Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
Open Access
Journal Browser
Turkish Journal of Analysis and Number Theory. 2019, 7(2), 50-55
DOI: 10.12691/tjant-7-2-4
Open AccessArticle

A Certain Type of Regular Diophantine Triples and Their Non-Extendability

Özen ÖZER1,

1Department of Mathematics, Faculty of Science and Arts, Kırklareli University, Kırklareli, 39100, Turkey

Pub. Date: April 13, 2019

Cite this paper:
Özen ÖZER. A Certain Type of Regular Diophantine Triples and Their Non-Extendability. Turkish Journal of Analysis and Number Theory. 2019; 7(2):50-55. doi: 10.12691/tjant-7-2-4


In the present paper, we consider some D(s) Diophantine triples for a prime integer s with its negative case/value although there exist infinitely many Diophantine triples. We give several properties of such Diophantine triples and prove that they are non – extendability to D(s) Diophantine quadruple using algebraic and elementary number theory structures.

non-extendable D(s) diophantine triples pell equations integral solutions quadratic reciprocity theorem modular arithmetic legendre/jacobi symbol

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/


[1]  Beardon A.F. and Deshpande M.N., Diophantine Triples, The Mathematical Gazette 86, 258-260, 2002.
[2]  Dujella, A., Jurasic, A., Some Diophantine Triples and Quadruples for Quadratic Polynomials, J. Comp. Number Theory, Vol.3, No.2, 123-141, 2011.
[3]  Gopalan M. A., Vidhyalakshmi S., Mallika S., Some special non-extendable Diophantine triples, Sch. J. Eng. Tech. 2, 159-160, 2014.
[4]  Gopalan M.A., Vidhyalaksfmi S., Özer Ö., “A Collection of Pellian Equation (Solutions and Properties)”, Akinik Publications, New Delhi, INDIA, 2018.
[5]  Özer Ö., A Note On The Particular Sets With Size Three, Boundary Field Problems and Computer Simulation Journal, 55, 56-59, 2016.
[6]  Özer Ö., On The Some Particular Sets, Kırklareli University Journal of Engineering and Science, 2, 99-108, 2016.
[7]  Özer Ö., Some Properties of The Certain Pt Sets, International Journal of Algebra and Statistics, Vol. 6; 1-2, 117-130, 2017.
[8]  Özer Ö., On The Some Nonextandable Regular P-2 Sets, Malaysian Journal of Mathematical Sciences 12(2): 255-266, 2018.
[9]  Özer Ö., Şahin Ç.Z., On Some Particular Regular Diophantine 3-Tuples, Mathematics in Natural Sciences, (Accepted).
[10]  Goldmakher L., Number Theory Lecture Notes, Legendre, Jacobi and Kronecker Symbols Section, 2018.
[11]  Kurur P. P (Instructor), Saptharishi R. (Scribe), Computational Number Theory, Lecture Notes, Quadratic Reciprocity (contd.) Section, 2017.
[12]  Larson, D. and Cantu J., Parts I and II of the Law of Quadratic Reciprocity, Texas A&M University, Lecture Notes, 2015.
[13]  Bashmakova I.G. (ed.), Diophantus of Alexandria, Arithmetics and The Book of Polygonal Numbers, Nauka, Moskow, 1974.
[14]  Biggs N.L., Discrete Mathematics, Oxford University Press, 2003.
[15]  Burton D.M., Elementary Number Theory, Tata McGraw-Hill Education, 2006.
[16]  Cohen H., Number Theory, Graduate Texts in Mathematics, vol. 239, Springer-Verlag, New York, 2007.
[17]  Dickson LE., History of Theory of Numbers and Diophantine Analysis, Vol.2, Dove Publications, New York, 2005.
[18]  Fermat, P. Observations sur Diophante, Oeuvres de Fermat, Vol.1 (P. Tonnery, C. Henry eds.), p.303, 1891.
[19]  Ireland K. and Rosen M., A Classical Introduction to Modern Number Theory, 2nd ed., Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, New York, 1990.
[20]  Mollin R.A., Fundamental Number Theory with Applications, CRC Press, 2008.
[21]  Silverman, J. H., A Friendly Introduction to Number Theory. 4th Ed. Upper Saddle River: Pearson, 141-157, 2013.
[22]  Kedlaya K.S., Solving constrained Pell equations, Math. Comp. 67, 833-842, 1998.