Positive density for Sun's $2^k+m$ conjecture

Jinbo Yu; Songlin Han

arxiv: 2605.15758 · v1 · pith:ZMLJHR4Snew · submitted 2026-05-15 · 🧮 math.NT

Positive density for Sun's 2^k+m conjecture

Songlin Han , Jinbo Yu This is my paper

Pith reviewed 2026-05-19 22:06 UTC · model grok-4.3

classification 🧮 math.NT

keywords Sun conjecturepositive densityprime representationpower of twosieve estimatesanalytic number theoryRomanov-type problem

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The pith

Natural numbers n that can be written as n = k + m with 2^k + m prime have positive asymptotic density at least 0.0734.

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

The paper proves unconditionally that a positive proportion of natural numbers greater than 1 can be expressed as n equals k plus m for positive integers k and m such that 2 to the power k plus m is a prime. This shows that Sun's 2013 conjecture holds for a large set of n, even if not yet for all of them. The lower bound of 0.0734 on the density is obtained through analytic and sieve estimates that count the numbers with at least one valid representation. The authors also establish an upper limit of roughly 0.5906 for any bound obtainable by their method under the uniform Hardy-Littlewood prime-pairs conjecture.

Core claim

We unconditionally prove that the natural numbers satisfying this property have a positive density. We compute this density to be at least 0.0734.

What carries the argument

Analytic and sieve estimates that produce a strictly positive lower bound on the count of n with at least one valid k and m.

If this is right

Infinitely many natural numbers admit such a representation n = k + m.
The full conjecture holds on a set of positive density.
The same method cannot yield a density bound larger than 1/(log 2 + 1) even under the uniform Hardy-Littlewood conjecture.
Further improvements to the unconditional lower bound require stronger estimates than those applied here.

Where Pith is reading between the lines

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

Numerical checks of the proportion up to moderate X could confirm whether the actual density lies well above 0.0734.
The positive-density result makes it plausible that exceptions, if any, are sparse and perhaps finite.
Similar sieve arguments might apply to other Romanov-type problems involving fixed powers plus a linear term.

Load-bearing premise

The analytic or sieve estimates used to obtain the unconditional lower bound of 0.0734 are valid and produce a strictly positive quantity.

What would settle it

A direct count of qualifying n up to 10^9 or larger showing the proportion falling below 0.01 and continuing to decrease.

read the original abstract

In 2013, Zhi-Wei Sun proposed a Romanov-type conjecture stating that every integer $n > 1$ can be written as $n = k + m$ with $k, m \ge 1$ such that $2^k + m$ is a prime. In this paper, we unconditionally prove that the natural numbers satisfying this property have a positive density. We compute this density to be at least $0.0734$. We also discuss the limitations of our method. Under a uniform Hardy-Littlewood prime pairs conjecture, we show that the lower bound of density obtained by this method cannot exceed $1/(\log 2 + 1) \approx 0.5906$.

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

Han and Yu get an unconditional positive lower density of 0.0734 for the set of n admitting n = k + m with 2^k + m prime, via sieves, but the explicit numerical bound needs checking against error terms.

read the letter

Hi, The key takeaway is that Han and Yu prove that the integers n for which there exists k and m with n = k + m and 2^k + m prime have lower asymptotic density at least 0.0734. This is unconditional and gives a concrete positive proportion for Sun's conjecture. What is new is the unconditional positive-density statement with an explicit lower bound. The literature on similar Romanov problems usually requires conjectures for density results, so this stands out as a step that avoids that. They also include a discussion of the method's limitations under the Hardy-Littlewood conjecture, capping the achievable density at roughly 0.59, which adds context. The paper does well in applying combinatorial sieves to control the exceptional set where 2^k + m is composite for the relevant k. The numerical bound suggests they optimized over the sifting parameters to maximize the surviving density. The potential soft spot is whether the computed lower bound of 0.0734 remains strictly positive once all error terms from the sieve are included. The optimization over the level D and the range of k could make the main term small, and one needs rigorous bounds showing the remainder does not overwhelm it. If the paper provides explicit constants or a verifiable computation for the product over primes, that would strengthen it. Otherwise, it might be worth checking if a slightly smaller bound is safer. The central claim looks plausible because the sieve is set up to show the bad set has density bounded away from 1. This work is aimed at researchers in analytic number theory interested in additive representations involving primes and powers of 2. A reader who follows sieve applications to conjectures like this would get value from the explicit constant and the limitation analysis. I think it deserves a serious referee. The result is substantive enough to warrant detailed review rather than a quick desk rejection.

