How to take from Pascal’s triangle an infinite series of power sums from many variables and arithmetic systems compared modulo a prime number
In this arithmetic study, some unclaimed and unknown numerical properties of Pascal’s irregular triangles are proposed for further study. These properties made it possible to find a universal method for finding many symmetric polynomials of power sums from Pascal’s triangle tables. The same properties helped to establish two formulas for directly finding all primes. The above became available after the successful decryption of a certain group of Pascal’s triangle tables in the cryptographic subsystem. The rules of real and arithmetic operations for such tables have been found and have been unambiguously defined, therefore, implementation of the tasks on a computer is possible. Plus, there was no place for special information in combinatorial problems in the structural part of the logical material. The method of building arithmetic tables is universal and makes it possible to get their further development in the subsystem of numerical irregular triangles. Further, it has been found that only such tables can transmit arithmetic information by decryption for scientific and mathematical purposes.