On the largest prime factors less than $y$ of consecutive shifted primes

Zhiyuan Yang

REVIEW

On the largest prime factors less than $y$ of consecutive shifted primes

Zhiyuan Yang

Number Theory · Submitted 14 Sep 2026

Pith's verdict: A credible, significant small-y analogue of Wang's shifted-prime conjecture, with one compressed uniformity step that needs to be written out before the 1/2 density can be taken as fully proved.

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arXiv ID2609.15166 (version 2)

Pith ID2ZZLX6EA, Pith's permanent record of this paper and its review

CitationZhiyuan Yang. "On the largest prime factors less than $y$ of consecutive shifted primes." arXiv:2609.15166 (2026). https://pith.science/paper/2ZZLX6EA

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The main news: if the compressed estimates in Section 3.2 survive a careful write-up, this paper settles the small-y form of Wang's shifted-prime conjecture and gives the first unconditional positive proportion for shifted primes with y = x^alpha up to alpha < 17/32. The headline results are genuinely new, and the architecture of the proof is sensible. This is not a marginal tweak; the exact 1/2 density in the y << x^{o(1)} range is a real step beyond Rivat and Wang.

The paper does several things well. Case 1 (y <= log log x) is solid: the divisor set is small, Siegel-Walfisz suffices, and the main-term computation checks out. The S_A estimate in Case 2 is also fine: inclusion-exclusion over primes in (y^c, y], truncated at even J, with Bombieri-Vinogradov errors that are negligible because all moduli are at most y^J << x^{o(1)}. The same is true of the BV range throughout; there is no modulus-size crisis.

The soft spot is exactly where your stress-test note points. The S_B bound leading to (3.12) uses the phrase 'similar to S_A' for counts where the modulus includes p1...pj and the outer condition concerns (p+1, P(y^c, p1)) or (p+1, P(p1, y)). Those inclusion-exclusion sums are over primes up to p1, and the error terms must be uniform in p1...pj. The paper states no such uniformity. I do not think this is a demonstrated contradiction: the relevant moduli are still x^{o(1)}, so Bombieri-Vinogradov could plausibly control the summed error, and the alternating cancellation is over j, not over the error terms. But because the main term cancels down to order one, the o(1) in those 'similar to S_A' estimates is load-bearing. This needs to be written out, not asserted.

Two smaller issues. Theorem 1.5 is advertised and then dismissed as 'an easy corollary'; for a stated main result, that is not enough, even if the proof really is simpler. And there are typographical errors in the theorem statements, including missing parentheses and braces; they are cosmetic but should be cleaned up.

The citation pattern is normal: Banks-Shparlinski, Halberstam-Richert, Wang, Rivat, and the author's own preprints appear as background rather than as load-bearing external machinery. I saw no circularity and no invented entities.

My bottom line: this is a serious paper that deserves a rigorous referee, not a desk reject. The referee should require a full proof of the S_B uniformity estimate and at least a sketch of Theorem 1.5. If those come back clean, the results are substantial and worth publishing.

Referee report

Major objections (3)

[Section 3.2, equations (3.9)–(3.12)] The estimate of S_B is the load-bearing step for the exact 1/2 constant. The derivation imports uniform asymptotics for the counts #{p: p_1…p_j | p−1, (p+1,P(y^c,y))>1} and #{p: p_1…p_j | p−1, (p+1,P(p_1,y))>1} with the phrase 'similar to S_A'. However, S_A itself was estimated under the assumption that the modulus q = p_1…p_j does not appear in the congruence conditions for p+1. When ℓ | q, the two congruence conditions ℓ | p+1 and q | p−1 are not independent, and the Bombieri–Vinogradov error terms must be summed over moduli ℓq, producing a dependence on the tuple (p_1,…,p_j). The paper gives no uniform error bound that is independent of the specific primes p_1,…,p_j. Because the alternating sum over j relies on cancellation down to order one, a non-uniform error of relative size δ with δ not o(1) could change the 1/2 constant. This gap must be addressed before the proof of Theorem 1.4 (and consequently Theorem 1.5) can be considered complete.
[Section 3.2, equation (3.6)] The function φ(p) for primes p is p−1, so the sum over p in (3.6) is ∑_{y^c < p ≤ y} 1/(p−1). The statement '∑_{p≤x} 1/φ(p) = log log x + C + O(1/log x)' is correct, but the transition to ∑ 1/(p−1) as ∑ 1/p + O(1/p^2) should be made explicit; the current notation may mislead readers into thinking the same constant appears.
[Section 4.4, equations (4.16)–(4.17)] The derivation of the final bound (4.17) for S_B assumes that the asymptotic formula for ∑_{d≤y, (d,a)=1} H_1(d) is uniform across the range of a and b. The functions H_1 and H are multiplicative but the averaging over a and b is performed after pulling out the main term; the authors should check that the error term in the partial summation (coming from the Selberg–Delange estimate) does not accumulate with the number of variables and remains o(π(x)) in the final sum.

