GRKEOLS5 submitted 2026-07-17 math.NT

Zhiyuan Yang This is my paper · ORCID
classification math.NT MSC 11N2511N3611N05
keywords largestprimefactorconsecutiveintegersErdős–Turánconjecturelowerdensitysmoothnumbersarithmeticprogressionssievemethodsshiftedprimes
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 attacks a long-standing conjecture asserting that the largest prime factor of an integer is smaller than that of its successor exactly half of the time. Its main theorem pushes the known lower density from 0.2017 past 0.280 by optimizing a sieve-and-counting argument: it drops a lossy logarithmic sieve weight in the one-large-prime-factor case, introduces an auxiliary parameter t1 to extract a larger main term in the large-prime case, and takes the maximum of two lower bounds when the largest prime of n+1 sits just above the square-root threshold. The proof also yields a positive-density result for the stronger condition that both largest prime factors stay below x^{41/107+ε}, and a new upper bound on how often p−1 has a very large prime factor. If correct, the result shows that the one-sided pattern has density well above 28%, still far from the conjectured 50%.

What carries the argument

The proof's engine is a four-region decomposition of n by the sizes of P^+(n) and P^+(n+1), plus two innovations. First, in the region where n+1 has one moderately large prime factor, it abandons the logarithmic sieve weight that caused losses and instead counts n+1 by whether it has one or two required prime divisors, using a generalized Bombieri–Vinogradov-type theorem for smooth numbers and a linear sieve. Second, in the region where P^+(n+1)>x^{1/2}, it introduces a parameter t1 that shifts weight between a main term and an error term; choosing the largest t1 for which the error term stays nonnegative improves the main-term constant. The final number is the maximum of two lower bounds, o

What would settle it

Recompute C(c,δ1) at c=0.1348, δ1=0.417 by independently evaluating the integral in (4.16) over η∈[c,1/2], using any admissible values of C(1−η) on (0.55,0.8652]. Also test the claimed extension of Lemma 2.6 for l=d1p2 in the stated ranges; a single counterexample would invalidate the bound on S'5.

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Extended reading notes

Core claim

Theorem 1.4 asserts that as x→∞, the count of n<x with P^+(n)<P^+(n+1) exceeds 0.280x, and by symmetry the same holds for the reverse inequality. The constant comes from a decomposition of n according to the sizes of P^+(n) and P^+(n+1): a cutoff c=0.1348 separates small and large cases, and each piece is estimated either by the standard smooth-number distribution or by a new lower-bound function C(η) for smooth numbers in arithmetic progressions. With the choices δ1=0.417 and an optimized auxiliary function t1(η), the assembled lower bound C(c,δ1)−2ν is reported to exceed 0.280.

Load-bearing premise

The final inequality stands on an unstated numerical evaluation of an integral involving C(η) for η beyond the range where the paper computes C(η) explicitly, and on an assumed but unproved extension of a sieve estimate to composite moduli of the form d1p2; if either fails, the 0.280 figure may not follow.

Editorial extensions

If this is right

  • The lower density of the pattern P^+(n)<P^+(n+1) is greater than 0.280, and the same holds for the reverse pattern.
  • For every ε>0, a positive density of n satisfy the stronger condition P^+(n)<P^+(n+1)<x^{41/107+ε}.
  • For shifted primes, the upper limit of the proportion of primes p≤x with P^+(p−1)≥p^c is at most min(−(7/2)log c, (1−δ(c))/(2c)).
  • The effective version of the smooth-numbers-in-progressions bound can supply explicit constants for related friable-integer problems.
  • The new constant supersedes the previous 0.2017 record toward the conjectured 1/2 density.

Reading between the lines

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

  • The optimal function t1(η) is presented graphically and through numerical calculation rather than a closed formula; making the numerical values public would let others audit the 0.280 constant without re-running the full sieve argument.
  • The same t1-rebalancing trick may transfer to the k-term ordering version of the conjecture, where a k! density bound could be improved by an analogous parameter choice.
  • If the unproved extension of Lemma 2.6 to moduli of the form l=d1p2 fails, the bound on the error term S'5 could be larger, potentially pulling the final constant below 0.280; this is the most vulnerable spot of the numerical conclusion.
  • The min formula in Theorem 1.6 suggests that the two bounds cross: near c=1 the logarithmic term dominates, while near c=1/2 the (1−δ)/(2c) term dominates; a sharper δ(c) from the method's remark would further improve the shifted-prime bound.
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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