Referee Report

2 major / 2 minor

Summary. The paper claims to unconditionally prove that the set of natural numbers n that admit a representation n = k + m (k, m ≥ 1) with 2^k + m prime has positive lower density, with an explicit lower bound of 0.0734 obtained via a combinatorial sieve. It further shows that the same method, under a uniform Hardy-Littlewood prime-pairs conjecture, cannot produce a lower bound larger than 1/(log 2 + 1) ≈ 0.5906, and discusses limitations of the approach.

Significance. If the unconditional lower bound is rigorously justified, the result would constitute a concrete quantitative advance on Sun's Romanov-type conjecture, establishing that a positive proportion of integers satisfy the required representation. The explicit numerical bound and the conditional upper limit on the method's reach provide useful quantitative information beyond a mere existence statement.

major comments (2)

[§3] §3 (Sieve setup and main theorem): the asserted lower density bound of 0.0734 is obtained by numerical optimization of a truncated product ∏_p (1 − ω(p)/p) together with a singular-series factor, but the manuscript provides neither the explicit choice of sifting level D nor a rigorous upper bound on the remainder term that would guarantee the main term remains strictly positive for all large X. Without these controls it is impossible to verify that the sifted set has upper density strictly less than 1 − 0.0734.
[Theorem 1.1] Theorem 1.1 and the paragraph following Eq. (2.4): the claim that the result is unconditional rests on the positivity of the sieve lower bound after all error terms are accounted for, yet no explicit verification is given that ω(p) < p for every prime p in the sifting range or that the local densities do not vanish at any small prime. An overlooked prime with ω(p) = p would drive the product to zero and invalidate the positivity assertion.

minor comments (2)

The numerical value 0.0734 is stated without an accompanying table or appendix showing the precise optimization parameters (range of k, level D, truncation point) used to obtain it.
Notation for the sifting function ω(p) is introduced without a clear definition of the admissible residue classes for each prime; a short clarifying sentence would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where additional explicit controls would strengthen the presentation. We address each major comment below. The revisions will consist of added details and verifications that make the argument fully rigorous while preserving the unconditional nature of the result.

read point-by-point responses

Referee: §3 (Sieve setup and main theorem): the asserted lower density bound of 0.0734 is obtained by numerical optimization of a truncated product ∏_p (1 − ω(p)/p) together with a singular-series factor, but the manuscript provides neither the explicit choice of sifting level D nor a rigorous upper bound on the remainder term that would guarantee the main term remains strictly positive for all large X. Without these controls it is impossible to verify that the sifted set has upper density strictly less than 1 − 0.0734.

Authors: We agree that the manuscript would benefit from an explicit statement of the sifting level and a concrete error bound. In the revised version we will fix D = X^{1/10} and invoke the standard upper-bound sieve estimates (as in Halberstam–Richert) to show that the remainder is O(X (log X)^{-2}) uniformly for X large. Because the main term is asymptotically c X / log X with c > 0.0734, the error is eventually smaller than half the main term, guaranteeing that the count is at least 0.0734 X for all sufficiently large X and hence that the lower density is at least 0.0734. revision: yes
Referee: Theorem 1.1 and the paragraph following Eq. (2.4): the claim that the result is unconditional rests on the positivity of the sieve lower bound after all error terms are accounted for, yet no explicit verification is given that ω(p) < p for every prime p in the sifting range or that the local densities do not vanish at any small prime. An overlooked prime with ω(p) = p would drive the product to zero and invalidate the positivity assertion.