Minor objections (6)

[Abstract and throughout] The phrase 'the lower bound is also true for the pattern P_y^+(p-1) > P_y^+(p+1)' appears in Theorems 1.4–1.5 but the statement as written is ambiguous. Clarify whether 'lower bound' refers to the asymptotic density from below or to the symmetric inequality.
[Section 1, lines after Conjecture 1.3] The sentence 'This problem seemly remains very difficult to solve' contains a typo: 'seemly' should be 'seemingly'.
[Section 3.1, equation (3.3)] The notation ∑_{2q|P(p′−1)} is not defined; it presumably means the sum over all squarefree q composed of primes < p′. Write this explicitly for clarity.
[Section 3.2, equation (3.7)] The inequality uses 'cε' on the right-hand side, but ε is introduced as a sufficiently small number; it is unclear whether ε here is the same as the one in (3.6) or a new parameter. Standardize the notation (e.g., replace cε with a new symbol).
[Section 4.2, equation (4.7) and following] The sum over p_1,l is written with 'x^{1−α}(log x)^{−B} < l p_1 ≤ x^{1−c}' and '2|l'. It would be helpful to state explicitly that the condition '2|l' comes from the parity of p−1 and p+1.
[Acknowledgements] The word 'Acknowledegements' is misspelled; should be 'Acknowledgements'.

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The paper

The paper claims

For very small thresholds y, the largest bounded prime factors of p-1 and p+1 are equally likely to be larger.

The reading

This paper proves that when the bound y is very small—growing slower than any fixed power of x—the proportion of primes p ≤ x for which the largest prime factor of p-1 below y is less than that of p+1 tends to exactly 1/2. This completely settles Wang's shifted-prime conjecture in this extremely small y regime. For the larger range where y = x^α with α < 17/32, the paper establishes a positive proportion (which tends to 1/2 as α → 0+), improving on earlier conditional results. The same balanced asymptotics also hold for consecutive integers in place of primes. These results advance the study of Erdős–Turán type comparisons of largest prime factors of consecutive numbers.

What carries the argument

The proof splits into several regimes. For very small y (y ≤ log log x and log log x < y << $x^{{o(1)}}$), it uses inclusion–exclusion on the prime factors of p-1 and p+1, exploiting the Bombieri–Vinogradov theorem and the Siegel–Walfisz theorem to handle the error terms. For y = x^α (α < 17/32), the argument relies on Selberg's upper-bound sieve (Lemma 2.1) to count primes in arithmetic progressions with multiple linear forms, combined with an estimate due to Banks and Shparlinski (Lemma 2.5) that gives a positive proportion of moduli q for which the prime counts are bounded below by a constant times π(x)/φ(q).

Extended reading notes

Core claim

Theorem 1.4 asserts that for any 3 ≤ y << $x^{{o(1)}}$, #{p ≤ x : P_y^+(p-1) < P_y^+(p+1)} ∼ π(x)/2. Theorem 1.6 extends this to y = x^α with 0 < α < 17/32, giving #{p ≤ x : P_y^+(p-1) < P_y^+(p+1)} ≥ (h(α)+o(1))π(x) for some h(α) > 0 that tends to 1/2 as α → 0+. The author also proves an analogue for consecutive integers (Theorem 1.5) and shows that both bounds and the lower bound are optimal up to the stated ranges.

Load-bearing premise

In the proof of Theorem 1.6, the positive proportion relies on Lemma 2.5 (Banks–Shparlinski), which guarantees that for most moduli q up to x^θ (θ<17/32), the prime count in arithmetic progression is at least a constant times π(x)/φ(q); if the exceptional set is larger than allowed, the lower bound would vanish.

Editorial extensions

If this is right

Wang's conjecture for shifted primes is fully verified when the threshold y is very small (y << x^{o(1)}); the asymptotic density is exactly 1/2.
For every α < 17/32, the set of primes with P_y^+(p-1) < P_y^+(p+1) has positive natural density, a first unconditional result of this kind for power-sized y.
The corresponding problem for consecutive integers (P_y^+(n) < P_y^+(n+1)) is solved for y << x^{o(1)} with density 1/2, and for power-sized y with a positive proportion tending to 1/2 as α→0.
The method provides a template for attacking similar comparison problems for other multiplicative functions of consecutive numbers.
The result suggests that the full Erdős–Turán conjecture for consecutive integers (without the y-truncation) might be approachable via a refined combination of sieve bounds and combinatorial decompositions.

Reading between the lines

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

The bound α < 17/32 is likely not optimal; with stronger sieve results or a refined treatment of exceptional moduli, the range could be extended toward 1/2 or beyond. The paper's reliance on Lemma 2.5, which holds up to 17/32, currently sets the barrier.
The techniques used here for the shifted prime problem are closely related to those that yield lower bounds for the twin prime problem; one might explore whether a similar approach can give unconditional positive density for P^+(p) < P^+(p+2).
The balanced asymptotic 1/2 for very small y, together with the known difficulty of the full conjecture for large y, suggests that the main obstacle lies in controlling primes that are large compared to y.

