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

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

$\begin{displaymath}k_{j+1} = \pi(n-k_j), \qquad k_0 = \pi(n),\end{displaymath}$

which converges to the solution of the equation, if it exists. We also analyze the equation n = ak + bp_k for fixed integer values $a \ne 0$ 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.

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

Juan Luis Varona Departamento de Matemáticas y Computación Universidad de La Rioja Logroño Spain

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

Juan Luis Varona
Departamento de Matemáticas y Computación
Universidad de La Rioja
Logroño
Spain