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

TL;DR

This paper investigates iterative methods to solve equations involving the n-th prime number and linear combinations of k and p_k, analyzing convergence and dynamics for different parameter cases.

Contribution

It introduces and analyzes iterative algorithms for solving prime-related equations, including convergence properties and dynamics for various parameter settings.

Findings

The iterative method converges for certain parameter cases.

Convergence is less effective when a<0.

The dynamics of the iterative methods are studied in detail.

Abstract

For fixed positive integers $n$, we study the solution of the equation $n = k + p_k$, where $p_k$ denotes the $k$th prime number, by means of the iterative method \[ k_{j+1} = \pi(n-k_j), \qquad k_0 = \pi(n), \] which converges to the solution of the equation, if it exists. We also analyze the equation $n = ak + bp_k$ for fixed integer values $a \ne 0$ 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.