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Jesmanowicz Conjecture and Gaussian Integer Ring
Version 1
: Received: 22 April 2024 / Approved: 23 April 2024 / Online: 24 April 2024 (09:32:50 CEST)
How to cite: Feng, N.; Cao, J.; Wang, Y. Jesmanowicz Conjecture and Gaussian Integer Ring. Preprints 2024, 2024041559. https://doi.org/10.20944/preprints202404.1559.v1 Feng, N.; Cao, J.; Wang, Y. Jesmanowicz Conjecture and Gaussian Integer Ring. Preprints 2024, 2024041559. https://doi.org/10.20944/preprints202404.1559.v1
Abstract
Let a,b,c be positive integers such that a2+b2=c2, 2|b,gcd(a,b)=1. In 1956, Jesmanowicz conjectured that for any positive integer w, the only solution of (aw)x+(bw)y=(cw)z in positive integers is (x, y, z) = (2, 2, 2). In this paper, based on Gaussian integer ring, we show that Je$\acute{s}$manowicz' conjecture is true for any positive integer
Keywords
Jesmanowicz conjecture; Diophantine equation; Gaussian integer ring; 4k+1 type prime number
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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