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April 3, 20260 citationsOpen Access

Diophantine approximation with two squares and three biquadrates of primes

YFYu FuEducation Department of Jiangxi ProvinceLHLiqun HuEducation Department of Jiangxi ProvinceSLSiqi LiuEducation Department of Jiangxi Province

Key Points

  • The research aims to understand the solvability of specific inequalities involving prime numbers using Diophantine approximation.
  • Introduce a sequence V under specific conditions on λ values and irrational ratios.
  • Examine the count of numbers v in V that satisfy the unsolvable inequality related to primes.
  • Analyze the behavior of the count as a function of N, δ, and ɛ.
  • The quantity of v satisfying the given condition is shown to be bounded by O(N^(6/7) + 2δ + ɛ).
  • Findings indicate that the unsolvability of the inequalities has specific constraints based on the arrangement of λ values.

Abstract

Let λ1, λ2, λ3, λ4, λ5 be nonzero real numbers, neither all positive nor all negative and λ1/λ2 be an irrational number. Let V be a well-spaced sequence and δ > 0. For arbitrary ɛ > 0, we show that the quantity of v ∈ V with v ≤ N making the inequality |λ1p12 + λ2p22 + λ3p34 + λ4p44 + λ5p54 − v| < v−δ unsolvable in primes p1,p2,p3,p4,p5 does not exceed O(N6/7 + 2δ + ε).

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Cite This Study

Fu et al. (2026) studied this question.

synapsesocial.com/papers/69cf5d775a333a821460b3efhttps://doi.org/10.1051/ita/2026013/pdf
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