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This work is made of two parts. In the first part, with the use of covering spaces, we study the Nielsen number of coincidence of functions. In the second part, we make some applications of the results of the first part. Being f,g: M → M continuous maps, with M an oriented, connected, closed manifold, we compute' A(f,g) when H*(M;Q) has a simple syStems of generators. If M is in addition an H-space with multiplication m, let us define, for x ∈ M, m₂(x) = m(x,x) and m_k(x)=m(x,m_m-1(x)) for k ≥ 2. We .show that the equation m_k(x) = m_s(x), k > s, has , at least (k-s)^β roots, where β is the first Betti number of M , and further we study the primitive roots of such equation.