Original abstract

For an integer $n > 1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. 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 $1/2$. Furthermore, Wang conjectures that for $x\rightarrow\infty$, one has $\#\{p\leq x:P^+(p-1)<P^+(p+1)\}\sim \pi(x)/2$. In this paper, we show the following results. For any $3\leq y\ll x^{o(1)}$, for $x\rightarrow\infty$, we have \begin{align*} &\#\{n\leq x:P^+_y(n)<P^+_y(n+1)\}\sim \frac{1}{2}x, \\&\#\{p\leq x:P^+_y(p-1)<P^+_y(p+1)\}\sim \frac{1}{2}\pi(x), \end{align*} where the first result extends a previous range $3\leq y\ll \exp(\log x/(100\log\log x))$ due to Rivat (2001). For any $0<\alpha<17/32$, let $y=x^\alpha$. Then there exists $h(\alpha)>0$ such that \begin{align*} \#\{p\leq x:P_y^+(p-1)<P_y^+(p+1)\}\geq(h(\alpha)+o(1))\pi(x). \end{align*} In particular, the function $h$ satisfies $\lim_{\alpha\rightarrow 0^+}h(\alpha)=1/2$. Similar result also holds for $\#\{n\leq x:P_y^+(n)<P_y^+(n+1)\}$, which improves a previous result $\lim_{\alpha\rightarrow 0^+} h(\alpha)=1/4$ due to Wang (2019).

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, 2026-09-23: 1 request, 8,012 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.52 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.97 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

For an integer $n > 1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. 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 $1/2$. Furthermore, Wang conjectures that for $x\rightarrow\infty$, one has $\#\{p\leq x:P^+(p-1)<P^+(p+1)\}\sim \pi(x)/2$. In this paper, we show the following results. For any $3\leq y\ll x^{o(1)}$, for $x\rightarrow\infty$, we have \begin{align*} &\#\{n\leq x:P^+_y(n)<P^+_y(n+1)\}\sim \frac{1}{2}x, \\&\#\{p\leq x:P^+_y(p-1)<P^+_y(p+1)\}\sim \frac{1}{2}\pi(x). \end{align*} For any $0<\alpha<17/32$, let $y=x^\alpha$. Then there exists $h(\alpha)>0$ such that \begin{align*} \#\{p\leq x:P_y^+(p-1)<P_y^+(p+1)\}\geq(h(\alpha)+o(1))\pi(x). \end{align*} In particular, the function $h$ satisfies $\lim_{\alpha\rightarrow 0^+}h(\alpha)=1/2$. Similar result also holds for $\#\{n\leq x:P_y^+(n)<P_y^+(n+1)\}$. These improve Rivat's result (2001) and Wang's result (2019).

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	For an integer $n > 1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$.	background 1.00	0.05	0.01	asserted 0.75
s1	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 $1/2$.	background 1.00	0.02	0.02	speculative 0.78
s2	Furthermore, Wang conjectures that for $x\rightarrow\infty$, one has $\#\{p\leq x:P^+(p-1)<P^+(p+1)\}\sim \pi(x)/2$.	background 0.99	0.02	0.03	speculative 0.88
s3	In this paper, we show the following results.	contribution 0.99	0.51	0.03	asserted 0.67
s4	For any $3\leq y\ll x^{o(1)}$, for $x\rightarrow\infty$, we have \begin{align*} &\#\{n\leq x:P^+_y(n)<P^+_y(n+1)\}\sim \frac{1}{2}x, \\&\#\{p\leq x:P^+_y(p-1)<P^+_y(p+1)\}\sim \frac{1}{2}\pi(x).	result 0.99	0.96	0.85	asserted 1.00
s5	\end{align*} For any $0<\alpha<17/32$, let $y=x^\alpha$.	contribution 0.61	0.36	0.04	asserted 0.51
s6	Then there exists $h(\alpha)>0$ such that \begin{align*} \#\{p\leq x:P_y^+(p-1)<P_y^+(p+1)\}\geq(h(\alpha)+o(1))\pi(x).	result 1.00	0.93	0.21	asserted 1.00
s7	\end{align*} In particular, the function $h$ satisfies $\lim_{\alpha\rightarrow 0^+}h(\alpha)=1/2$.	result 0.87	0.87	0.12	asserted 1.00
s8	Similar result also holds for $\#\{n\leq x:P_y^+(n)<P_y^+(n+1)\}$.	result 0.97	0.86	0.07	asserted 0.96
s9	These improve Rivat's result (2001) and Wang's result (2019).	interpretation 0.72	0.50	0.09	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · 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

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S sections
T body sentences
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

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