Complete solution of a family of simultaneous Pellian equations. (English) Zbl 1024.11014

A set of positive integers \(\{a_1,a_2,\ldots ,a_m\}\) is called a Diophantine \(m\)-tuple with the property \(D(n)\) if \(a_ia_j+n\) is a perfect square for all \(1\leq i < j \leq m\). The author obtains a new result concerning Diophantine quadruples. He proves that the pair \(\{1,2\}\) cannot be extended to a Diophantine quadruple with the property \(D(-1)\).
This result follows directly from the main theorem of the paper, which asserts that for all positive integers \(k\) all solutions of the system of the simultaneous Pellian equations \[ z^2-c_kx^2=c_k-1, \quad 2z^2-c_ky^2=c_k-2 \] (where \(c_k=P_{2k}^2+1\) and \(P_{2k}\) denotes the \(k\)th Pell number) are given by \((x,y,z)=(0, \pm 1, \pm P_{2k})\).


11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
Full Text: EuDML


[1] Baker A.: The diophantine equation \(y^2 = ax^3 + bx^2 + cx + d\). J. London Math. Soc 43 (1968), 1-9. · Zbl 0157.09801
[2] Baker A., Davenport H.: The equations \(3x^2 - 2 = y^2\) and \(8x^2 - 7 = z^2\). Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137. · Zbl 0177.06802
[3] Bennett M.A.: On the number of solutions of simultaneous Pell equations. J. Reine Angew. Math., to appear. · Zbl 1165.11034
[4] Brown E.: Sets in which xy + k is always a square. Math. Comp. 45 (1985), 613-620. · Zbl 0577.10015
[5] Cohn J. H. E.: Lucas and Fibonacci numbers and some Diophantine equations. Proc. Glasgow Math. Assoc. 7 (1965), 24-28. · Zbl 0127.01902
[6] Dickson L. E.: History of the Theory of Numbers, Vol. 2. Chelsea, New York, 1966, pp. 518-519.
[7] Diophantus of Alexandria: Arithmetics and the Book of Polygonal Numbers. (I.G. Bashmakova, Nauka, Moscow, 1974 (in Russian), pp. 103-104, 232.
[8] Dujella A.: Generalization of a problem of Diophantus. Acta Arith. 65 (1993), 15-27. · Zbl 0849.11018
[9] Dujella A.: The problem of the extension of a parametric family of Diophantine triples. Publ. Math. Debrecen 51 (1997), 311-322. · Zbl 0903.11010
[10] Dujella A.: A proof of the Hoggatt-Bergum conjecture. Proc. Amer. Math. Soc., to appear. · Zbl 0937.11011
[11] Dujella A.: An extension of an old problem of Diophantus and Euler. · Zbl 1125.11308
[12] Dujella A., Petho A.: Generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxford Ser. (2), to appear.
[13] Gupta H. K., Singh K.: On k-triad sequences. Internat. J. Math. Math. Sci. 5 (1985), 799-804. · Zbl 0585.10006
[14] Kedlaya K. S.: Solving constrained Pell equations. Math. Comp., to appear. · Zbl 0945.11027
[15] Mohanty S. P., Ramasamy A. M. S.: The simultaneous Diophantine equations \(5y^2 - 20 = x^2\) and \(2y^2 + 1 = z^2\). J. Number Theory 18 (1984), 356-359. · Zbl 0534.10012
[16] Mohanty S. P., Ramasamy A.M.S.: On \(P_{r,k}\) sequences. Fibonacci Quart. 23 (1985), 36-44. · Zbl 0554.10003
[17] Nagell T.: Introduction to Number Theory. Almqvist, Stockholm, Wiley, New York, 1951. · Zbl 0042.26702
[18] Rickert J. H.: Simultaneous rational approximations and related diophantine equations. Math. Proc. Cambridge Philos. Soc. 113 (1993), 461-472. · Zbl 0786.11040
[19] SIMATH Manual. Universität des Saarlandes, Saarbrücken, 1993.
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