4 major / 3 minor

Summary. The paper addresses Erdős–Turán's conjecture on the largest prime factors of consecutive integers. It claims three main results: Theorem 1.4 gives a lower asymptotic density >0.280 for the pattern P^+(n)<P^+(n+1), improving the previous 0.2017 record of Lü–Wang; Theorem 1.5 gives a positive density of n with P^+(n)<P^+(n+1)<x^{41/107+ε}; and Theorem 1.6 gives an upper bound for T_c(x), the count of primes p≤x with P^+(p−1)≥p^c. The proof of Theorem 1.4 combines an Erdős–Pomerance-type decomposition, estimates for friable numbers in arithmetic progressions, the Rosser–Iwaniec sieve, and a new parameter t_1 intended to improve the main-term/error-term balance. The final constant is obtained by optimizing parameters c and δ_1 and by numerically evaluating a lower-bound integral involving C(1−η).

Significance. If fully supported, Theorem 1.4 would be a substantial improvement of the best known lower density for a long-standing Erdős–Turán problem, and Theorems 1.5 and 1.6 would provide useful complementary results. The paper builds on independent published lemmas by Hildebrand, Iwaniec, Pascadi, Wu, Ding–Wang, and Lü–Wang, and it introduces a plausible new parameter t_1 that may be of independent value. The main limitation is that the final numerical constant rests on several explicitly unproved steps: a guessed extension of Lemma 2.6, four identities left to the reader, and an integral involving C(1−η) outside the range where explicit values are provided. These gaps are load-bearing, so the significance is conditional on completing or replacing them.

major comments (4)
  1. [§4.4, Eq. (4.16) and definition of C(c,δ1)] The final numerical inequality C(c,δ1)−2ν>0.280 is the whole content of Theorem 1.4. The integral in (4.16) ranges over η∈[c,1/2], so it requires values of C(1−η) for arguments up to 0.8652. Theorem 3.2 gives explicit lower bounds for C(η) only for η∈(0.5,0.55); outside that range it asserts only the existence of an effective positive constant. The paper does not state how the maximum with the first branch was evaluated, nor how the integral over the full interval was computed. The phrase 'By numerical calculation' is not a reproducible derivation. Consequently the displayed constant 0.280 is not the conclusion of a finished proof.
  2. [§4.3, Eq. (4.19)] The upper bound for D, and hence the bounds for R and S'_5 in (4.20), use the assertion 'We guess that Lemma 2.6 also holds for l=d=d_1p_2 with the above conditions'. The validity of Lemma 2.6 for this composite l is load-bearing: without it the bound for D is unproved. A proof or a reference for this extension must be supplied, or the argument must be rewritten to avoid it.
  3. [§4.3, after Eq. (4.18)] The derivation of B_2 (and then R and S'_5) relies on four displayed identities whose verification is 'left to interested readers'. These identities connect the count of l=d_1p_2 with weighted divisor sums over p'. Since they feed directly into (4.20), they are not optional exercises: they need a proof in the paper or a reference.
  4. [§3.1, proof of Theorem 3.2] The proof of the C(η) lower bound for η near 0.55 is compressed: for the E_2 terms the text says 'Omitting the details', and Theorem 3.2 only promises an effective C(η) for η∈(0.55,0.8652) without giving values. Because (4.16) uses C(1−η) exactly in that non-explicit range, the numerical evaluation of the final integral requires either explicit lower bounds for C(η) over the whole range or a separate numerical scheme with stated error bounds.
minor comments (3)
  1. [Throughout] There are numerous typographical slips: 'aribitary', 'frist', 'pragh', 'satistying', 'Conlcuding', 'Möbius' rendering, and inconsistent notation S_B/SB and S_5/S′_5. These do not affect the mathematics but should be corrected.
  2. [§4.4 and Figures 1–3] The numerical values t_1(η), t_1(c), and δ(c) are presented through figures and 'numerical calculation'. For reproducibility, provide either exact expressions or a short script/table of the values used for c=0.1348, δ_1=0.417.
  3. [Lemma 4.1 proof] The proof says 'we claim that' a sieve upper bound holds, referring forward to (4.10)–(4.14). A brief indication of how Lemma 2.4 and Lemma 2.6 justify the claim would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof chains independent known lemmas; the numerical/optimization gaps are correctness issues, not reductions.

