Sequences with Power-Law Properties
https://doi.org/10.18384/2949-5067-2025-4-100-109
Abstract
Aim. The purpose of this paper is to prove theorems on the existence, nonexistence, and divisibility of power sequences.
The power sequences considered in this paper consist of elements that generalize the properties of well-known Diophantine equations, such as the unsolvable equation in Fermat's Last Theorem or the equation relating the lengths of the sides of a right triangle using the Pythagorean theorem.
Methodology. In this work, the methods of elementary number theory were mainly used. In this work, the methods of elementary number theory were mainly used.
Results. The result of the work is completely proven theorems on sequences generalizing Diophantine equations of high degrees.
Research implications. The theoretical significance of this study lies in its generalization of the concept of Diophantine equations to a set of sequences. The work is purely theoretical in nature.
Keywords
Fermat's Great Theorem,
The Diophantine equation,
Pythagorean triples,
sequence,
The Pythagorean Theorem
Supporting agencies: This research was supported by an RNF grant No. 25-21-00273 “A study of the scientific foundations of the derivation of the Lamé and Bourgain-Kontorovich theorems with the aim of refining them and generalizing them to wider classes of argument sets”
About the Authors
I. Kan
Moscow Aviation Institute (National Research University)
Russian Federation
Russian Federation
Igor D. Kan, Dr. Sci. (Phys.-Math.), Leading Researcher, Prof.
Department No. 311 “Applied Software and Mathematical Methods”
Moscow
N. Zverev
Moscow Aviation Institute (National Research University)
Russian Federation
Russian Federation
Nikolay A. Zverev, Cand. Sci. (Phys.-Math.), Researcher, Assoc. Prof.
Department No. 311 “Applied Software and Mathematical Methods”
Moscow
E. Davidenko
Moscow Aviation Institute (National Research University)
Russian Federation
Russian Federation
Ekaterina V. Davidenko, Technician
NIO-311 (Research Department of the Department No. 311 “Applied Software and Mathematical methods”)
Moscow
References
1. Alvarez, L. F. (2015). The Hardest Problem in the World. Fermat. Fermat's Last Theorem. Moscow: De Agostini publ. (Series: Science. Greatest Theories. Vol. 18) (in Russ.).
2. Violant i Holz, A. (2014). Fermat's Riddle. A Three-Century Challenge to Mathematics. Moscow: De Agostini publ. (Series: The World of Mathematics. Vol. 9) (in Russ).
3. Manin, Yu. I. & Panchishkin, A. A. (2009). Introduction to Modern Number Theory. Moscow: Moscow: Moscow Center for Continuous Mathematical Education publ. (in Russ.).
4. Postnikov, M. M. (1982). Introduction to the Theory of Algebraic Numbers. Moscow: Nauka publ. (in Russ.).
5. Ribenboim, P. (2003). Fermat's Last Theorem for Amateurs. Moscow: Mir publ. (in Russ.).
6. Singh, S. (2000). Fermat's Last Theorem. Moscow: Moscow Center for Continuous Mathematical Education publ. (in Russ.).
7. Khinchin, A. Ya. (1934). Fermat's Last Theorem. Moscow: ONTI (United Scientific and Technical Publishing House) publ. (in Russ.).
8. Edwards, G. (1980). Fermat's Last Theorem. Moscow: Mir publ. (in Russ.).
9. Darmon, H. (1999). A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced. In: Notices of the American Mathematical Society, 46 (11), 1397–1406.
10. Conrad, B., Diamond, F. & Taylor, R. (1999). Modularity of certain potentially Barsotti-Tate Galois representation. In: Journal of the American Mathematical Society, 12 (2), 521–567.
11. Stewart, I. (2016). The Greatest Mathematical Problems. Moscow: Alpina Non-Fiction publ. (in Russ.).
12. Wiles, A. (1995). Modular elliptic curves and Fermat’s Last Theorem. In: Annals of Mathematics, 141 (3), 443–551. DOI: 10.2307/2118559.
13. Soloviev, Yu. P. (1998). Taniyama's Conjecture and Fermat's Last Theorem. In: Soros Educational Journal, 2, 135–138 (in Russ.).
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