A Diophantine problem with prime variables

We study the distribution of the values of the form $\lambda_1 p_1 + \lambda_2 p_2 + \lambda_3 p_3^k$,where$\lambda_1$,$\lambda_2$and$\lambda_3$arenon-zerorealnumbernotallofthesamesign,with$\lambda_1 / \lambda_2$irrational,and$p_1$,$p_2$and$p_3$ are prime numbers. We prove that, when $1<k<4\mathrm{/}3$, these value approximate rather closely any prescribed real number.