LAUSR

Abstract Let λ 1 , λ 2 , λ 3 , λ 4 , λ 5 be non-zero real numbers, not all negative. Let V be a well-spaced sequence, δ > 0 . If λ 1 / λ 2 is irrational and algebraic, then we prove that E ( V , X , δ ) ≪ X 17 / 18 + 2 δ + e , where E ( V , X , δ ) denotes the number of v ∈ V with v ≤ X such that the inequality | λ 1 p 1 3 + λ 2 p 2 3 + λ 3 p 3 3 + λ 4 p 4 3 + λ 5 p 5 3 − v | v − δ has no solution in primes p 1 , p 2 , p 3 , p 4 , p 5 . Further, we assume that except for one, all other the ratios λ k / λ l ( 1 ≤ k l ≤ 5 ) are irrational and algebraic, then 17/18 can be replaced by 11/12. These improve the earlier results.