Arithmetic table as an integral part of all computational mathematics

The paper is devoted to studying the following issue as a statement. What do we know and what we don’t know about arithmetic tables. Perhaps there is no mathematical problem as naive or as simple as finding a method for creating arithmetic tables. We confirm that the general method has not been found yet. This study provides a nonterminal solution to this problem. Why? The presentation of arithmetic material in essence, plus some accompanying ideas, makes it possible to develop them further in the system. Materials and methods . The system looks like this: a numerical table as Pascal's triangle and a symmetric polynomial in two or three variables. Some arithmetic properties of such tables will be found, studied, and proved. All this was made possible only after the successful decryption of the entire class of numeric tables of truncated triangles in the cryptographic system. Results. For example, the arithmetic properties of truncated Pascal’s triangle for finding all prime numbers have been found and presented, and then their formulas have been placed. In addition to elementary addition and subtraction tables, unlimited “comparison” tables of numbers are given and presented for the first time. Conclusions. For computer implementation of the objectives set, the rules of real actions that should exist for tables have been laid down. Only the recurrent numeric series should be used for this purpose.