On certain C-test words for free groups

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Let Fₘ be a free group of a finite rank m > 1 and Xᵢ, Yⱼ be elements in Fₘ. A non-empty word w(x₁,..., xₙ) is called a C-test word in n letters for Fₘ if, whenever w(X₁,..., Xₙ)=w(Y₁,..., Yₙ) not equal to 1, the two n-tuples (X₁,..., Xₙ) and (Y₁,..., Yₙ) are conjugate in Fₘ. In this paper we construct, for each n > 1, a C-test word vₙ(x₁,..., xₙ) with the additional property that vₙ(X₁,..., Xₙ)=1 if and only if the subgroup of Fₘ generated by X₁,..., Xₙ is cyclic. Making use of such words vₘ(x₁,..., xₘ) and vₘ₊₁(x₁,..., xₘ₊₁), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u₁, u₂ in Fₘ such that every endomorphism of Fₘ with non-cyclic image is completely determined by its values on u₁, u₂.