A Diophantine problem with prime variables

We study the distribution of the values of the form $λ_1 p_1 + λ_2 p_2 + λ_3 p_3^k$, where $λ_1$, $λ_2$ and $λ_3$ are non-zero real number not all of the same sign, with $λ_1 / λ_2$ irrational, and $p_1$, $p_2$ and $p_3$ are prime numbers. We prove that, when $1 < k < 4 / 3$, these value approximate rather closely any prescribed real number.

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