Authors: This observation is correct and we will remedy the omission. We will insert a short lemma (or appendix table) that, for every prime p ≤ 10^4, records the exact value of ω(p) and confirms ω(p) ≤ p−1; for p > 10^4 we note that the admissible residues modulo p are determined by the condition that 2^k + (n−k) ≢ 0 (mod p) for at least one admissible k in the range, which leaves at least one residue class unsifted. Consequently the local density is strictly positive at every prime and the infinite product converges to a positive constant. revision: yes

Circularity Check

0 steps flagged

No circularity: standard sieve lower bound for positive density

full rationale

The paper applies combinatorial sieve methods to show that the sifted set has positive lower density, with the explicit numerical lower bound 0.0734 obtained from optimizing the main term product over local densities and verifying the remainder is controlled. This is a self-contained analytic argument relying on external sieve estimates rather than any self-definitional loop, fitted input renamed as prediction, or load-bearing self-citation chain. The derivation does not reduce to its own inputs by construction and remains independent of the target density result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of unspecified analytic estimates that deliver a strictly positive lower bound; the numerical value 0.0734 is presented as computed but its derivation is not shown.

free parameters (1)

lower density bound
The paper states a computed lower bound of 0.0734 for the density.

axioms (1)

standard math Standard results from analytic number theory or sieve theory sufficient to bound the density from below
The unconditional claim presupposes such background theorems.

pith-pipeline@v0.9.0 · 5642 in / 1243 out tokens · 61310 ms · 2026-05-19T22:06:02.726941+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

By applying explicit estimates from the Selberg upper bound sieve and Theorem 1, we refine the second-moment method to obtain a positive lower bound on the density... |R(N)| ≥ δN with δ ≥ 0.0734
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

D(N) ≤ 8Φ(N)/(log 2)^2 N (1+o(1)) ... S2(N) ≤ (1/log 2 + 8Φ(N)/(log 2)^2) N (1+o(1))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

[1]

MR 4412547

Kevin Broughan,Bounded gaps between primes: The epic breakthroughs of the early twenty-first century, Cambridge University Press, Cambridge, 2021. MR 4412547

work page 2021
[2]

2, 275–284

Yong-Gao Chen and Xue-Gong Sun,On Romanoff’s constant, Journal of Number Theory106(2004), no. 2, 275–284

work page 2004
[3]

1, 103–107

Roger Crocker,On the sum of a prime and of two powers of two, Pacific Journal of Mathematics36 (1971), no. 1, 103–107

work page 1971
[4]

3-4, 713–724

Christian Elsholtz and Jan-Christoph Schlage-Puchta,On Romanov’s constant, Mathematische Zeitschrift288(2018), no. 3-4, 713–724

work page 2018
[5]

8, 113–123

Paul Erdős,On integers of the form2k +pand some related problems, Summa Brasiliensis Mathe- maticae2(1950), no. 8, 113–123

work page 1950
[6]

Gallagher,On the distribution of primes in short intervals, Mathematika23(1976), no

Patrick X. Gallagher,On the distribution of primes in short intervals, Mathematika23(1976), no. 1, 4–9

work page 1976
[7]

1, 45–50

Laurent Habsieger and Xavier-François Roblot,On integers of the formp+ 2 k, Acta Arithmetica 122(2006), no. 1, 45–50

work page 2006
[8]

Halberstam and H.-E

H. Halberstam and H.-E. Richert,Sieve methods, L.M.S. Monographs, No. 4, Academic Press, London-New York, 1974

work page 1974
[9]

Montgomery and Robert C

Hugh L. Montgomery and Robert C. Vaughan,Multiplicative number theory I: Classical theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2006

work page 2006
[10]

Nathanson,Additive number theory: The classical bases, Graduate Texts in Mathematics, vol

Melvyn B. Nathanson,Additive number theory: The classical bases, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996. MR 1395371

work page 1996
[11]

1-2, 1–14

János Pintz,A note on Romanov’s constant, Acta Mathematica Hungarica112(2006), no. 1-2, 1–14

work page 2006
[12]

N. P. Romanoff,Über einige Sätze der additiven Zahlentheorie, Mathematische Annalen109(1934), no. 1, 668–678. MR 1512916

work page 1934
[13]

3, 261–275

Zhi-Wei Sun,On integers not of the formc(2a + 2b) +p, Acta Arithmetica93(2000), no. 3, 261–275

work page 2000
[14]

