Двоїстий алгоритм пошуку простих чисел на відрізках великих розмірностей

Abstract

A matrix model of subsequence of natural numbers with a multiplicative basis from the first prime numbers is proposed. The matrix model is a square matrix, where vector columns are arithmetic progressions with difference and number of progressions equal to product of base elements. By crossing out arithmetic progressions with first terms which are multiples of base elements, we obtain a symmetric sparse matrix that contains all the prime numbers of subsequence of natural numbers, except for the base ones, which increases the density of prime numbers in sparse matrices. Sparse matrices are not formed clearly, only the vector of first terms of arithmetic progressions is formed. Properties of sparse matrices have been proven. Formulas that speed up the calculation of compound numbers in arithmetic progressions have been derived, the structures of vector elements of first terms of arithmetic progressions have been determined, the connectivity of symmetric parts of sparse matrices has been investigated. With the expansion of the base the number of pairs of elements with the difference equal to the power of two («twins», «fours», etc.) increases in the vector of first terms. This is a necessary condition for the existence of constants for which linear equations of two variables can have an infinite set of solutions in prime numbers. The irregularity of distribution of prime numbers in subsequences of natural numbers is related to the structure of elements of the vector of first terms. An algorithm for finding prime numbers on segments of large dimensions with a parallel calculation process has been built. The proposed algorithm is binary to algorithms for sifting subsequences of natural numbers by prime divisors. In these algorithms, it is not possible to parallelize the calculation process, since screening procedure requires storing the numerical information from the preceding steps (vector model of array processing). The binary algorithm calculates compound numbers in each pair of arithmetic progressions with symmetric first terms simultaneously, using only vector of the first terms of arithmetic progressions, which makes possible processing of large-dimensional arrays