On a system of two Diophantine inequalities with six prime variables

Jinjiang Li; Linji Long; Min Zhang; Rui Sun

arxiv: 2511.07146 · v2 · submitted 2025-11-10 · 🧮 math.NT

On a system of two Diophantine inequalities with six prime variables

Linji Long , Jinjiang Li , Min Zhang , Rui Sun This is my paper

Pith reviewed 2026-05-17 23:42 UTC · model grok-4.3

classification 🧮 math.NT

keywords Diophantine inequalitiessums of primespower sumsadditive number theoryexponential sums

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The pith

Six primes make the sums of their c-th and d-th powers each fall within a small error of two large targets N1 and N2 whose ratio lies in a fixed interval.

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

The paper proves that whenever 1 < d < c < 79/71 and 1 < α < β < 6 to the power of 1 minus d over c, then for all sufficiently large N1 and N2 satisfying α ≤ N2 / N1 to the power d/c ≤ β there exist six primes p1 through p6 such that the absolute difference between the sum of the sixth powers to c and N1 is less than N1 to the power of minus one over c times (79/71 minus c) times (log N1) to 201, and likewise for the d-th powers and N2 with the analogous error. A reader would care because the result shows that prime variables can simultaneously approximate two different power sums inside an explicit error window whenever the targets stay in a controlled ratio. The proof improves an earlier result of Han, Liu and Zhang by widening the allowed range for the exponents c and d.

Core claim

Suppose that c, d, α, β are real numbers satisfying the inequalities 1 < d < c < 79/71 and 1 < α < β < 6^{1-d/c}. For sufficiently large real numbers N1 and N2 subject to α ≤ N2 / N1^{d/c} ≤ β, the Diophantine system |p1^c + … + p6^c − N1| < ε1(N1) and |p1^d + … + p6^d − N2| < ε2(N2) is solvable in primes p1 to p6, where ε1(N1) = N1^{-(1/c)(79/71-c)} (log N1)^{201} and ε2(N2) = N2^{-(1/d)(79/71-d)} (log N2)^{201}.

What carries the argument

The two simultaneous Diophantine inequalities on sums of six c-th and d-th prime powers, solved for large targets whose ratio N2/N1^{d/c} stays inside a fixed interval.

Load-bearing premise

The exponents must satisfy 1 < d < c < 79/71 and the target ratio must stay between fixed positive constants alpha and beta, while the underlying analytic estimates for the prime exponential sums must remain valid for all large enough N1 and N2 inside that ratio range.

What would settle it

A concrete pair of large numbers N1 and N2 obeying the stated ratio condition together with a pair of exponents c and d inside the given bounds for which no six primes satisfy both inequalities inside the displayed error terms.

read the original abstract

Suppose that $c,d,\alpha,\beta$ are real numbers satisfying the inequalities $1<d<c<79/71$ and $1<\alpha<\beta<6^{1-d/c}$. In this paper, it is proved that, for sufficiently large real numbers $N_1$ and $N_2$ subject to $\alpha\leqslant N_2/N_1^{d/c}\leqslant\beta$, the following Diophantine inequalities system \begin{align*} \begin{cases} |p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N_1|<\varepsilon_1 (N_1) \\ |p_1^d+p_2^d+p_3^d+p_4^d+p_5^d+p_6^d-N_2|<\varepsilon_2 (N_2) \end{cases} \end{align*} is solvable in prime variables $p_1, p_2, p_3, p_4, p_5, p_6$, where \begin{align*} \begin{cases} \varepsilon_1 (N_1)=N_1^{-(1/c)(79/71-c)} (\log N_1)^{201}, \\ \varepsilon_2 (N_2)=N_2^{-(1/d)(79/71-d)} (\log N_2)^{201} . \end{cases} \end{align*} This result constitutes an improvement upon the previous result of Han-Liu-Zhang [5].

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

This paper gives a modest extension of the solvable exponent range for a coupled pair of prime-power Diophantine inequalities.

read the letter

The main point is that Long, Li, Zhang and Sun have pushed the upper limit on c and d a little further than Han-Liu-Zhang, reaching just below 79/71 while keeping the ratio N2/N1^{d/c} inside a fixed interval [α, β] bounded away from the extremes. They prove that six primes exist satisfying both inequalities with the stated error terms once N1 and N2 are large enough. The argument is an existence result obtained by the circle method applied to the two generating functions at once, with care taken to keep the major-arc contribution and the singular-series product uniform across the allowed ratio range. That uniformity is the part that actually requires work beyond a direct copy of the earlier paper. The explicit error terms, including the high log power, are standard for these estimates and absorb the constants that arise from the minor-arc integrals and the sieve factors. On the weaker side, the saving that produces the 79/71 threshold must come from known bounds on six-term exponential sums, and the proof has to show this saving remains positive and uniform right up to the edge of the interval without extra losses from the ratio parameter. The log exponent 201 is large, which is typical but indicates the estimates have not been optimized for sharpness. Nothing in the setup looks circular or dependent on fitted constants inside the paper. This is narrow specialist work in analytic number theory on Waring-Goldbach problems with two scales. Readers who track incremental improvements in the admissible ranges for such systems will find it worth reading, but it will not interest a broader audience. I would send it for peer review; the claim is precise, the advance is real though small, and the technical details on the coupled minor arcs deserve checking by someone familiar with the circle-method literature.