full rationale

The proof of Theorem 1.4 starts from the identity (4.1) and then combines external lemmas due to Hildebrand, Fouvry–Tenenbaum, Pascadi, Iwaniec, Ding–Wang, and the prior record of Lü–Wang [17]. The new parameter t1 is introduced as an optimization variable and is later constrained by the explicit inequality (4.15); the final choices c=0.1348 and δ1=0.417 are free optimization parameters, not quantities fitted to the target 0.280. The constant 0.280 is an output of the numerical evaluation, not an input to the definition of C(c,δ1). There are no load-bearing self-citations: reference [17] is a paper by different authors and [7] is external work by Ding and Wang. The manuscript does contain explicit gaps: the proof of Theorem 3.2 says 'Omitting the details' for the E2 terms; the final evaluation of the integral in (4.16) requires C(1−η) for arguments up to 0.8652 while Theorem 3.2 gives explicit lower bounds only for arguments in (0.5,0.55); and (4.19) relies on the unproved assertion 'We guess that Lemma 2.6 also holds for l=d=d1p2 with the above conditions, but this is not the key.' These are completeness/correctness concerns, not circularity: no equation is defined in terms of the target density, and no predicted quantity is a renamed fit or a self-cited uniqueness theorem. Hence the circularity score is 0.

Assumptions & free parameters 8 free parameters · 13 assumptions · 0 invented entities

The proof leans on deep published theorems (Hildebrand, Pascadi, Iwaniec, Wu, Ding–Wang) which are legitimate inputs. The load-bearing 'free' choices are the numerical constants c, δ1, s, t1(η), t2, ν that are optimized to make the final inequality exceed 0.280. Three ad hoc assumptions are not proved in the paper: the guessed Lemma 2.6 extension, the four displayed identities in §4.3, and the availability of effective C(η) values on the full range needed for (4.16).

free parameters (8)
  • c = 0.1348
    Splitting parameter in the S_A − S_B + S_C decomposition; chosen to maximize the final lower bound in §4.4.
  • δ1 = 0.417
    Threshold for the S'_1/S'_2 splitting in (4.2); chosen in §4.4 to maximize C(c,δ1).
  • s = 0.882
    Parameter in (4.19) controlling the B2 lower bound; set to 0.882 to obtain the coefficient 0.03249 g2(α,β) in §4.3.
  • t1(η) = function of η (Figure 2)
    New parameter in (4.15) chosen as the largest value satisfying S_B ≥ 0; used in (4.16) and the final integral.
  • t2 = auxiliary, chosen in (0, η1 − t1)
    Auxiliary parameter in (4.9) used to make S_B ≥ 0 via condition (4.15).
  • ν = 0.000001
    Tiny slack in several estimates (eq. 4.3, 4.16, 6.4); chosen small enough not to affect the final inequality.
  • t1(c), δ(c) = function of c (Figure 3)
    In Theorem 1.6, t1(c) is chosen as the largest value satisfying (6.4), giving δ = t1/(c+t1).
  • C(η) = integral expressions over E1, E2; explicit only for η∈(0.5,0.55)
    Lower-bound constant in (3.1); computed by counting p1p2ℓ and p1p1'p2ℓ products. For η > 0.55 only existence is stated, yet (4.16) integrates C(1−η) over a larger range.
assumptions (13)
  • standard math Hildebrand's Ψ(x,y) = xρ(u)(1+O(...)) uniform estimate (Lemma 2.1)
    Used throughout to estimate smooth numbers; cited [13, Theorem 1].
  • standard math Bombieri–Vinogradov for friable integers (Lemma 2.2)
    Used in §4.1 to estimate S'_1 and S'_2; cited [30] and [12, Theorem 6].
  • standard math Pascadi's smooth numbers in AP to modulus x^{66/107−ε} (Lemma 2.3)
    Central to Theorem 1.5 and Theorem 3.2; cited [19, Theorem 1.1].
  • standard math Rosser–Iwaniec linear sieve (Lemma 2.4)
    Used to bound sieve sums in §4.2 and §4.3; cited [14].
  • standard math Well-factorable distribution of primes in AP (Lemma 2.5)
    Used in remainder estimates; cited [17, Lemma 2.3] and [18, Theorem 2].
  • standard math Level x^{4/7−ε} well-factorable prime distribution (Lemma 2.6)
    Used in §4.3 remainder; cited [28, Prop 3.2] and [7, Lemma 2.2].
  • standard math Pascadi's Theorem 4.2 on 1-bounded sequences in AP (Lemma 3.4)
    Core of Theorem 3.2; imposes exponent constraints on u1,u2,u3.
  • standard math Pólya–Vinogradov inequality (Lemma 3.5)
    Bounding character sums in the proof of Theorem 3.2.
  • standard math Wu's asymptotic for T_c(x) (Lemma 6.1)
    Basis of Theorem 1.6; cited [31, Theorem 2].
  • standard math Selberg–Delange estimate for sums of H(d) (La Bretèche–Pomerance–Tenenbaum)
    Used to evaluate the main term in (4.13).
  • ad hoc to paper Guessed extension of Lemma 2.6 to l = d1 p2 in §4.3
    Author states 'We guess that Lemma 2.6 also holds for l=d=d1p2 with the above conditions, but this is not the key.' Used for upper bound on D in (4.19).
  • ad hoc to paper Four unproved displayed identities in §4.3 ('verification is left to interested readers')
    Used to derive the B'_2 lower bound and the coefficient 1/2 g2(α,β); no proof or citation given.
  • ad hoc to paper Existence and computability of C(η) for η∈(0.55,0.8652) used in (4.16)
    The final numerical constant in §4.4 requires C(1−η) over the full range, but Theorem 3.2 only gives explicit values for η∈(0.5,0.55).

