On two-parametric family of quartic Thue equations

Borka Jadrijević[1]

  • [1] FESB, University of Split R. Boškovića bb 21000 Split, Croatia

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 161-167
  • ISSN: 1246-7405

Abstract

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We show that for all integers m and n there are no non-trivial solutions of Thue equation x 4 - 2 m n x 3 y + 2 m 2 - n 2 + 1 x 2 y 2 + 2 m n x y 3 + y 4 = 1 , satisfying the additional condition gcd ( x y , m n ) = 1 .

How to cite

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Jadrijević, Borka. "On two-parametric family of quartic Thue equations." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 161-167. <http://eudml.org/doc/249420>.

@article{Jadrijević2005,
abstract = {We show that for all integers $m$ and $n$ there are no non-trivial solutions of Thue equation\begin\{equation*\} x^\{4\}-2mnx^\{3\}y+2\left( m^\{2\}-n^\{2\}+1\right) x^\{2\}y^\{2\}+2mnxy^\{3\}+y^\{4\}=1, \end\{equation*\}satisfying the additional condition $\gcd (xy,mn)=1$.},
affiliation = {FESB, University of Split R. Boškovića bb 21000 Split, Croatia},
author = {Jadrijević, Borka},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {1},
pages = {161-167},
publisher = {Université Bordeaux 1},
title = {On two-parametric family of quartic Thue equations},
url = {http://eudml.org/doc/249420},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Jadrijević, Borka
TI - On two-parametric family of quartic Thue equations
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 161
EP - 167
AB - We show that for all integers $m$ and $n$ there are no non-trivial solutions of Thue equation\begin{equation*} x^{4}-2mnx^{3}y+2\left( m^{2}-n^{2}+1\right) x^{2}y^{2}+2mnxy^{3}+y^{4}=1, \end{equation*}satisfying the additional condition $\gcd (xy,mn)=1$.
LA - eng
UR - http://eudml.org/doc/249420
ER -

References

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  1. A. Baker, Contributions to the theory of Diophantine equations I. On the representation of integers by binary forms. Philos. Trans. Roy. Soc. London Ser. A 263 (1968), 173–191. Zbl0157.09702MR228424
  2. A. Baker, H. Davenport, The equations 3 x 2 - 2 = y 2 and 8 x 2 - 7 = z 2 . Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. Zbl0177.06802MR248079
  3. A. Baker, G. Wüstholz, Logarithmic forms and group varieties. J. Reine Angew. Math. 442 (1993), 19–62. Zbl0788.11026MR1234835
  4. M. A. Bennett, On the number of solutions of simultaneous Pell equations. J. Reine Angew. Math. 498 (1998), 173–199. Zbl1044.11011MR1629862
  5. Yu. Bilu, G. Hanrot, Solving Thue equations of high degree. J. Number Theory, 60 (1996), 373–392. Zbl0867.11017MR1412969
  6. J. H. Chen, P. M. Voutier, Complete solution of the Diophantine equation X 2 + 1 = d Y 4 and a related family of Thue equations. J. Number Theory 62 (1996), 273–292. Zbl0869.11025
  7. A. Dujella, B. Jadrijević, A parametric family of quartic Thue equations. Acta Arith. 101 (2002), 159–170. Zbl0987.11017MR1880306
  8. A. Dujella, B. Jadrijević, A family of quartic Thue inequalities. To appear in Acta Arith. Zbl1050.11036
  9. A. Dujella, A. Pethő, A generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxford Ser. (2) 49 (1998), 291–306. Zbl0911.11018MR1645552
  10. C. Heuberger, A. Pethő, R. F. Tichy, Complete solution of parametrized Thue equations. Acta Math. Inform. Univ. Ostraviensis 6 (1998), 93–113. Zbl1024.11017MR1822519
  11. B. Jadrijević, A two-parametric family of quartic Thue equations. PhD thesis, University of Zagreb, 2001. (in Croatian) Zbl1162.11327
  12. B. Jadrijević, A system of Pellian equations and related two-parametric family of quartic Thue equations. Rocky Mountain J. Math. 35 no. 2 (2005), 547–571. Zbl1090.11021MR2135585
  13. G. Lettl, A. Pethő, Complete solution of a family of quartic Thue equations. Abh. Math. Sem. Univ. Hamburg 65 (1995), 365–383. Zbl0853.11021MR1359142
  14. W. Ljunggren, Über die Gleichung x 4 - D y 2 = 1 . Arch. Math. Naturvid. 45 (1942), 1–12. Zbl0026.29604MR12619
  15. M. Mignotte, A. Pethő, R. Roth, Complete solutions of quartic Thue and index form equations. Math. Comp. 65 (1996), 341–354. Zbl0853.11022MR1316596
  16. A. Pethő, Complete solutions to families of quartic Thue equations. Math. Comp. 57 (1991), 777–798. Zbl0738.11028MR1094956
  17. A. Pethő, R. Schulenberg, Effectives Lösen von Thue Gleichungen. Publ. Math. Debrecen 34 (1987), 189–196. Zbl0657.10015MR934900
  18. A. Pethő, R. T. Tichy, On two-parametric quartic families of Diophantine problems. J. Symbolic Comput. 26 (1998), 151–171. Zbl0926.11016MR1635234
  19. E. Thomas, Complete solutions to a family of cubic Diophantine equations. J. Number Theory 34 (1990), 235–250. Zbl0697.10011MR1042497
  20. A. Thue, Über Annäherungswerte algebraischer Zahlen. J. Reine Angew. Math. 135 (1909), 284–305. 
  21. A. Togbé, On the solutions of a family of quartic Thue equations. Math. Comp. 69 (2000), 839–849. Zbl0963.11018MR1648411
  22. N. Tzanakis, Explicit solution of a class of quartic Thue equations. Acta Arith. 64 (1993), 271–283. Zbl0774.11014MR1225429
  23. N. Tzanakis, B. M. M. de Weger, On the practical solution of the Thue equation. J. Number Theory 31 (1989), 99–132. Zbl0657.10014MR987566
  24. I. Wakabayashi, On a family of quartic Thue inequalities. J. Number Theory 66 (1997), 70–84. Zbl0884.11021MR1467190
  25. I. Wakabayashi, On a family of quartic Thue inequalities, II. J. Number Theory 80 (2000), 60–88. Zbl1047.11030MR1735648
  26. P. G. Walsh, A note a theorem of Ljunggren and the Diophantine equations x 2 - k x y 2 + y 4 = 1 , 4 . Arch Math. 73, No.2, (1999), 119–125. Zbl0941.11012MR1703679

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