Some Notes on the Extendibility of an Especial Family of Diophantine P_2 Pairs

Yıl 2023, Cilt: 6 Sayı: 2, 1 - 7, 17.05.2023

Özen Özer

Öz

Although it is known that there are an infinite number of Diophantine P_1 triples, there is no complete characterization for these triples.
This paper is a continuation and a generalization of one of the recent papers (see [ ref. 35 ]) in which several numerical results are demonstrated and some properties are given for special Diophantine P_2 pairs and triples. Here, the expansion of the single-element set {2} into a Diophantine P_2 binary special family as {2, s} (with s values expressed as a recurrence/iteration of natural numbers) is obtained firstly. Then, binary special family {2, s} is extended as {2, s, a_s} Diophantine P_2 triples ( a_s is determined in the terms of s ) using solutions of Diophantine equations. Lastly, it is proved that {2, s, a_s} can not be extended Diophantine P_2 quadruples using elementary and algebraic methods different from other works in the literaure.

Anahtar Kelimeler

Diophantine P_2 sets, System of Equations, Integral Solutions, Non- extendable Diophantine P_2 triples, Elementary Number Theory, Natural Numbers.

Destekleyen Kurum

Kırklareli Üniversitesi BAPKO

Proje Numarası

KLUBAP-233

Teşekkür

The study is supported by Scientific Research Project with number KLUBAP-233 of Kırklareli University.

Kaynakça

• CITATIONS 1. Adžaga, N., Dujella, A., Kreso, D. and Tadič, P.: Triples which are D(n) ‐sets for several ns, J. Number Theory 184, 330‐341 (2018).
• 2. Arkin, J. Hoggatt, V.E. and Strauss, E.G.: On Euler’s solution of a problem of Diophantus, Fibonacci Quart. 17, 333–339 (1979).
• 3. Baker, A. and Davenport, H.: The equations 3x2 - 2=y2 and 8x2 - 7=z2 , Quart. J. Math. Oxford Ser. (2) 20, 129‐137 (1969).
• 4. Bashmakova I.G. (ed.) : Diophantus of Alexandria, Arithmetics and The Book of Polygonal Numbers, Nauka , Moskow. (1974).
• 5. Beardon, A.F. and Deshpande, M.N.: Diophantine triples, The Mathematical Gazette, 86,253-260 (2002).
• 6. Bokun, M. and Soldo, I.: Pellian equations of special type, Math. Slovaca 71, 1599-1607. 2021.
• 7. Brown, E. : Sets in which xy+k is always a square, Math.Comp.45, 613-620 (1985).
• 8. Burton D.M. : Elementary Number Theory. Tata McGraw-Hill Education. (2006).
• 9. Cipu, M., Filipin, A. and Fujita, Y. : Diophantine pairs that induce certain Diophantine triples, J. Number Theory 210, 433-475 (2020) .
• 10. Cohen H., : Number Theory, Graduate Texts in Mathematics, vol. 239, Springer-Verlag, New York (2007).
• 11. Deshpande, M.N. : One interesting family of Diophantine Triples, Internet.J. Math.Ed.Sci.Tech, 33, 253-256 (2002).
• 12. Deshpande, M.N.: Families of Diophantine Triplets, Bulletin of the Marathawada Mathematical Society, 4, 19-21 (2003).
• 13. Dickson LE.: History of Theory of Numbers and Diophantine Analysis, Vol 2, Dove Publications, New York (2005).
• 14. Dujella, A. : Diophantine m-tuples, http://web.math.pmf.unizg.hr/~duje/dtuples.html .
• 15. Dujella, A. : Generalization of a problem of Diophantus, Acta Arith. 65, 15–27 (1993).
• 16. Dujella, A. : On the size of Diophantine m ‐tuples, Math. Proc. Cambridge Phiıos. Soc. 132, 23‐33 (2002).
• 17. Dujella, A. : Pethő, A. A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49, 291–306 (1998).
• 18. Dujella, A.: Some polynomial formulas for Diophantine quadruples, Grazer Math. Ber. 328, 25–30 (1996).
• 19. Dujella, A.: An absolute bound for the size of Diophantine m-tuples, J. Number Theory 89 126–156 (2001).
• 20. Dujella, A.: Bounds for the size of sets with the property D(n) , Glas. Mat. Ser. III 39,199‐205 (2004).
• 21. Dujella, A. : Generalization of a problem of Diophantus, Acta Arith. 65, 15‐27 (1993).
• 22. Dujella, A. : On the size of Diophantine m-tuples, Math. Proc. Cambridge Philos. Soc. 132, 23–33 (2002).
• 23. Dujella, A. : The problem of the extension of a parametric family of Diophantine triples, Publ. Math. Debrecen 51, 311–322 (1997).
• 24. Dujella, A., Jurasic, A. : Some Diophantine Triples and Quadruples for Quadratic Polynomials, J. Comp. Number Theory, Vol.3, No.2, 123-141 (2011).
• 25. Fermat, P.: Observations sur Diophante, Oeuvres de Fermat, Vol.1 (P. Tonnery, C. Henry eds.), (1891).
• 26. Filipin, A., Fujita, Y. and Togbé, A.: The extendibility of Diophantine pairs I: the general case, Glas. Mat. Ser. III 49 (1) 25–36 (2014).
• 27. Fujita, Y.: The extensibility of Diophantine pairs {k − 1, k + 1}, J. Number Theory 128, 322–353 (2008).
• 28. Gopalan M.A., Vidhyalaksfmi S., Özer Ö. : A Collection of Pellian Equation ( Solutions and Properties) , Akinik Publications, New Delh, INDIA (2018).
• 29. He, B. and Togbé, A. : On the family of Diophantine triples {k + 1, 4k, 9k + 3}, Period. Math. Hungar. 58, 59–70 (2009). 30. 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).
• 31. Kihel, A. and Kihel, O. :On the intersection and the extendibility of Pt ‐sets, Far East J. Math. Sci. 3, 637‐643 (2001). 32. Mollin R.A.: Fundamental Number theory with Applications, CRC Press (2008).
• 33. Özer Ö.: A Certain Type of Regular Diophantine Triples and Their Non-Extendability, Turkish Journal of Analysis & Number Theory, 7(2), 50-55 (2019).
• 34. Özer Ö.: On The Some Nonextandable Regular P-2 Sets, Malaysian Journal of Mathematical Science (MJMS), 12(2), 255–266 (2018).
• 35. Özer Ö.: One of the Special Type of D(2) Diophantine Pairs (Extendibility of Them and Their Properties) 6th International Conference on Mathematics: An Istanbul Meeting for World Mathematicians ( ICOM 2022)”,
• 21-24 June 2022, Proceeding Book ISBN: 978-605-67964-8-6, , pp. 433-442, 2022. Fatih Sultan Mehmet University, İstanbul (2022).
• 36. Park, J.: The extendibility of Diophantine pairs with Fibonacci numbers and some conditions, J. Chungcheong Math. Soc. 34, 209-219 (2021).
• 37. Silverman, J. H.: A Friendly Introduction to Number Theory. 4th Ed. Upper Saddle River: Pearson, 141-157 (2013).
• 38. Thamotherampillai, N. : The set of numbers {1,2,7}, Bull. Calcutta Math.Soc.72, 195-197 (1980).
• 39. Zhang, Y. and Grossman, G. On Diophantine triples and quadruples, Notes Number Theory Discrete Math. 21, 6–16 (2015).