Almost primes and primes that are sums of two squares plus one

Kunjakanan Nath; Likun Xie

REVIEW 4 major objections 4 minor 26 references

Almost primes and primes that are sums of two squares plus one

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that infinitely many primes $p=m^2+n^2+1$ satisfy $\Omega(p+2)\le 11$, with count $\gg x/(\log x)^{5/2}$.

desk verdict A genuine new result in sieve theory, but the load-bearing vector-sieve proposition is only sketched and needs a real proof before I'd trust the constant. read the letter →

arxiv 2501.16723 v2 pith:IPFAWTLA submitted 2025-01-28 math.NT

Kunjakanan Nath , Likun Xie This is my paper · ORCID

classification math.NT MSC 11N0511N3511N36

keywords almostprimessumsoftwosquaressemi-linearsievelinearvectorlogarithmicweightmeanvaluetheoremforinarithmeticprogressionsrepresentedbyquadraticforms

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

This paper proves that infinitely many primes $p$ can be written as $p=m^2+n^2+1$ with $m,n\in\mathbb{N}$ and, at the same time, have $p+2$ with at most 11 prime factors counted with multiplicity. The quantitative form is $$\#\{p\le x: p \text{ prime},\ p=$m^{2}$+$n^{2}$+1,\ m,n\in\mathbb{N},\ \$\Omega$(p+2)\le 11\}\gg \frac{x}{(\log x)^{5/2}},$$ which matches the order predicted heuristically even for the stronger condition that $p+2$ be prime. This matters because the family $p=m^2+n^2+1$ is genuinely sparse: an upper-bound sieve already gives only $O(x/(\log x)^{3/2})$ primes in it, so the almost-prime condition on the shift does not cost an extra logarithmic factor. The proof combines a semi-linear sieve for the condition that $p-1$ be a sum of two squares with a linear sieve for $p+2$, inside a vector sieve with two different dimensions.

What carries the argument

The carrying object is the vector-sieve fundamental lemma for two $\beta$ sieves of different dimensions (Proposition 3.10). A $\beta$ sieve of dimension $\kappa$ is a combinatorial sieve whose weights are Möbius functions restricted to specified divisors, with $\kappa=1/2$ giving the semi-linear sieve and $\kappa=1$ giving the linear sieve. The lemma counts pairs $(m,n)$ surviving two different prime sets, $\mathcal P_1$ and $\mathcal P_2$, by pre-sieving out small primes below $z_0=(\log z_1z_2)^{1/3}$ and then bounding the doubly-sifted count by the $\beta$-sieve functions $F_1,F_2$ and $f_1,f_2$, combined through the inf/sup formulas (3.13) and (3.14). The same framework is used in two modes: a lower-bound mode for the main sifting term, and an upper-bound weighted mode in which a weight $1-\log p/\log y$ is summed over prime divisors of $p+2$. The remaining error terms are controlled by mean-value estimates for primes in arithmetic progressions.

What would settle it

Recompute the numerical value of the main-term constant $H(0.14,0.23,0.449,0.011)$ from the displayed formulas; if the computed value is not positive, the proof's parameter search fails. Separately, test Proposition 3.10 on a model two-dimensional sequence with the same local densities: any concrete violation of its claimed upper or lower inequality would refute the lemma the theorem rests on.

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Extended reading notes

Core claim

The paper's central claim is a lower-bound, sieve-theoretic result: for all large $x$, the number of primes $p\le x$ of the form $p=m^2+n^2+1$ whose shift $p+2$ has at most 11 prime factors is $\gg x/(\log x)^{5/2}$. The argument decomposes the sieving by the set of primes congruent to $3$ modulo $4$: the setup forces $p-1$ to have exactly one factor of $2$, so the representation condition reduces to the absence of $3\pmod 4$ prime factors in the odd part. A vector-sieve fundamental lemma supplies lower and upper bounds for two simultaneous $\beta$ sieves of dimensions $1/2$ and $1$; the Buchstab term is handled by a switching principle; and a logarithmic weight function is optimized numerically with the parameters $\theta_2=0.011$, $\theta_1=0.449$, $\theta=0.23$, and $\lambda=0.14$, producing the bound 11.

Load-bearing premise

The proof relies on a vector-sieve lemma that joins a half-dimensional sieve with a linear sieve, and the paper only sketches why that combination works; if the mixed-dimension lemma fails, the lower bound collapses.

