On some particular regular Diophantine 3-tuples
Volume 3, Issue 1, pp 29--38
http://dx.doi.org/10.22436/mns.03.01.04
Publication Date: August 02, 2019
Submission Date: November 14, 2018
Revision Date: January 03, 2019
Accteptance Date: January 30, 2019
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Authors
O. Ozer
- Department of Mathematics, Faculty of Science and Arts , Kirklareli University, Kirklareli, 39100, Turkey.
Z. C. Sahin
- Department of Mathematics, Faculty of Science and Arts, Suleyman Demirel University, Isparta, Turkey.
Abstract
Diophantine n-tuple where n=3 is called as a Diophantine triple. It means that Diophantine triple is a set of three positive integers satisfying special condition. For example, \(\{a,b,c\}\) is called a \(D(k)\)-Diophantine triple if multiplying of any two different of them plus k is a perfect square integer where k is an integer.
In this work, we take in consideration some kind of regular \(D(\pm 3^3 )\)-Diophantine triples. We demonstrate that such sets can not be extendible to \(D(\pm 3^3 )\)-Diophantine quadruple by using algebraic methods such as classical Pell equation’s solutions, solutions of \(ux^2+ vy^2=w\) Diophantine equations where \(u,v,w \in \mathbb Z\), factorization in the set of integers, and so on. Besides, we obtain some notable characteristic properties for such sets.
Share and Cite
ISRP Style
O. Ozer, Z. C. Sahin, On some particular regular Diophantine 3-tuples, Mathematics in Natural Science, 3 (2018), no. 1, 29--38
AMA Style
Ozer O., Sahin Z. C., On some particular regular Diophantine 3-tuples. Math. Nat. Sci. (2018); 3(1):29--38
Chicago/Turabian Style
Ozer, O., Sahin, Z. C.. "On some particular regular Diophantine 3-tuples." Mathematics in Natural Science, 3, no. 1 (2018): 29--38
Keywords
- Diophantine Triple
- Pell equations
- Diophantine equations
- modular arithmetic
- reciprocity theorem
- Legendre symbol
MSC
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