A prime number generator for devices for noise-resistant transmission of information along a radio channel (Q1363888)

A matrix recursion for the generation of prime numbers is studied. The equation is of the form: \[ A_i^{(N_k )} = \left[ A_n^{(N_k )} P_{k-1} + B_i^{(N_k )} N_k \right]\pmod{P_k} \] where \(N_k\) is the \(k\)-th prime number, \(P_k = \prod_{m=1}^k N_m\), \(A_n^{(N_k )}\) is a matrix of dimension \((N_k -1) \cdot (N_{k-1} -1)\) and \(B_i^{(N_k )}\) is a matrix of dimension \((N_k -1) \times (N_{k-1} -1)\). The matrices \(A_i^{(N_k )}\) are viewed as initial elements of another recursion and its properties are studied. It is shown, in particular, that all the elements of these matrices \(A_i^{(N_k )}\) that are less than \(N_{k+1}^2\) are primes if the product \(P_k\) contains all the prime numbers from \(1\) to \(N_k\).