Rocky Mountain Journal of Mathematics

On Differences of Two Squares in Some Quadratic Fields

Andrej Dujella and Zrinka Franušić

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Rocky Mountain J. Math., Volume 37, Number 2 (2007), 429-453.

First available in Project Euclid: 5 June 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D09: Quadratic and bilinear equations 11E25: Sums of squares and representations by other particular quadratic forms 11R11: Quadratic extensions

Difference of squares quadratic fields Pellian equations


Dujella, Andrej; Franušić, Zrinka. On Differences of Two Squares in Some Quadratic Fields. Rocky Mountain J. Math. 37 (2007), no. 2, 429--453. doi:10.1216/rmjm/1181068760.

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  • A. Baker and H. Davenport, The equations $3x^2-2=y^2$ and $8x^2-7=z^2$, Quart. J. Math. Oxford 20 (1969), 129-137.
  • E. Brown, Sets in which $xy+k$ is always a square, Math. Comp. 45 (1985), 613-620.
  • L.E. Dickson, History of the theory of numbers, Chelsea, New York, 1966.
  • A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65 (1993), 15-27.
  • --------, The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser. III 32 (1997), 1-10.
  • --------, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.
  • A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. 71 (2005), 33-52.
  • H. Gupta and K. Singh, On $k$-triad sequences, Internat. J. Math. Math. Sci. 8 (1985), 799-804.
  • P. Kaplan and K.S. Williams, Pell's equations $x^2-my^2=-1,-4$ and continued fractions, J. Number Theory 23 (1986), 169-182.
  • F. Lemmermeyer, Higher descent on Pell conics I. From Legendre to Selmer, preprint, available at: math.NT /0311309.
  • C. Madonna and V.V. Nikulin, On a classical corespodence between K3 surfaces, Tr. Mat. Inst. Steklova 241 (2003), Teor. Chisel, Algebra i Algebr. Geom., 132-168 (in Russian); Proc. Steklov Inst. Math. 241 (2003), 120-153 (in English).
  • S.P. Mohanty and M.S. Ramasamy, On $P_r,k$ sequences, Fibonacci Quart. 23 (1985), 36-44.
  • R.A. Mollin, A continued fraction approach to the Diophantine equation $ax^2 - by^2 = \pm 1$, J. Algebra Number Theory Appl. 4 (2004), 159-207.
  • I. Niven, Integers of quadratic fields as sums of squares, Trans. Amer. Math. Soc. 48 (1940), 405-417.
  • O. Perron, Die Lehre von den Kettenbruchen, Teubner, Stuttgart, 1954.
  • W. Sierpiński, Elementary theory of numbers, PWN, Warszawa; North Holland, Amsterdam, 1987.
  • A.J. Stephens and H.C. Williams, Some computational results on a problem of Eisenstein, in Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 869-886.