Algorithm for investigating the unsolvability of the equation zn=xn+yn, n≥3 in integers

Authors

DOI:

https://doi.org/10.31652/3041-1955-2025-02-01-01

Keywords:

necessary conditions, classes of parameters, irrational numbers, Fermat’s sequence, matrix model of powers

Abstract

The necessary conditions under which an equation can have a solution in positive integers are determined. The parameters of the equation  are consistent with  and belong to bounded closed sets. The exponents and variables are divided into classes. It is proved that in the space of mixed variables connected by an equation, where one of the variables is real and the others are integers, the value of the real variable is irrational, which is a sufficient condition for the unsolvability of the equation in positive integers for all exponents of powers greater than three. A matrix (linear) model of powers of positive integers is constructed.

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Author Biographies

  • Vasyl Abramchuk, Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University

    Vasyl Abramchuk, Candidate of Science in Physіcs and Mathematics, Professor, Department of Mathematics and Informatics, Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University, 32 Ostrozkyi Str., Vinnytsia 21001, Ukraine

  • Ihor Abramchuk, Vinnytsia National Technical University

    Ihor Abramchuk, senior lecturer,  Department of Higher Mathematics, Vinnytsia National Technical University, 95 Khmelnytske highway Str., Vinnytsia 21000, Ukraine

References

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Абрамчук В.С., Абрамчук І.В. Алгоритми розкладу цілих чисел і гладкого наближення функцій. Математичне та комп’ютерне моделювання. Серія фіз.-мат. науки. 2022. Випуск 23. С. 6-13. DOI: https://doi.org/10.32626/2308-5878.2022-23.5-13

Абрамчук В.С., Абрамчук І.В. Двоїстий алгоритм пошуку простих чисел на відрізках великих розмірностей. Математичне та комп’ютерне моделювання. Серія фіз.-мат. науки. 2024. Випуск 25. С. 6-19. DOI: https://doi.org/10.32626/2308-5878.2024-25.6-19

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Published

2025-05-21

Issue

Vol. 2 No. 1 (2025)

Section

Actual problems of mathematics

License

Copyright (c) 2025 Василь Абрамчук, Ігор Абрамчук

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Algorithm for investigating the unsolvability of the equation zn=xn+yn, n≥3 in integers. (2025). Mathematics, Informatics, Physics: Science and Education, 2(1), 1-8. https://doi.org/10.31652/3041-1955-2025-02-01-01