Editorial extensions

If this is right

There are infinitely many primes $p$ with $p=m^2+n^2+1$ and $\Omega(p+2)\le 11$.
The lower bound has the same shape $x/(\log x)^{5/2}$ as the conjectured count for the case $p+2$ prime, so the almost-prime condition does not cost an extra logarithmic factor.
Under a strong equidistribution conjecture for primes in arithmetic progressions, the argument can be adapted to force $\Omega(p+2)\le 3$.
The same vector-sieve framework extends to products such as $(p+2)(p+6)$, and a multi-variable version would handle simultaneous almost-prime conditions on several shifts with count $\gg x/(\log x)^{7/2}$ for suitable optimizable bounds.
Numerical optimization of the parameters $\theta,\theta_1,\theta_2,\lambda$ could lower the number 11 within the same proof framework.

Reading between the lines

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

The value 11 appears to be a computational artifact of a coarse parameter search rather than a structural limit; a finer optimization or a small improvement to the sieve lemma could plausibly reduce it.
The mixed-dimension vector-sieve lemma is likely reusable for other sparse prime sets defined by one representation condition and one almost-prime condition, provided the corresponding local density and prime-distribution estimates are available.
A natural test of the architecture is to compare the proven lower bound with the heuristic constant on finite ranges; if the ratio settles near $x/(\log x)^{5/2}$, the remaining gap to an asymptotic formula looks mostly technical.
The multi-shift extension discussed in the paper is where the method faces its sharpest test, since a genuine multi-variable vector sieve would require distribution estimates beyond those used here.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is the problem and the mixed-dimension vector sieve: combining a semi-linear sieve (kappa=1/2, primes 3 mod 4) with a linear sieve (kappa=1, all primes) to handle the simultaneous conditions on p-1 and p+2. Heath-Brown and Li did two linear sieves; this extends the framework to mixed dimensions and applies it to primes of the form m^2+n^2+1 with p+2 almost prime. The target set is sparse (expected order x/(log x)^{5/2}), and the claimed lower bound with at most 11 prime factors is a plausible, nontrivial improvement over what a naive combination of Iwaniec and Chen would give. The numerical optimization (H=1.2471 at the chosen parameters) is standard and not circular; the parameters are chosen to make the main term positive, which is fine.

The paper does several things well. The structure is clear: the switching-principle upper bound (Prop 4.2) is worked out in detail, the Bombieri-Vinogradov input is cited appropriately, and the treatment of the error terms through Cauchy-Schwarz is standard and looks correct. The explicit Euler products and the Mertens-type estimates in Prop 4.1 and 4.3 are checked carefully.

The soft spots are real, though I think they are fixable. Proposition 3.10 is the load-bearing step: both the lower bound in Prop 4.1 and the upper bound in Prop 4.3 depend on its main-term functions F and f. The proof is a sketch, and the critical part—the lower-bound combination delta_1^- delta_2^+ + delta_1^+ delta_2^- - delta_1^+ delta_2^+—is simply asserted to work "as in [8, Section 3.3]." That reference is for two linear sieves with the same prime set. Here the two sieves have different dimensions and different prime sets, and for common primes p in P1∩P2, axiom (A2) does not give h(d1,d2)=h1(d1)h2(d2). So the evaluation of the sum Sigma, and hence the formula (3.14), needs an independent verification. There's also a small but worrying mismatch: Proposition 3.10 fixes z0=e^{(log z1 z2)^{1/3}}, but the proof of Proposition 4.1 uses z0=e^{(log z1 z2)^{1/2}}. That could be a typo, but it's exactly in the step where the size of the pre-sieving error depends on z0.

I would not desk-reject this. The result is interesting, the strategy is sound, and the gaps are localized. But as it stands, the central lower bound is not fully established: the leading constant of the main term in Prop 4.1 could change sign if the mixed-dimension f and F from Prop 3.10 are wrong by an O(1) factor. The authors need to write out the proof of Proposition 3.10 in full, or state it as a conditional lemma with a clear reference to a complete treatment.

Who is this for? Sieve theorists and anyone working on sparse prime sets. It deserves a serious referee, but the referee should insist on a complete proof of Prop 3.10 before publication.

Referee Report

4 major / 4 minor

Summary. The paper claims Theorem 1.1: #{p ≤ x : p prime, p = m^2+n^2+1, m,n ∈ N, Ω(p+2) ≤ 11} ≫ x/(log x)^{5/2}. The proof combines Iwaniec's semi-linear sieve for the condition that p−1 is a sum of two squares with a linear sieve for p+2, using a new vector sieve (Proposition 3.10) for two beta sieves of mixed dimensions, together with Richert's weighted sieve to control the number of prime factors of p+2. The final arithmetic is reduced to a numerical optimization of four parameters (θ1, θ2, θ, λ) for which H(0.14, 0.23, 0.449, 0.011) = 1.2471 > 0.