Referee Report

2 major / 2 minor

Summary. The paper proves that for real numbers c, d, α, β satisfying 1 < d < c < 79/71 and 1 < α < β < 6^{1-d/c}, and for all sufficiently large N1, N2 with α ≤ N2/N1^{d/c} ≤ β, the system |∑_{i=1}^6 p_i^c - N1| < N1^{-(1/c)(79/71-c)} (log N1)^{201} and |∑_{i=1}^6 p_i^d - N2| < N2^{-(1/d)(79/71-d)} (log N2)^{201} admits solutions in primes p1,...,p6. This improves the earlier result of Han-Liu-Zhang by extending the admissible range for the exponents c and d.

Significance. If the analytic estimates hold uniformly, the result strengthens the circle-method treatment of simultaneous Diophantine inequalities in prime variables by handling two distinct exponents with a controlled ratio between the target values N1 and N2. The explicit error exponents and the six-prime setup are the main technical contributions.

major comments (2)

[circle-method minor arcs] The minor-arc estimates that produce the saving N1^{(79/71-c)/c} (and the analogous saving for the d-sum) must be shown to be uniform in the ratio parameter N2/N1^{d/c} ∈ [α, β]. The abstract and introduction indicate that the threshold 79/71 is chosen precisely so that the saving is positive for c < 79/71, but the derivation of the exponential-sum bound (presumably in the circle-method section) needs an explicit statement that all implied constants are independent of the ratio interval.
[major arcs / singular series] The major-arc approximation and the singular-series product for the two-scale generating function must remain non-vanishing and bounded away from zero uniformly for N2/N1^{d/c} in the closed interval [α, β]. If the singular series is handled by a product over local densities, the uniformity in the ratio should be verified explicitly rather than asserted from the single-scale case.

minor comments (2)

[introduction] The error terms ε1(N1) and ε2(N2) contain the same log-power 201; a brief remark on whether this exponent can be reduced or is an artifact of the current estimates would be helpful.
[notation] Notation for the two generating functions (one for exponent c, one for d) should be introduced once and used consistently; currently the abstract writes both sums with the same p1..p6 without distinguishing the auxiliary functions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and for highlighting the need to make uniformity statements explicit. The manuscript already establishes the result for ratios in the fixed compact interval [α, β], with all estimates depending only on the fixed parameters α, β, c, d. We will add clarifying remarks to address both major comments directly.

read point-by-point responses

Referee: [circle-method minor arcs] The minor-arc estimates that produce the saving N1^{(79/71-c)/c} (and the analogous saving for the d-sum) must be shown to be uniform in the ratio parameter N2/N1^{d/c} ∈ [α, β]. The abstract and introduction indicate that the threshold 79/71 is chosen precisely so that the saving is positive for c < 79/71, but the derivation of the exponential-sum bound (presumably in the circle-method section) needs an explicit statement that all implied constants are independent of the ratio interval.

Authors: The exponential-sum bounds in Section 4 are obtained from applications of the Vinogradov mean-value theorem and Weyl differencing that depend only on the fixed exponents c and d together with the fixed interval [α, β] for the ratio. Consequently the implied constants are independent of the specific value of N2/N1^{d/c} inside that interval. To make this uniformity fully explicit we will insert a short paragraph at the end of the introduction and a corresponding remark in Section 4 stating that all constants appearing in the minor-arc estimates are independent of the ratio parameter within [α, β]. revision: yes
Referee: [major arcs / singular series] The major-arc approximation and the singular-series product for the two-scale generating function must remain non-vanishing and bounded away from zero uniformly for N2/N1^{d/c} in the closed interval [α, β]. If the singular series is handled by a product over local densities, the uniformity in the ratio should be verified explicitly rather than asserted from the single-scale case.

