# Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 39. no. 2. pp. 222–236. ISSN 2079-6641

**MSC 11A63 **

**Research Article**

**How to take from Pascal’s triangle an infinite series of power sums from many variables and arithmetic systems compared modulo a prime number**

**V. L. Shcherban’**

Center for Additional Mathematical Education, 640000, Kurgan, Tomina str. 53, Russia

E-mail: sherba-q@ya.ru

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.

*Key words: recurrent numerical sequences; symmetric polynomials; prime numbers; Pascal’s triangle.*

DOI: 10.26117/2079-6641-2022-39-2-222–236

Original article submitted: 05.07.2022

Revision submitted: 22.08.2022

**For citation.** Shcherban’ V. L. How to take from Pascal’s triangle an infinite series of power sums from many variables and arithmetic systems compared modulo a prime number. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 39: 2, 222–236. DOI: 10.26117/2079-6641-2022-39-2-222–236

**Competing interests.** The author declares that there are no conflicts of interest with respect to authorship and publication.**Contribution and responsibility.** The author contributed to the writing of the article and is solely responsible for submitting the final version of the article to the press. The final version of the manuscript was approved by the author.

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Shsherban’ V. L., 2022

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** Scherban’ V. L.** – Head of the educational Department, Center for Additional Mathematical Education, Kurgan, Russia, ORCID 0000-0002-5631-9681.