Significance. If the result is correct, it is a genuine advance: it gives an unconditional lower bound of the expected order of magnitude for primes of a sparse quadratic form with a bounded shift, extending Iwaniec's theorem on primes p = m^2+n^2+1. The vector sieve for two beta sieves of different dimensions is an interesting technique with potential applications to other sparse prime problems. The paper is clearly written and the overall strategy is standard, following Heath-Brown and Li. However, the load-bearing Proposition 3.10 is only sketched, and there are inconsistencies between its statement and its application that need to be resolved before the argument is complete.

major comments (4)

[Section 3.3, Proposition 3.10] The proof of the lower-bound part of Proposition 3.10 is only sketched and refers to [8, Section 3.3] for the main-term evaluation. That reference treats two linear sieves with a multiplicative density h(d1,d2) = h1(d1)h2(d2), whereas the present application uses κ1 = 1/2 and κ2 = 1 with P1 = {p ≡ 3 mod 4} and P2 = all primes. For p ∈ P1 ∩ P2, axiom (A2) does not imply h(p,p) = h(p,1)h(1,p), so the lower-bound combination δ1^-δ2^+ + δ1^+δ2^- − δ1^+δ2^+ on page 13 requires an independent evaluation. The formula (3.14) for f(σ1,σ2) is asserted without derivation, and Propositions 4.1 and 4.3 depend directly on this formula. This is a load-bearing gap.
[Section 3.3 / Section 5] Proposition 3.10 explicitly assumes log z1 ≍ log z2, but the numerical parameters chosen in Section 5, θ1 = 0.449 and θ2 = 0.011, give log z1 / log z2 = θ1/θ2 ≈ 40.8, which is not bounded by an absolute constant. Propositions 4.1 and 4.3 apply Proposition 3.10 with z1 = x^{θ1} and z2 = x^{θ2}, so the main theorem is obtained outside the stated range of validity. Either the optimization must be restricted to θ1 ≍ θ2, or Proposition 3.10 must be proven without this condition.
[Section 4.1, proof of Proposition 4.1] There is an inconsistency in the choice of z0. Proposition 3.10 states z0 = e^{(log z1 z2)^{1/3}}, but the proof of Proposition 4.1 (after equation (4.4)) uses z0 = e^{(log z1 z2)^{1/2}}. These correspond to different pre-sieving levels, and the error term E2 in the proof of Proposition 3.10 depends on |log z0|. The claimed bound (4.4) is therefore not justified by the stated proposition. The authors should specify the correct z0 and verify that the error terms remain of the claimed size.
[Section 4.1, equation (4.4)] The lower bound in (4.4) is written with a '+' sign before the error term: '≥ ... + x/(log x)^{10}'. Since the lower-bound vector sieve (3.12) has an error term that is bounded in absolute value, the displayed inequality should have a subtracted error term. As written, the sign is incorrect, although the asymptotic conclusion (4.9) would be unaffected if the error is negative. Please correct the sign or clarify the convention.

minor comments (4)

[Abstract] The abstract contains a typo: 'semi-lin ear' should be 'semi-linear'.
[Introduction, page 2] The reference to 'Matom¨aki, Radziwi/suppress l/suppress l, and Tao' should read 'Matomäki, Radziwiłł, and Tao'.
[Section 5, numerical verification] The proof relies on the numerical inequality H(0.14, 0.23, 0.449, 0.011) > 0. To make the proof fully rigorous, the authors should provide a verified computation (e.g., interval arithmetic) or a rigorous analytic bound, rather than a stepwise Matlab search without error bounds.
[Proposition 4.2 statement] The constant in the statement is written as '2c1c2^2c3C(θ1)' without parentheses around the product; consider writing it as '(2c1 c2^2 c3 C(θ1) + o(1))' for clarity, matching the proof.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the central lower bound is a genuine vector-sieve derivation from external sieve theorems, and the numerical optimization is standard; the main concerns are a sketched mixed-dimension lemma and an exponent mismatch, which are correctness risks rather than circular reductions.

