Glasnik Matematicki, Vol. 49, No. 1 (2014), 37-46.

ON D(W)-QUADRUPLES IN THE RINGS OF INTEGERS OF CERTAIN PURE NUMBER FIELDS

Ljerka Jukić Matić

Abstract.   The purpose of this paper is to show the non-existence of D(w)-quadruples in number fields of odd degree whose rings of integers are of the special form. We derive some elements which can not be represented as difference of squares in such rings and comment the non-existence of corresponding Diophantine quadruples. This relies on the non-solvability of system of congruences which we prove in some low-degree cases.

2010 Mathematics Subject Classification.   11D09, 11R16, 11D79.

Key words and phrases.   Diophantine quadruples, pure number fields, cyclic bases.

DOI: 10.3336/gm.49.1.04

References:

1. E. Brown, Sets in which xy+k is always a square, Math. Comp. 45 (1985), 613-620.
   MathSciNet     CrossRef

2. R. Dedekind, Ueber die Anzahl der Idealklassen in reinen kubischen Zahlkörpern, J. Reine Angew. Math. 121 (1900), 40-123.

3. A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65 (1993), 15-27.
   MathSciNet    

4. A. Dujella, The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser. III 32 (1997), 1-10.
   MathSciNet    

5. A. Dujella and Z. Franušić, On differences of two squares in some quadratic fields, Rocky Mountain J. Math. 37 (2007), 429-453.
   MathSciNet     CrossRef

6. A. Dujella and C. Fuchs, A polynomial variant of a problem of Diophantus and Euler, Rocky Mountain J. Math. 33 (2003), 797-811
   MathSciNet     CrossRef

7. A. Dujella, C. Fuchs and F. Luca, A polynomial variant of a problem of Diophantus for pure powers, Int. J. Number Theory 4 (2008), 57-71.
   MathSciNet     CrossRef

8. A. Dujella and I. Soldo, Diophantine quadruples in Z[√-2], An. Ştiinţ. Univ. "Ovidius" Constanţa Ser. Mat. 18 (2010), 81-97.
   MathSciNet    

9. Z. Franušić, A Diophantine problem in Z[(1+√d)/2], Studia Sci. Math. Hungar. 46 (2009), 103-112.
   MathSciNet     CrossRef

10. Z. Franušić, Diophantine quadruples in Z[√(4k+3)], Ramanujan J. 17 (2008), 77-88.
    MathSciNet     CrossRef

11. Z. Franušić, Diophantine quadruples in the ring Z[√ 2], Math. Commun. 9 (2004), 141-148.
    MathSciNet    

12. Lj. Jukić Matić, Non-existence of certain Diophantine quadruples in rings of integers of pure cubic fields, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), 163-167.
    MathSciNet     CrossRef

13. H. Gupta and K. Singh, On k-triad sequences, Internat. J. Math. Math. Sci. 8 (1995), 799-804.
    MathSciNet     CrossRef

14. M. J. Lavallee, B. K. Spearman, K. S. Williams and Q. Yang, Dihedral quintic fields with a power basis, Math. J. Okayama Univ. 47 (2005), 75-79.
    MathSciNet    

15. S. P. Mohanty and A. M. S. Ramasamy, On P_r,k sequences, Fibonacci Quart. 23 (1985), 36-44.
    MathSciNet    

16. A. Muriefah and A. Al-Rashed, Some Diophantine quadruples in the ring Z[√-2], Math. Commun. 9 (2004), 1-8.
    MathSciNet    

17. I. Soldo, On the existence of Diophantine quadruples in Z[√-2], Miskolc Math. Notes 14 (2013), 265-277.
    MathSciNet