Diophantine approximation with two squares and three biquadrates of primes

Nanchang, Jiangxi 330031, P.R. China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 14 January 2026
Accepted: 26 February 2026

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(N^{6/7 + 2δ + ε}).