On the Solution of the Equation n = ak + bp_k by Means of an Iterative Method

11/28/2021

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by   Juan Luis Varona, et al.

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For fixed positive integers n, we study the solution of the equation n = k + p_k, where p_k denotes the kth prime number, by means of the iterative method k_j+1 = π(n-k_j), k_0 = π(n), which converges to the solution of the equation, if it exists. We also analyze the equation n = ak + bp_k for fixed integer values a 0 and b>0, and its solution by means of a corresponding iterative method. The case a>0 is somewhat similar to the case a=b=1, while for a<0 the convergence and usefulness of the method are less satisfactory. The paper also includes a study of the dynamics of the iterative methods.