,Ona n +bnmodulom, arXiv preprint arXiv:1312.1166, 2013

work page internal anchor Pith review Pith/arXiv arXiv 2013
[15]

,Sequence A231201: Number of ways to writen=x+y(x, y >0)with2 x +yprime, The On-Line Encyclopedia of Integer Sequences, 2013

work page 2013
[16]

3, 215–273

Jie Wu,Chen’s double sieve, Goldbach’s conjecture and the twin prime problem, Acta Arithmetica 114(2004), no. 3, 215–273. Graduate School of Mathematics, Kyushu University, Fukuoka, Japan. Email address:han.songlin.638@s.kyushu-u.ac.jp Graduate School of Mathematics, Nagoya University, Nagoya, Japan. Email address:jinbo.yu.e6@math.nagoya-u.ac.jp

work page 2004

[1] [1]

MR 4412547

Kevin Broughan,Bounded gaps between primes: The epic breakthroughs of the early twenty-first century, Cambridge University Press, Cambridge, 2021. MR 4412547

work page 2021

[2] [2]

2, 275–284

Yong-Gao Chen and Xue-Gong Sun,On Romanoff’s constant, Journal of Number Theory106(2004), no. 2, 275–284

work page 2004

[3] [3]

1, 103–107

Roger Crocker,On the sum of a prime and of two powers of two, Pacific Journal of Mathematics36 (1971), no. 1, 103–107

work page 1971

[4] [4]

3-4, 713–724

Christian Elsholtz and Jan-Christoph Schlage-Puchta,On Romanov’s constant, Mathematische Zeitschrift288(2018), no. 3-4, 713–724

work page 2018

[5] [5]

8, 113–123

Paul Erdős,On integers of the form2k +pand some related problems, Summa Brasiliensis Mathe- maticae2(1950), no. 8, 113–123

work page 1950

[6] [6]

Gallagher,On the distribution of primes in short intervals, Mathematika23(1976), no

Patrick X. Gallagher,On the distribution of primes in short intervals, Mathematika23(1976), no. 1, 4–9

work page 1976

[7] [7]

1, 45–50

Laurent Habsieger and Xavier-François Roblot,On integers of the formp+ 2 k, Acta Arithmetica 122(2006), no. 1, 45–50

work page 2006

[8] [8]

Halberstam and H.-E

H. Halberstam and H.-E. Richert,Sieve methods, L.M.S. Monographs, No. 4, Academic Press, London-New York, 1974

work page 1974

[9] [9]

Montgomery and Robert C

Hugh L. Montgomery and Robert C. Vaughan,Multiplicative number theory I: Classical theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2006

work page 2006

[10] [10]

Nathanson,Additive number theory: The classical bases, Graduate Texts in Mathematics, vol

Melvyn B. Nathanson,Additive number theory: The classical bases, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996. MR 1395371

work page 1996

[11] [11]

1-2, 1–14

János Pintz,A note on Romanov’s constant, Acta Mathematica Hungarica112(2006), no. 1-2, 1–14

work page 2006

[12] [12]

N. P. Romanoff,Über einige Sätze der additiven Zahlentheorie, Mathematische Annalen109(1934), no. 1, 668–678. MR 1512916

work page 1934

[13] [13]

3, 261–275

Zhi-Wei Sun,On integers not of the formc(2a + 2b) +p, Acta Arithmetica93(2000), no. 3, 261–275

work page 2000

[14] [14]

,Ona n +bnmodulom, arXiv preprint arXiv:1312.1166, 2013

work page internal anchor Pith review Pith/arXiv arXiv 2013

[15] [15]

,Sequence A231201: Number of ways to writen=x+y(x, y >0)with2 x +yprime, The On-Line Encyclopedia of Integer Sequences, 2013

work page 2013

[16] [16]

3, 215–273

Jie Wu,Chen’s double sieve, Goldbach’s conjecture and the twin prime problem, Acta Arithmetica 114(2004), no. 3, 215–273. Graduate School of Mathematics, Kyushu University, Fukuoka, Japan. Email address:han.songlin.638@s.kyushu-u.ac.jp Graduate School of Mathematics, Nagoya University, Nagoya, Japan. Email address:jinbo.yu.e6@math.nagoya-u.ac.jp

work page 2004