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April 3, 20260 citationsOpen Access

Diophantine approximation with two squares and three biquadrates of primes

YFYu FuLHLiqun HuSLSiqi Liu

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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