For k ≥ 5 the exceptional set in a well-spaced sequence V where |λ1 p1² + λ2 p2³ + λ3 p3³ + λ4 p4³ + λ5 p5³ + λ6 p6⁴ + λ7 p7^k − υ| < υ^{-δ} has no prime solutions is bounded above for υ ≤ X.
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The exceptional set for Diophantine inequality with mixed powers of primes
For k ≥ 5 the exceptional set in a well-spaced sequence V where |λ1 p1² + λ2 p2³ + λ3 p3³ + λ4 p4³ + λ5 p5³ + λ6 p6⁴ + λ7 p7^k − υ| < υ^{-δ} has no prime solutions is bounded above for υ ≤ X.