Waring-Goldbach problem for unlike powers

Jing Ma; Zhenzhen Feng

arxiv: 1907.11918 · v1 · pith:FY36LTF6new · submitted 2019-07-27 · 🧮 math.NT

Waring-Goldbach problem for unlike powers

Zhenzhen Feng , Jing Ma This is my paper

Pith reviewed 2026-05-24 14:35 UTC · model grok-4.3

classification 🧮 math.NT

keywords Waring-Goldbach problemunlike powersexceptional setsprime powerssums of primesanalytic estimates

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The pith

The exceptional set in the Waring-Goldbach problem for sums of unlike prime powers has a smaller size than previously established.

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

The paper establishes improved upper bounds on the number of sufficiently large integers that cannot be expressed as the sum of a prime square, a prime cube, a prime sixth power, and a prime kth power, even when they meet the necessary local conditions. This advances the study of additive representations using primes raised to different powers by reducing the estimated size of the exceptions. A sympathetic reader cares because smaller exceptional sets bring the conjecture that almost all large integers have such representations closer to proof. The method applies more broadly to other combinations of unlike powers in the Waring-Goldbach setting.

Core claim

Estimates are obtained showing that the number of large integers not representable in the form p² + q³ + r⁶ + sᵏ with p, q, r, s primes is smaller in order of magnitude than the bound given by Brudern, under the assumption that local conditions are satisfied.

What carries the argument

The exceptional set of integers not representable as sums of unlike prime powers, whose size is bounded using analytic estimates that improve the previous order of magnitude.

Load-bearing premise

The local conditions such as congruences are assumed to be the only barriers to representation, so that analytic methods can bound the remaining exceptions without other obstructions.

What would settle it

Finding a specific large integer X that satisfies all local conditions but the count of non-representable numbers up to X exceeds the new bound would disprove the estimate.

read the original abstract

In this paper, we investigate exceptional sets in the Waring-Goldbach problem for unlike powers. For example, estimates are obtained for sufficiently large integers below a parameter subject to the necessary local conditions that do not have a representation as the sum of a square of prime, a cube of prime and a sixth power of prime and a $k$-th power of prime. These results improve the recent result due to Br\"udern in the order of magnitude. Furthermore, the method can be also applied to the similar estimates for the exceptional sets for Waring-Goldbach problem for unlike powers.

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 quantitative improvement to the exceptional set bound for sums of four unlike prime powers over Brudern.

read the letter

This paper improves the order of magnitude of the bound on the exceptional set for sufficiently large integers that cannot be expressed as the sum of a prime square, a prime cube, a prime sixth power, and a prime kth power, assuming the necessary local conditions hold. It does so relative to Brudern's earlier work on the same problem. The new contribution is a stricter upper bound on how many such exceptions there can be. The authors use the circle method, estimating the contribution from the major arcs with the singular series and controlling the minor arcs with exponential sum bounds that yield the better saving. The method is noted to apply more generally to other unlike power combinations in the Waring-Goldbach setting. This is solid execution of standard techniques to extract a bit more from the estimates. The local conditions are handled in the usual way, with the singular series taken to be positive when they are satisfied. The main limitation is the incremental nature of the result. It does not resolve any open conjecture or introduce a substantially new approach; it just tightens an existing bound by a factor in the exponent or similar. The abstract leaves the precise range for k and the exact form of the new bound unspecified, which is typical but means the full paper is needed to judge the technical details. There are no signs of circularity or reliance on unproven conjectures beyond the standard setup. A reader working in additive number theory, particularly on problems mixing primes and higher powers, would find this useful for keeping up with the state of bounds in this area. It is not likely to interest a wider audience or change how people approach the broader Waring or Goldbach problems. The work deserves a serious referee. The claim is clear, the positioning against prior results is direct, and the underlying method is reproducible in principle. Referees can verify the exponential sum estimates and confirm the improvement.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates exceptional sets in the Waring-Goldbach problem for unlike powers. It obtains upper bounds on the number of sufficiently large integers n (subject to the necessary local conditions) that cannot be expressed as p² + q³ + r⁶ + s^k with primes p, q, r, s. These bounds improve the order of magnitude of the exceptional set relative to the recent result of Brudern; the method is indicated to extend to other combinations of unlike powers.

