The authors exhibit a class \({\mathcal N} {\mathcal R}\) of compact solvmanifolds such that for any \(S \in {\mathcal N} {\mathcal R}\) and any selfmap \(f : S \to S\) the Nielsen number \(N(f)\) equals the absolute value \(|L(f) |\) of the Lefschetz number. A solvmanifold is a homogeneous space of a solvable Lie group \(G\), \(S = G/ \Delta\), and the class \({\mathcal N} {\mathcal R}\) (``no roots'') consists roughly speaking of those compact solvmanifolds \(G/ \Delta\) such that, for all \(g \in G\), one is the only eigenvalue of \(\text{Ad} (g)\) which is a root of unity. Every compact exponential solvmanifold belongs to \({\mathcal N} {\mathcal R}\), but the Klein bottle does not (and in fact it is known that \(N(f) \neq |L(f) |\) may occur in this case). The precise definition of the class \({\mathcal N} {\mathcal R}\) involves the minimal Mostow fibration \(N \to S \to T\) of the solvmanifold \(S\) with \(N\) a nilmanifold and \(T\) a torus.