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


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.


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.

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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