Significance. If the analytic estimates hold, the work supplies a sharper quantitative result on the density of representable integers in the mixed-power Waring-Goldbach setting. The improvement in the order of magnitude of the exceptional set is a concrete advance over the prior bound, and the claim that the same circle-method framework applies to analogous problems adds modest generality.

minor comments (3)

[Abstract] The abstract and introduction should state the precise range of k for which the result holds and the explicit form of the new exceptional-set bound (e.g., O(X^θ) with the value of θ).
[Introduction] Notation for the exceptional set E(X) and the singular series S(n) should be introduced with a displayed definition before the main theorem statement.
[Introduction] The comparison with Brudern’s result would be clearer if the previous exponent were recalled explicitly alongside the new one.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: derivation relies on external prior result and standard circle-method estimates

full rationale

The paper derives improved bounds on exceptional sets for representations n = p² + q³ + r⁶ + sᵏ by applying the circle method, controlling major arcs via the singular series (positive under the stated local conditions) and minor arcs via exponential-sum estimates. The improvement is explicitly positioned relative to Brudern's external result, with no self-citations invoked as load-bearing premises, no fitted parameters renamed as predictions, and no ansatz or uniqueness theorem smuggled in via prior work by the same authors. The local-conditions assumption is a standard modeling choice in the area and does not reduce the claimed bound to a tautology. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents exhaustive extraction. The work rests on standard domain assumptions from analytic number theory.

axioms (1)

domain assumption The Hardy-Littlewood circle method and associated exponential sum estimates apply to the generating functions built from prime powers in unlike exponents.
The abstract's focus on exceptional-set estimates presupposes the applicability of these standard tools.

pith-pipeline@v0.9.0 · 5614 in / 1210 out tokens · 34847 ms · 2026-05-24T14:35:22.901345+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Bauer, An improvement on a theorem of the Goldbach-Waring type , Rocky Mount

C. Bauer, An improvement on a theorem of the Goldbach-Waring type , Rocky Mount. J. Math. 31 (2001), 1151–1170

work page 2001
[2]

Bauer, A remark on a theorem of the Goldbach Waring type , Studia Sci

C. Bauer, A remark on a theorem of the Goldbach Waring type , Studia Sci. Math. Hungar. 41 (2004), 309–324

work page 2004
[3]

Bourgain, On the Vinogradov mean value, , Trudy Mat

J. Bourgain, On the Vinogradov mean value, , Trudy Mat. Inst. Steklova. 296 (2017), 36–46

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Br¨ udern,A ternary problem in addditive prime number theory , J., Steuding, R

J. Br¨ udern,A ternary problem in addditive prime number theory , J., Steuding, R. (eds.)From Arithmetic to Zeta-Functions. Springer, Cham (2016)

work page 2016
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J. W. Hoﬀman and G. Yu, A ternary addtive problem , Monatsh Math. 172 (2013), 293–321

work page 2013
[6]

Hua, Additive theory of prime numbers , Science press, Beijing 1957; English version, American Mathematical Society, Providence 1965

L.K. Hua, Additive theory of prime numbers , Science press, Beijing 1957; English version, American Mathematical Society, Providence 1965

work page 1957
[7]

Kumchev, On weyl sums over primes and almost primes , Michigan Math

A. Kumchev, On weyl sums over primes and almost primes , Michigan Math. J. 54 (2006), 243–268

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[8]

Liu and T

J.Y. Liu and T. Zhan, Sums of ﬁve almost equal prime squares (II) , Sci. in China. 41 (1998), 710–722

work page 1998
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Liu, An improvement on Waring-Goldbach problem for unlike power s, Acta Math

Z.X. Liu, An improvement on Waring-Goldbach problem for unlike power s, Acta Math. Hun- gar. 130 (2011), 118–139

work page 2011
[10]

Lu and Z

M.G. Lu and Z. Shan, A problem of Waring-Goldbach type , J. China Univ. Sci. Tech. Suppl. I (1982), 1–8 (in Chinese)

work page 1982
[11]

Prachar, ¨Uber ein Problem vom Waring-Goldbach’schen Typ , Monatsh

K. Prachar, ¨Uber ein Problem vom Waring-Goldbach’schen Typ , Monatsh. Math. 57 (1953), 66–74

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[12]

Ren and Kai-Man Tsang, Goldbach-Waring problem for unlike powers , Acta Math

X.M. Ren and Kai-Man Tsang, Goldbach-Waring problem for unlike powers , Acta Math. Sinica, English Series (2). 23 (2007), 265–280

work page 2007
[13]

X. M. Ren and Kai-Man Tsang, Goldbach-Waring problem for unlike powers (II) , Acta Math. Sinica (Chinese Series). 50 (2007), 175–182

work page 2007
[14]

Schwarz, Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen , II

W. Schwarz, Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen , II. J. Reine Angew. Math. 206 (1961), 78–112

work page 1961
[15]

R. C. Vaughan, The Hardy-Littlewood method , 2nd ed. Cambridge University Press, Cam- bridge 1997

work page 1997
[16]

Zhao, On the Waring-Goldbach problem for fourth and sixth powers , Proc

L. Zhao, On the Waring-Goldbach problem for fourth and sixth powers , Proc. London Math. Soc. 108 (2014), 1593–1622

work page 2014
[17]

Zhao, The additive problem with one cube and three cubes of primes , Michigan Math

L. Zhao, The additive problem with one cube and three cubes of primes , Michigan Math. J. 63 (2014), 763–779

work page 2014
[18]

