Mkrtumyan E.L. Matrix Mathematics

Bilingual (english, russian) ed. — Independently published, 2019. — 299 p.A new method of research of mathematical problems in which the concept of number is used is developed. It is based on the concept of "residue matrix". The elements of these matrices are the residuals of dividing some sets of positive integers by Prime numbers.
The structure of such matrices is studied in detail, the regularities of distribution of zero and non-zero residues in them are established. The connections between the residue matrices corresponding to different sets of natural numbers are studied. The residue matrix method is applied to solve a number of problems in number theory. Proved a number of theorems describing the distribution of Prime numbers on the number line.
The formulas Mersen and Fermat prime numbers, and also obtained other formulas for primes. The problems of Shinzel and Goldbach. A method for fnding solutions to the Serpinsky-Erdos equation is proposed.
Inside...
On some problems of distribution of prime numbers.
On the estimation of the numbers of primes in certain intervals.
About some problems of prime numbers.
Prime and relatively prime numbers in special intervals I.
Prime and relatively prime numbers in special intervals II.
Adjacent residual matrices and sieve method.
The matrices pf residues and the Schinzel problem.
The Goldbach problem.
Properties of matrices of residues.
Square matrices. Focuses of matrices of residues.
Average matrix of residues.
Prime numbers in certain intervals of numbers,
and an estimate of their number on two sides.
Power matrices of residues.
Focuses and focal distances in matrices of residues.
The problem of prime numbers - twins.
Division of numbers on a number axis.
Function π(n) and the problem of cryptography.
Commensurable and incommensurable segments.
Opposing and shifted matrices of residues.
Double sieve.
Range of columns in matrices of residues.
Numbering on the number axis.
The main and side focuses of matrices of residues.
Decay of the numeric axis.
Prime figurative numbers and combinations.О некоторых задачах распределения простых чисел.
Об оценке количеств простых чисел в определенных промежутках.
О некоторых проблемах простых чисел.
О простых и взаимнопростых числах в специальных промежутках I.
О простых и взаимнопростых числах в специальных промежутках II.
Смежные матрицы остатков и метод решета.
Матрицы остатков и проблема Шинцеля.
Проблема Гольдбаха.
Свойства матриц остатков.
Квадратные матрицы. Фокусы матриц остатков.
Средняя матрица остатков.
Простые числа в определенных промежутках чисел, и оценка их количества с двух сторон.
Степенные матрицы остатков.
Фокусы и фокусные расстояния в матрицах остатков.
Проблемы простых чисел – близнецов.
Деление чисел на числовой оси.
Функция π(n) и проблема криптографии.
Соизмеримые и несоизмеримые отрезки.
Противолежащие и сдвинутые матрицы остатков.
Двойное решето.
Дальность столбцов в матрицах остатков.
Числообразование на числовой оси.
Главные и побочные фокусы матриц остатков.
Распад числовой оси.
Простообразные числа и комбинаторика.