Glasnik Matematicki, Vol. 46, No. 2 (2011), 283-309.

DIOPHANTINE m-TUPLES FOR QUADRATIC POLYNOMIALS

Ana Jurasić

Department of Mathematics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia
e-mail: ajurasic@math.uniri.hr


Abstract.   In this paper, we prove that there does not exist a set with more than 98 nonzero polynomials in Z[X], such that the product of any two of them plus a quadratic polynomial n is a square of a polynomial from Z[X] (we exclude the possibility that all elements of such set are constant multiples of a linear polynomial pinZ[X] such that p2|n). Specially, we prove that if such a set contains only polynomials of odd degree, then it has at most 18 elements.

2000 Mathematics Subject Classification.   11C08, 11D99.

Key words and phrases.   Diophantine m-tuples, polynomials.


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DOI: 10.3336/gm.46.2.02


References:

  1. A. Baker and H. Davenport, The equations 3x2-2=y2 and 8x2-7=z2, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137.
    MathSciNet     CrossRef

  2. Diophantus of Alexandria, Arithmetics and the Book of Polygonal Numbers, (I. G. Bashmakova, Ed.) (Nauka 1974), 85-86, 215-217.
    MathSciNet    

  3. A. Dujella, On Diophantine quintuples, Acta Arith. 81 (1997), 69-79.
    MathSciNet    

  4. A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge Philos. Soc. 132 (2002), 23-33.
    MathSciNet     CrossRef

  5. A. Dujella, Bounds for the size of sets with the property D(n), Glas. Mat. Ser. III 39 (2004), 199-205.
    MathSciNet     CrossRef

  6. A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.
    MathSciNet     CrossRef

  7. 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

  8. A. Dujella and C. Fuchs, Complete solution of the polynomial version of a problem of Diophantus, J. Number Theory 106 (2004), 326-344.
    MathSciNet     CrossRef

  9. A. Dujella, C. Fuchs and R. F. Tichy, Diophantine m-tuples for linear polynomials, Period. Math. Hungar. 45 (2002), 21-33.
    MathSciNet     CrossRef

  10. A. Dujella, C. Fuchs and G. Walsh, Diophantine m-tuples for linear polynomials. II. Equal degrees, J. Number Theory, 120 (2006), 213-228.
    MathSciNet     CrossRef

  11. A. Dujella and A. Jurasić, On the size of sets in a polynomial variant of a problem of Diophantus, Int. J. Number Theory 6 (2010), 1449-1471.

  12. P. Gibbs, Some rational sextuples, Glas. Mat. Ser. III 41 (2006), 195-203.
    MathSciNet     CrossRef

  13. B. W. Jones, A variation of a problem of Davenport and Diophantus, Quart. J. Math. Oxford Ser.(2) 27 (1976), 349-353.
    MathSciNet     CrossRef

  14. B. W. Jones, A second variation of a problem of Davenport and Diophantus, Fibonacci Quart. 15 (1978), 155-165.
    MathSciNet    

  15. R. C. Mason, Diophantine equations over function fields, London Mathematical Society Lecture Notes Series, vol. 96, Cambridge University Press, Cambridge, 1984.
    MathSciNet    


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