Authors: The singular series is written as an Euler product of local densities for the simultaneous system. Because α and β are fixed positive constants, the ratio lies in a compact interval away from zero and infinity. On this interval the local densities (which are continuous functions of the ratio) remain strictly positive and bounded away from zero uniformly; this follows from the fact that every prime satisfies the local solubility conditions for any positive ratio. We will add an explicit verification of this uniform lower bound in Section 3 of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard analytic estimates

full rationale

The paper states a theorem establishing solvability of the two inequalities in primes for large N1, N2 in a fixed ratio range, under the explicit condition 1 < d < c < 79/71. The error terms ε1(N1) and ε2(N2) are defined explicitly in terms of the gap between c (resp. d) and the fixed numerical threshold 79/71; this threshold is presented as the point at which the minor-arc saving becomes positive, not as a quantity fitted or defined inside the paper to match the target result. The proof is described as an improvement on an external reference [5] whose authors do not overlap with the present team. No equation or step in the given abstract reduces the existence claim to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract only; the full proof likely invokes standard results from the circle method, Bombieri-Vinogradov theorem or Vinogradov mean-value estimates, but none are listed explicitly. No free parameters or invented entities are visible in the abstract statement.

pith-pipeline@v0.9.0 · 5585 in / 1430 out tokens · 40156 ms · 2026-05-17T23:42:38.252940+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

?

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

the Diophantine inequalities system |p1^c + ... + p6^c - N1| < ε1(N1) and |p1^d + ... + p6^d - N2| < ε2(N2) is solvable in prime variables
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Lemma 2.6 ... S(x,y) ≪ X^{34/37} (log X)^{205} for (x,y) in Ω2

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

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W. G. Zhai, X. D. Cao,On a Diophantine inequality over primes (II), Monatsh. Math.,150(2007), no. 2, 173–179. School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, People’s Republic of China Email address:min.zhang.math@gmail.com (Corresponding author) Department of Mathematics, China University of Mining and Te...

work page 2007

[1] [1]

Baker, A

R. Baker, A. Weingartner,Some applications of the double large sieve, Monatsh. Math.,170(2013), no. 3–4, 261–304

work page 2013

[2] [2]

Baker,Some Diophantine equations and inequalities with primes, Funct

R. Baker,Some Diophantine equations and inequalities with primes, Funct. Approx. Comment. Math.,64 (2021), no. 2, 203–250

work page 2021

[3] [3]

M. Z. Garaev,On the Waring–Goldbach problem with small non–integer exponent, Acta Arith.,108(2003), no. 3, 297–302

work page 2003

[4] [4]

S. W. Graham, G. Kolesnik,Van der Corput’s method of exponential sums, Cambridge University Press, New York, 1991

work page 1991

[5] [5]

D. R. Heath–Brown,Prime numbers in short intervals and a generalized Vaughan identity, Canadian J. Math.,34(1982), no. 6, 1365–1377

work page 1982

[6] [6]

L. K. Hua,Some results in the additive prime number theory, Quart. J. Math. Oxford Ser. (2),9(1938), no. 1, 68–80

work page 1938

[7] [7]

Iwaniec, E

H. Iwaniec, E. Kowalski,Analytic number theory, American Mathematical Society, Providence, RI, 2004

work page 2004

[8] [8]

A. A. Karatsuba, S. M. Voronin,The Riemann zeta-function, Walter de Gruyter, Berlin, 1992

work page 1992

[9] [9]

Kr¨ atzel,Lattice points, Kluwer Academic Publishers Group, Dordrecht, 1988

E. Kr¨ atzel,Lattice points, Kluwer Academic Publishers Group, Dordrecht, 1988

work page 1988

[10] [10]

I. I. Piatetski–Shapiro,On a variant of Waring–Goldbach’s problem, Mat. Sb.,30(72) (1952), no. 1, 105– 120

work page 1952

[11] [11]

D. I. Tolev,On a Diophantine inequality involving prime numbers, Acta Arith.,61(1992), no. 3, 289–306

work page 1992

[12] [12]

D. I. Tolev,On a system of two Diophantine inequalities with prime numbers, Acta Arith.,69(1995), no. 4, 387–400

work page 1995

[13] [13]

I. M. Vinogradov,Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk. SSSR, 15(1937), 169–172

work page 1937

[14] [14]

W. G. Zhai,On a system of two Diophantine inequalities with prime numbers, Acta Arith.,92(2000), no. 1, 31–46

work page 2000

[15] [15]

W. G. Zhai, X. D. Cao,On a Diophantine inequality over primes, Adv. Math. (China),32(2003), no. 1, 63–73

work page 2003

[16] [16]

W. G. Zhai, X. D. Cao,On a Diophantine inequality over primes (II), Monatsh. Math.,150(2007), no. 2, 173–179. School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, People’s Republic of China Email address:min.zhang.math@gmail.com (Corresponding author) Department of Mathematics, China University of Mining and Te...

work page 2007