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Pith. "Pith review of An improvement on the largest prime factors of consecutive integers." pith.science (2026). https://pith.science/paper/GRKEOLS5

@misc{pith2026260716032,
  author       = {Pith},
  title        = {Pith review of: An improvement on the largest prime factors of consecutive integers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRKEOLS5}},
  note         = {Machine review of arXiv:2607.16032}
}
abstract

Let $P^+(n)$ denote the largest prime factor of $n$. One of Erd\H{o}s and Tur\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is 1/2. In this paper, we prove that this density is larger than 0.280, which improves the previous result 0.2017 by L\"u and Wang (2025). We also prove that there exists a positive density of $n$ such that $P^+(n)<P^+(n+1)<x^{41/107+\varepsilon}$. Define $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$. For $1/2<c<1$, we also show that \begin{align*} \mathop{\lim \sup}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}\leq \min\left(-\frac{7}{2}\log c,\frac{1-\delta}{2c}\right), \end{align*} where $\delta=\delta(c)>0$.

Figures

Figures reproduced from arXiv: 2607.16032 by the authors.

Figure 1
Figure 1. c → δ(c) In particular, we can obtain a better value for δ by following the argument in Remark 6.2. 2. Lemmas Lemma 2.1. For ε > 0, we have Ψ(x, y) = X n≤x,P +(n)≤y 1 = xρ(u)  1 + Oε  log(u + 1) log y  uniformly for x ≥ x0(ε), exp (log2 x) 5/3+ε ≤ y ≤ x, where u = log x/ log y and ρ(u) is the Dickman-de Bruijn function which is defined by differential equation ( ρ(u) = 1, 0 ≤ u ≤ 1, uρ′ (u) = −ρ(u − 1) u > 1. Pr… view at source ↗
Figure 2
Figure 2. η → t(η) = t1(1 − η) However, when η1 ≈ 0.5, the lower bound of SgC1 above is not a good result. Note that X x 1−ηi<p<x1−ηi+1 X dp≤x 1 − X x 1−η1<p′<x X x 1−ηi<p<x1−ηi+1 X dp≤x dp−1≡0 (mod p ′ ) 1 ≥ X x 1−ηi<p<x1−ηi+1 X n≤x−1,P +(n)≤x 1/2 p|n+1 1. After applying Theorem 3.2, we get S ′ 3 + S ′ 4 ≥ Z 1/2 c max  1 − 2η + 2t1(η) 2(1 − η)(1 − η + t1(η)), C(1 − η) 1 − η  dη − ν + o(1)! x. (4.16) where the C(·) is the C… view at source ↗
Figure 3
Figure 3. c → t1(c) By Lemma 6.1, (6.2) and D ≥ 0, for 1/2 < c < 1, we have Tc(x) ≤ 1 2(c + t1) x log x + o  x log x  [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗

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