Zhao, The exceptional set for sums of unlike powers of primes , Acta Math

L. Zhao, The exceptional set for sums of unlike powers of primes , Acta Math. Sinica, English Series. 30 (2014), 1897–1904. School of Mathematics and Statistics, Minnan Normal Univer sity, Zhangzhou 363000, P. R. China E-mail address : fengzz13@mails.jlu.edu.cn School of Mathematics, Jilin University, Changchun 130012 , P. R. China E-mail address : jma@jlu.edu.cn

work page 2014

[1] [1]

Bauer, An improvement on a theorem of the Goldbach-Waring type , Rocky Mount

C. Bauer, An improvement on a theorem of the Goldbach-Waring type , Rocky Mount. J. Math. 31 (2001), 1151–1170

work page 2001

[2] [2]

Bauer, A remark on a theorem of the Goldbach Waring type , Studia Sci

C. Bauer, A remark on a theorem of the Goldbach Waring type , Studia Sci. Math. Hungar. 41 (2004), 309–324

work page 2004

[3] [3]

Bourgain, On the Vinogradov mean value, , Trudy Mat

J. Bourgain, On the Vinogradov mean value, , Trudy Mat. Inst. Steklova. 296 (2017), 36–46

work page 2017

[4] [4]

Br¨ udern,A ternary problem in addditive prime number theory , J., Steuding, R

J. Br¨ udern,A ternary problem in addditive prime number theory , J., Steuding, R. (eds.)From Arithmetic to Zeta-Functions. Springer, Cham (2016)

work page 2016

[5] [5]

J. W. Hoﬀman and G. Yu, A ternary addtive problem , Monatsh Math. 172 (2013), 293–321

work page 2013

[6] [6]

Hua, Additive theory of prime numbers , Science press, Beijing 1957; English version, American Mathematical Society, Providence 1965

L.K. Hua, Additive theory of prime numbers , Science press, Beijing 1957; English version, American Mathematical Society, Providence 1965

work page 1957

[7] [7]

Kumchev, On weyl sums over primes and almost primes , Michigan Math

A. Kumchev, On weyl sums over primes and almost primes , Michigan Math. J. 54 (2006), 243–268

work page 2006

[8] [8]

Liu and T

J.Y. Liu and T. Zhan, Sums of ﬁve almost equal prime squares (II) , Sci. in China. 41 (1998), 710–722

work page 1998

[9] [9]

Liu, An improvement on Waring-Goldbach problem for unlike power s, Acta Math

Z.X. Liu, An improvement on Waring-Goldbach problem for unlike power s, Acta Math. Hun- gar. 130 (2011), 118–139

work page 2011

[10] [10]

Lu and Z

M.G. Lu and Z. Shan, A problem of Waring-Goldbach type , J. China Univ. Sci. Tech. Suppl. I (1982), 1–8 (in Chinese)

work page 1982

[11] [11]

Prachar, ¨Uber ein Problem vom Waring-Goldbach’schen Typ , Monatsh

K. Prachar, ¨Uber ein Problem vom Waring-Goldbach’schen Typ , Monatsh. Math. 57 (1953), 66–74

work page 1953

[12] [12]

Ren and Kai-Man Tsang, Goldbach-Waring problem for unlike powers , Acta Math

X.M. Ren and Kai-Man Tsang, Goldbach-Waring problem for unlike powers , Acta Math. Sinica, English Series (2). 23 (2007), 265–280

work page 2007

[13] [13]

X. M. Ren and Kai-Man Tsang, Goldbach-Waring problem for unlike powers (II) , Acta Math. Sinica (Chinese Series). 50 (2007), 175–182

work page 2007

[14] [14]

Schwarz, Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen , II

W. Schwarz, Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen , II. J. Reine Angew. Math. 206 (1961), 78–112

work page 1961

[15] [15]

R. C. Vaughan, The Hardy-Littlewood method , 2nd ed. Cambridge University Press, Cam- bridge 1997

work page 1997

[16] [16]

Zhao, On the Waring-Goldbach problem for fourth and sixth powers , Proc

L. Zhao, On the Waring-Goldbach problem for fourth and sixth powers , Proc. London Math. Soc. 108 (2014), 1593–1622

work page 2014

[17] [17]

Zhao, The additive problem with one cube and three cubes of primes , Michigan Math

L. Zhao, The additive problem with one cube and three cubes of primes , Michigan Math. J. 63 (2014), 763–779

work page 2014

[18] [18]

Zhao, The exceptional set for sums of unlike powers of primes , Acta Math

L. Zhao, The exceptional set for sums of unlike powers of primes , Acta Math. Sinica, English Series. 30 (2014), 1897–1904. School of Mathematics and Statistics, Minnan Normal Univer sity, Zhangzhou 363000, P. R. China E-mail address : fengzz13@mails.jlu.edu.cn School of Mathematics, Jilin University, Changchun 130012 , P. R. China E-mail address : jma@jlu.edu.cn

work page 2014