# On some particular regular Diophantine 3-tuples

Volume 3, Issue 1, pp 29--38
Publication Date: August 02, 2019 Submission Date: November 14, 2018 Revision Date: January 03, 2019 Accteptance Date: January 30, 2019
• 2022 Views

### 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

•  11Dxx
•  11C08
•  11G99

### References

• [1] A. Baker, H. Davenport, The equations $3x^2-2=y^2$ and $8x^2-7=z^2$, Quart. J. Math. Oxford Ser. (2), 20 (1969), 129--137

• [2] I. G. Bashmakova, Diophantus of Alexandria: Arithmetics and The Book of Polygonal Numbers, Nauka, Moskow (1974)

• [3] A. F. Beardon, M. N. Deshpande, Diophantine triples, The Mathematical Gazette, 86 (2002), 258--260

• [4] E. Brown, Sets in which $xy+k$ is Always a Square, Math. Comp., 45 (1985), 613--620

• [5] M. N. Deshpande, On interesting family of Diophantine triples, Int. J. Math. Ed. Sci. Tech., 33 (2002), 253--256

• [6] L. E. Dickson, History of Theory of Numbers: Diophantine Analysis, Dove Publ., New York (2005)

• [7] A. Dujella, A. Jurasić, Some Diophantine Triples and Quadruples for Quadratic Polynomials, J. Comb. Number Theory, 3 (2011), 123--141

• [8] P. Fermat, Observations sur Diophante, Oeuvres de Fermat, Paris (1891)

• [9] A. Fillipin, Non-extend ability of D(-1) triples of the form $\{1, 10, c\}$, Int. J. Math. Math. Sci., 35 (2005), 2217--2226

• [10] Y. Fujita, The D(1)-extensions of D(-1)-triples $\{1, 2, c\}$ and integer points on the attached elliptic curves, Acta Arith., 128 (2007), 349--375

• [11] L. Goldmakher, Legendre, Jacobi and Kronecker Symbols Section, Number Theory Lecture Notes, (4 pages)

• [12] M. A. Gopalan, V. Sangeetha, M. Somnath, Construction of the Diophantine Triple involving polygonal numbers, Sch. J. Eng. Tech., 2 (2014), 19--22

• [13] M. A. Gopalan, G. Srividhya, Two special Diophantine triples, Diophantus J. Math., 1 (2012), 23--27

• [14] M. A. Gopalan, S. Vidhyalakshmi, S. Mallika, Some special non-extendable Diophantine triples, Sch. J. Eng. Tech., 2 (2014), 159--160

• [15] M. A. Gopalan, S. Vidhyalaksfmi, Ö. Özer, A Collection of Pellian Equation (Solutions and Properties), Akinik Publications, New Delhi (2018)

• [16] K. S. Kedlaya, Solving constrained Pell equations, Math. Comp., 67 (1998), 833--842

• [17] P. Kurur (Instructor), R. Saptharishi (Scribe), Computational Number Theory, Lecture Notes, Quadratic Reciprocity (contd.) Section (3 pages)

• [18] R. A. Mollin, Fundamental Number theory with Applications, Chapman \& Hall/CRC, Boca Raton (2008)

• [19] Ö. Özer, A Note On The Particular Sets With Size Three, Boundary Field Prob. Comput. Simul. J., 55 (2016), 56--59

• [20] Ö. Özer, On The Some Particular Sets, Kirklareli Univer. J. Eng. Sci., 2 (2016), 99--108

• [21] Ö. Özer, Some Properties of The Certain Pt Sets, Int. J. Algebra Stat., 6 (2017), 117--130

• [22] Ö. Özer, On The Some Non Extandable Regular $P_{-2}$ Sets, Malaysian J. Math. Sci., 12 (2018), 255--266

• [23] J. Roberts, Lure of the Integers, Mathematical Association of America, Washington, DC (1992)