Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE DIOPHANTINE EQUATION (5pn2 - 1)x + (p(p - 5)n2 + 1)y = (pn)z

  • Received : 2019.04.17
  • Accepted : 2019.10.23
  • Published : 2020.03.25
Download PDF
⟨ Previous Next ⟩

Abstract

Let p be a prime number with p > 3, p ≡ 3 (mod 4) and let n be a positive integer. In this paper, we prove that the Diophantine equation (5pn2 - 1)x + (p(p - 5)n2 + 1)y = (pn)z has only the positive integer solution (x, y, z) = (1, 1, 2) where pn ≡ ±1 (mod 5). As an another result, we show that the Diophantine equation (35n2 - 1)x + (14n2 + 1)y = (7n)z has only the positive integer solution (x, y, z) = (1, 1, 2) where n ≡ ±3 (mod 5) or 5 | n. On the proofs, we use the properties of Jacobi symbol and Baker's method.

Keywords

References

  1. Cs. Bertok, The complete solution of the Diophantine equation $(4m^2+1)^x + (5m^2 -1)^y=(3m)^z$, Period. Math. Hung. 72 (2016), 37-42. https://doi.org/10.1007/s10998-016-0111-x
  2. W. Bosma, J. Cannon, Handbook of Magma functions, Department of Mathematics, University of Sydney, http://magma.math.usyd.edu.au/magma/
  3. Y. Bugeaud, Linear forms in p-adic logarithms and the Diophantine equation $(x^{n}-1)/(x-1)=y^{q}$, Math. Proc. Camb. Philos. Soc. 127 (1999), 373-381. https://doi.org/10.1017/S0305004199003692
  4. Z. Cao, A note on the Diophantine equation $a^x+b^y=c^z$, Acta Arith. 91 (1999), 85-93. https://doi.org/10.4064/aa-91-1-85-93
  5. Z. Cao and X. Dong, An application of a lower bound for linear forms in two logarithms to the Terai-Jesmanowicz conjecture, Acta Arith. 110 (2003), 153-164. https://doi.org/10.4064/aa110-2-5
  6. M. Cipu and M. Mignotte, On a conjecture exponential Diophantine equations, Acta Arith. 140 (2009), 251-270. https://doi.org/10.4064/aa140-3-3
  7. H. Darmon and L. Merel, Winding quotients and some variants of Fermat's Last Theorem, J. Reine. Angew. Math. 490 (1997), 81-100.
  8. J. Ellenberg, Galois representations attached to Q-curves and generalized Fermat equation $A^4+B^2=C^p$, Amer. J. Math. 126 (2004), 763-787. https://doi.org/10.1353/ajm.2004.0027
  9. R. Fu, H. Yang, On the exponential Diophantine equation $(am^2+1)^x+(bm^2-1)^y=(cm)^z$ with c $\mid$ m, Period. Math. Hung. 75 (2017), 143-149. https://doi.org/10.1007/s10998-016-0170-z
  10. A. O. Gel'fond, Sur la divisibilite de la difference des puissances de deux nombres entiers par une puissance d'un ideal premier, Math. Sb. 7 (1940), 7-25.
  11. T. Hadano, On the Diophantine equation $a^x=b^y+c^z$, Math. J. Okayama University 19 (1976), 1-5.
  12. L. Jesmanowicz, Some remarks on Pythagorean numbers, Wiadom Math. 1 (1955-1956), 196-202.
  13. A. Y. Khincin, Continued Fractions 3rd edition, P. Noordhoff Ltd, Graningen 1963.
  14. E. Kizildere, T. Miyazaki, G. Soydan, On the Diophantine equation $((c+1)m^2+1)^x+(cm^2-1)^y=(am)^z$, Turkish J. of Math. 42 (2018), 2690-2698. https://doi.org/10.3906/mat-1803-14
  15. M. Laurent, Linear forms in two logarithms and interpolation determinants II, Acta Arith. 133 (2008), 325-348. https://doi.org/10.4064/aa133-4-3
  16. M. Le, On Terai's conjecture concerning Diophantine equation $a^x+b^y=c^z$, Acta Math. Sinica, Eng. Ser. 46 (2003), 245-250.
  17. M. Le, A conjecture concerning the pure exponential Diophantine equation $a^x+b^y=c^z$, Acta Math. Sinica, Eng. Ser. 21 (2005), 943-948. https://doi.org/10.1007/s10114-004-0436-x
  18. K. Mahler, Zur approximation algebraischer Zahlen I: Uber den grossten Primteiler binaer formen, Math. Ann. 107 (1933), 691-730. https://doi.org/10.1007/BF01448915
  19. T. Miyazaki, Exceptional cases of Terai's conjecture on Diophantine equations, Arch. Math. (Basel) 95 (2010), 519-527. https://doi.org/10.1007/s00013-010-0201-6
  20. T. Miyazaki, Terai's conjecture on exponential Diophantine equations, Int. J. Number Theory 7 (2011), 981-999. https://doi.org/10.1142/S1793042111004496
  21. T. Miyazaki, N. Terai, On the exponential Diophantine equation $(m^2+1)^x+(cm^2-1)^y=(am)^z$, Bull. Aust. Math. Soc. 90 (2014), 9-19. https://doi.org/10.1017/S0004972713000956
  22. T. Nagell, Sur une classe d'equations exponentielles, Ark. Math. 3 (1958), 569-582. https://doi.org/10.1007/BF02589518
  23. X. Pan, A note on the exponential Diophantine equation $(am^2+1)^x+(bm^2-1)^y=(cm)^z$, Colloq. Math. 149 (2017), 265-273. https://doi.org/10.4064/cm6878-10-2016
  24. G. Soydan, M. Demirci, I. N. Cangul, A. Togbe, On the conjecture of Jesmanowicz, Int. J. of Appl. Math. and Stat. 56 (2017), 46-72.
  25. J. Su, X. Li, The exponential Diophantine equation $(4m^2+1)^x+(5m^2-1)^y=(3m)^z$, Abstr. Appl. Anal., 2014 (2014), 1-5.
  26. N. Terai, The Diophantine equation $a^x+b^y=c^z$, Proc. Japan. Ser. A Math. Sci. 70 (1994), 22-26.
  27. N. Terai, Applications of a lower bound for linear forms in two logarithms to exponential Diophantine equations, Acta. Arith. 90 (1999), 17-35. https://doi.org/10.4064/aa-90-1-17-35
  28. N. Terai, On the exponential Diophantine equation $(4m^2+1)^x+(5m^2-1)^y=(3m)^z$, Int. J. Algebra 6 (2012), 1135-1146.
  29. N. Terai, T. Hibino, On the exponential Diophantine equation $(12m^2+1)^x+(13m^2-1)^y=(5m)^z$, Int. J. Algebra 9 (2015), 261-272. https://doi.org/10.12988/ija.2015.5529
  30. N. Terai, T. Hibino, On the exponential Diophantine equation $(3pm^2-1)^x+(p(p-3)m^2+1)^y=(pm)^z$, Period. Math. Hung. 74 (2017), 227-234. https://doi.org/10.1007/s10998-016-0162-z
  31. S. Uchiyama, On the Diophantine equation $2^x=3^y+13^z$, Math. J. Okayama Univ. 19 (1976), 31-38.
  32. J. Wang, T. Wang and W. Zhang, A note on the exponential Diophantine equation $(4m^2+1)^x+(5m^2-1)^y=(3m)^z$, Colloq. Math. 139 (2015), 121-126. https://doi.org/10.4064/cm139-1-7