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Let q be a prime power and n ≥ 2 an integer. We denote by F_q the finite field of q elements and by F_ {q^n} its extension of degree n. An element of F^∗_{q^n} of order (q^n−1)/r, where r | q^n−1, is called r-primitive, while, if r = 1, we simply call it primitive. If θ is a generator of the extension F_{q^n} /F_q, i.e., is such that F_ {q^n} = F_q(θ), then T_θ := {θ + x : x ∈ F_q} is the set of translates of θ over Fq and, if α ∈ F^∗_{q^n} , L_{α,θ} := {α(θ + x) : x ∈ F_q} is the line of α and θ over F_q. It is known that, given n, if q is large enough, every set of translates and every line contain a primitive element, while effective versions for these existence results are known for just a few small values of n. In this work, we extend these existence results to r-primitive elements and we provide effective results for the case r = n = 2. This work is still in progress and is in collaboration with Stephen D. Cohen.