full rationale

The paper's central claim, Theorem 1.1, is obtained by combining lower and upper vector-sieve estimates (Propositions 4.1, 4.2, and 4.3) into a positivity condition H(lambda, theta, theta1, theta2) > 0, and then choosing the sieve parameters numerically. The coefficient H is not fitted to the output: it is an explicit expression built from the beta-sieve functions F1, f1, F2, f2, Mertens constants, and convergent Euler products, all of which come from Friedlander-Iwaniec's Opera de Cribro and from Heath-Brown and Li's vector sieve framework. The positivity check H(0.14, 0.23, 0.449, 0.011) = 1.2471 is a numerical evaluation of this expression, not a parameter tuned to force the final count. No step in the proof defines the object being counted in terms of the bound being proved, and no fitted quantity is renamed as a prediction. The only self-citation, Nath [19], appears in a list of related generalizations in the introduction and is not load-bearing for Theorem 1.1. The paper's principal weakness is that Proposition 3.10, the vector-sieve fundamental lemma for mixed dimensions kappa1 = 1/2 and kappa2 = 1, is only sketched, with the core sum and error estimates delegated to Heath-Brown and Li [8, Section 3.3], which treats a different mixed situation; additionally, Proposition 3.10 states z0 = exp((log z1 z2)^{1/3}) while Proposition 4.1 uses z0 = exp((log z1 z2)^{1/2}). These are genuine rigor gaps that could invalidate the proof as written, but they are not circularity: the missing verification does not assume the target theorem or reduce to the paper's own inputs. Consequently, the appropriate circularity score is low, with the caveat that the proof's completeness is questionable on external grounds.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The central claim depends on four numerically chosen sieve parameters and on the correctness of a new vector sieve proposition. There are no invented physical or mathematical entities. The main axioms are standard theorems from sieve theory; the mixed-dimension vector sieve is a new but not fully proven assumption.

free parameters (4)

θ1 = 0.449 (terminal; 0.431 in an intermediate G-search)
Sifting exponent for the semi-linear sieve; chosen by Matlab search to make H>0 and to optimize the almost-prime count.
θ2 = 0.011
Sifting exponent for the linear sieve; chosen by numerical search to balance error terms.
θ = 0.23
Exponent in the Richert weight y=x^θ; chosen to minimize 1/θ+1/λ.
λ = 0.14
Weight multiplier in the weighted sieve; chosen by numerical search to yield ω(p+2) ≤ 11.

assumptions (3)

standard math Bombieri-Vinogradov type mean value theorems (Pan's theorem, Lemma 3.2) hold in the ranges used.
These are established theorems in analytic number theory, quoted from [20] and [8].
standard math The beta sieve weights satisfy the estimates of Lemma 3.6 with κ1=1/2 and κ2=1.
Standard beta sieve theory from Friedlander and Iwaniec, Chapter 11, is applied.
domain assumption The vector sieve fundamental lemma, Proposition 3.10, is valid as stated for mixed dimensions.
This is the paper's new tool; its proof is only sketched and is not independently verified in the manuscript.

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Cite this review

Pith. "Pith review of Almost primes and primes that are sums of two squares plus one." pith.science (2026). https://pith.science/paper/IPFAWTLA

@misc{pith2026250116723,
  author       = {Pith},
  title        = {Pith review of: Almost primes and primes that are sums of two squares plus one},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPFAWTLA}},
  note         = {Machine review of arXiv:2501.16723}
}

abstract

In this paper, we obtain a lower bound for the number of primes $p\leq x$ such that $p-1$ is a sum of two squares and $p+2$ has a bounded number of prime factors. The proof uses the vector sieve framework, involving a semi-linear sieve and a linear sieve.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Cai, Almost prime triples and Chen’s theorem

Y. Cai, Almost prime triples and Chen’s theorem. Acta Arith. 179 (2017), no. 3, 233–250

work page 2017

[2] [2]

J. R. Chen, On the representation of a larger even integer as the sum of a p rime and the product of at most two primes. Sci. Sinica 16 (1973), 157–176

work page 1973

[3] [3]

J. B. Friedlander and H. Iwaniec, Hyperbolic prime number theorem. Acta Math. 202 (2009), no. 1, 1–19

work page 2009

[4] [4]

Friedlander and H

J. Friedlander and H. Iwaniec, Opera de cribro. Amer. Math. Soc. Colloq. Publ., 57 American Mathe- matical Society, Providence, RI, 2010

work page 2010

[5] [5]

D. A. Goldston, J. Pintz, and C. Y. Yıldırım, Primes in tuples I , Ann. of Math. (2) 170 (2009), no. 2, 819–862

work page 2009

[6] [6]

Halberstam and H.-E

H. Halberstam and H.-E. Richert, Sieve methods. London Mathematical Society Monographs, No. 4. Academic Press [Harcourt Brace Jovanovich, Publishers], L ondon-New York, 1974

work page 1974

[7] [7]

Harman, Prime-detecting sieves

G. Harman, Prime-detecting sieves. London Math. Soc. Monogr. Ser., 33 Princeton University Pre ss, Princeton, NJ, 2007

work page 2007

[8] [8]

Heath-Brown and X

R. Heath-Brown and X. Li, Almost prime triples and Chen’s theorem. J. Number Theory 169 (2016), 265–294

work page 2016