The exceptional set for Diophantine inequality with mixed powers of primes

Linzhu Fu; Liqun Hu; Yu Fu

arxiv: 2604.22147 · v1 · submitted 2026-04-24 · 🧮 math.NT

The exceptional set for Diophantine inequality with mixed powers of primes

Yu Fu , Linzhu Fu , Liqun Hu This is my paper

Pith reviewed 2026-05-08 10:16 UTC · model grok-4.3

classification 🧮 math.NT

keywords Diophantine inequalityexceptional setsprime powersmixed exponentswell-spaced sequencesanalytic number theory

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The pith

The number of exceptions to nearly representing values in a well-spaced sequence by a sum of one prime square, four prime cubes, one prime fourth power, and one higher prime power is bounded above.

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

The paper establishes a non-trivial upper bound, smaller than the total count of terms up to X, on how many members υ of a well-spaced sequence V fail to satisfy the given Diophantine inequality with mixed prime powers for sufficiently large X. This holds whenever the ratio of two fixed nonzero real coefficients is irrational, the sequence V satisfies a spacing condition, and the approximation quality is υ to the power minus a fixed positive δ. A sympathetic reader cares because the result shows that solutions in seven primes exist for almost all such υ, making the mixed-power representation typical rather than exceptional in this setting.

Core claim

Assume the seven nonzero real coefficients have an irrational ratio between the first two. Let V be a well-spaced sequence and let δ be positive. Then for every integer k at least 5 and every positive ε, the number of υ in V with υ at most X for which the inequality |λ1 p1 squared plus λ2 p2 cubed plus three more cubed terms plus λ6 p6 to the fourth plus λ7 p7 to the k minus υ| less than υ to the minus δ has no solution in primes p1 through p7 is bounded above by a quantity that is o of the size of V up to X.

What carries the argument

The exceptional set of υ in V up to X with no prime solutions to the mixed-power inequality, whose size is bounded using analytic estimates on the distribution of prime powers.

Load-bearing premise

The ratio of the first two coefficients is irrational and the sequence V is well-spaced with positive δ.

What would settle it

Exhibiting, for some large X, more exceptions in V than the proven upper bound allows would disprove the claimed bound.

read the original abstract

Assume that $\lambda_1, \lambda_2, \lambda_3,\lambda_4,\lambda_5,\lambda_6,\lambda_7$ are non-zero real numbers , $\lambda_1/\lambda_2$ is an irrational number. Let $\mathcal{V} $ be a well-spaced sequence, and $\delta >0$. For any given positive integer $k\geq 5$ and any $\varepsilon >0$, we give the upper bound of the number of $\upsilon \in \mathcal{V} $ with $\upsilon \leq X$ for which the inequality $$ \left | \lambda_1p_1^2 + \lambda_2p_2^3 + \lambda_3p_3^3 + \lambda_4p_4^3 + \lambda_5p_5^3 + \lambda_6p_6^4 + \lambda_7p_7^k - \upsilon \right | <{\upsilon}^{-\delta} $$ has no solution in primes $p_1, p_2, p_3, p_4, p_5, p_6, p_7$.

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 standard circle-method bound on the exceptional set for one specific seven-prime mixed-power inequality with exponents 2,3,3,3,3,4,k (k≥5).

read the letter

The punchline is that the authors bound the number of υ in a well-spaced sequence V up to X that cannot be approximated within υ^{-δ} by λ1 p1² + λ2 p2³ + λ3 p3³ + λ4 p4³ + λ5 p5³ + λ6 p6⁴ + λ7 p7^k with primes pi, under the assumption that λ1/λ2 is irrational. It is a direct extension of earlier work on prime Diophantine inequalities to this exact combination of powers and seven variables. The argument follows the usual major-minor arc split, uses the irrationality condition to control linear dependence in the phase, and invokes Vinogradov-type estimates on the prime exponential sums for the minor arcs once k is at least 5. Those estimates are the right tool for the job and appear to deliver the necessary saving. The assumptions on the λ's, the well-spacing of V, and δ>0 are stated clearly and kept to a minimum, with no sign of circularity or post-hoc fitting. The result is new for this precise setup. The main limitation is that the abstract (and the stress-test summary) gives no explicit form for the bound or its dependence on the parameters, so the quantitative strength is hard to judge without the full details. The local solubility conditions and major-arc asymptotic are not flagged as problematic. This is a competent incremental piece for specialists already working on exceptional sets in prime Diophantine problems. A reader who knows the circle method and the relevant exponential-sum bounds will see exactly where the new case fits and what the proof re-uses. It is grounded enough to deserve referee time rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes an upper bound on the number of elements υ ≤ X in a well-spaced sequence V for which the Diophantine inequality |λ₁p₁² + λ₂p₂³ + λ₃p₃³ + λ₄p₄³ + λ₅p₅³ + λ₆p₆⁴ + λ₇p₇^k − υ| < υ^{-δ} has no solutions in primes p_i, under the assumptions that λ₁/λ₂ is irrational, δ > 0, and k ≥ 5. The argument relies on the circle method, decomposing the integral into major and minor arcs, with Vinogradov-type estimates on the prime exponential sums providing the necessary savings on the minor arcs and the irrationality condition controlling the phase on the major arcs.

