Mathematics, Informatics, Physics: Science and Education

Electronic scientific journal

Vol. 2 No. 1 (2025)
Actual problems of mathematics

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

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  • Vasyl Abramchuk,
  • Ihor Abramchuk
Vasyl Abramchuk
Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University
Bio
Ihor Abramchuk
Vinnytsia National Technical University
Bio

Published 2025-05-21

Keywords

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

Copyright (c) 2025 Василь Абрамчук, Ігор Абрамчук

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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
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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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