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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).

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

Özen Özer

Öz

Proje Numarası

KLUBAP-233

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).
Toplam 38 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik, Uygulamalı Matematik
Bölüm Some Notes on the Extendibility of an Especial Family of Diophantine 𝑷𝟐 Pairs
Yazarlar

Özen Özer 0000-0001-6476-0664

Proje Numarası KLUBAP-233
Yayımlanma Tarihi 17 Mayıs 2023
Gönderilme Tarihi 10 Şubat 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 6 Sayı: 2

Kaynak Göster

APA Özer, Ö. (2023). Some Notes on the Extendibility of an Especial Family of Diophantine P_2 Pairs. Journal of Advanced Mathematics and Mathematics Education, 6(2), 1-7.
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