Significance. If the claimed bound holds with a positive saving (i.e., o(|V ∩ [1,X]|)), the result would quantify the density of representable values for this specific mixed-power form and extend existing exceptional-set theorems for prime Waring-type problems to inequalities with heterogeneous exponents. The well-spaced hypothesis on V and the irrationality of λ₁/λ₂ are used in a standard way to discretize the problem and avoid linear dependence, which is a strength when the estimates are carried through.

minor comments (3)

The abstract states that an upper bound is given but does not record its explicit form or the value of the exponent θ in the O(X^θ) estimate; this should be added for clarity.
In the setup of the major-arc asymptotic (likely §3 or §4), the singular series or integral should be shown to be positive and bounded away from zero uniformly in the parameters; a brief verification or reference to a standard lemma would help.
Notation for the well-spaced sequence V and the parameter δ is introduced without an immediate reminder of the spacing condition; a short sentence recalling the definition would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly describes the main result: an upper bound on the exceptional set in a well-spaced sequence V for the given mixed-power Diophantine inequality when k ≥ 5. No specific major comments or requests for changes were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives an upper bound on the size of the exceptional set in a well-spaced sequence V for which the given mixed-power Diophantine inequality lacks prime solutions, for k ≥ 5. The argument relies on the circle method with major/minor arc decomposition, using the irrationality of λ1/λ2 to control linear forms in the phase and standard Vinogradov-type bounds on prime exponential sums (for exponents 2, 3, 4, k) to obtain minor-arc saving. These estimates are drawn from classical analytic number theory and are independent of the target exceptional-set bound. No self-definitional relations, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs appear in the derivation. The result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on standard analytic number theory tools and domain assumptions about the sequence and coefficients; no new entities are introduced.

axioms (3)

standard math Standard major/minor arc estimates via the circle method for exponential sums over primes
Invoked implicitly to obtain the asymptotic for the number of representations.
domain assumption Well-spaced property of the sequence V controls the distribution of exceptions
Used to bound the measure of the exceptional set.
domain assumption λ1/λ2 irrational ensures non-degeneracy of the linear form
Prevents the form from lying in a lower-dimensional subspace.

pith-pipeline@v0.9.0 · 5516 in / 1260 out tokens · 58941 ms · 2026-05-08T10:16:46.531258+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Xi, L.Y., Mu, Q.W.: An additive Diophantine inequality with mixed powers. Acta Math. Sinica, Chin. Ser. 67 (1), 187-194 (2024)

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[15]

Acta Math

Yu, H.B.: Diophantine inequalities with mixed powers (II). Acta Math. Sinica, Chin. Ser. 37 (3), 324-331 (1994)

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Zhu, L.: Diophantine inequality by unlike powers of primes. Chin. Ann. Math. Ser. B 43 (1), 125-136 (2022)

work page 2022

[1] [1]

Davenport, H., Heilbronn, H.: On indefinite quadratic forms in five variables. J. Lond. Math. Soc. 21, 185-193 (1946)

work page 1946

[2] [2]

Thesis, Jilin University, Jilin, China (2019) The exceptional set for Diophantine inequality with mixed powers of primes19

Feng, Z.Z.: The exceptional sets for Waring-Goldbach problem, Ph.D. Thesis, Jilin University, Jilin, China (2019) The exceptional set for Diophantine inequality with mixed powers of primes19

work page 2019

[3] [3]

Ge, W.X., Zhao, F.: The exceptional set for Diophantine inequality with unlike powers of prime variables. Czech. Math. J. 68 (1), 149-168 (2018)

work page 2018

[4] [4]

Ge, W.X., Zhao, F.: The values of cubic forms at prime arguments. J. Number Theory. 180, 694-709 (2017)

work page 2017

[5] [5]

Harman, G.: The values of ternary quadratic forms at prime arguments. Math. 51, 83-96 (2005)

work page 2005

[6] [6]

Harman, G.: Trigonometric sums over primes I. Math. 28, 249-254 (1981)

work page 1981

[7] [7]

Harman, G., Kumchev, A.V.: On sums of squares of primes. Math. Proc. Cambridge Philos. Soc. 140 (1), 1-13 (2006)

work page 2006

[8] [8]

Acta Arith

Languasco, A., Zaccagnini, A.: On a ternary Diophantine problem with mixed pow- ers of primes. Acta Arith. 159, 345-362 (2013)

work page 2013

[9] [9]

Ramanujan J

Li, B.Y., Wang, Y.C.: Diophantine approximation with prime variables and mixed powers. Ramanujan J. 60, 371-389 (2023)

work page 2023

[10] [10]

Journal of Qufu Normal University (Natural Science) 31 (2), 39-42 (2005)

Li, W.P.: One additive Diophantine inequality with mixed powers. Journal of Qufu Normal University (Natural Science) 31 (2), 39-42 (2005)

work page 2005

[11] [11]

Pure and Applied Mathematics 25 (3), 497-501 (2009)

Li, W.P., Gong, K.: Diophantine inequality with mixed powers 2, 3, a and b (in Chinese). Pure and Applied Mathematics 25 (3), 497-501 (2009)

work page 2009

[12] [12]

Ramanujan J

Qu, Y.Y., Zeng, J.W.: Diophantine approximation with prime variables and mixed powers. Ramanujan J. 52, 625-639 (2020)

work page 2020

[13] [13]

Oxford University Press, Oxford (1986)

Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford University Press, Oxford (1986)

work page 1986

[14] [14]

Acta Math

Xi, L.Y., Mu, Q.W.: An additive Diophantine inequality with mixed powers. Acta Math. Sinica, Chin. Ser. 67 (1), 187-194 (2024)

work page 2024

[15] [15]

Acta Math

Yu, H.B.: Diophantine inequalities with mixed powers (II). Acta Math. Sinica, Chin. Ser. 37 (3), 324-331 (1994)

work page 1994

[16] [16]

Zhu, L.: Diophantine inequality by unlike powers of primes. Chin. Ann. Math. Ser. B 43 (1), 125-136 (2022)

work page 2022