Abstract
The Greek mathematician Diophantus of Alexandria noted that the numbers x,x + 2, 4x + 4 and 9x + 6, where x = 1/16, have the following property: the product of any two of them increased by 1 is a square of a rational number (see [4]). Fermat first found a set of four positive integers with the above property, and it was {1,3,8,120}. Later, Davenport and Baker [3] showed taht if d is a positive integer such taht the set {1,3,8,d} has the property of Diophantus, then d has to be 120.
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References
Arkin, J. and Bergum, G.E. “More on the problem of Diophantus.” Applications of Fibonacci Numbers. Vol. 2. Edited by A.N. Philippou, A.F. Horadam and G.E. Bergum. Kluwer Academic Publishers (1988): pp. 177–181.
Brown, E. “Sets in which xy + k is always a square.” Mathematics of Computation, Vol. 45 (1985): pp. 613–620.
Davenport, H. & Baker, A. “The equations 3x 2 — 2 = y 2 and 8x 2 — 7 = z 2.” Quart. J. Math. Oxford Ser. (2), Vol. 20 (1969): pp. 129–137.
Diofant, Aleksandriiskii Arifmetika i kniga o mnogougol’nvh chislakh. Moscow: Nauka, 1974.
Dujella, A. “Generalization of a problem of Diophantus.” Acta Arithmetica, Vol. 65 (1993): pp. 15–27.
Dujella, A. “Diophantine quadruples for squares of Fibonacci and Lucas numbers.” Portugaliae Mathematica, Vol. 52 (1995): pp. 305–318.
Dujella, A. “Some polynomial formulas for Diophantine quadruples.” Grazer Mathematishe Berichte, Vol. 328 (1996): pp. 25–30.
Gupta, H. & Singh, K. “On k-triad sequences.” Internat. J. Math. Math. Sci., Vol. 8 (1985): pp. 799–804.
Jones, B. W. “A variation on a problem of Davenport and Diophantus.” Quart. J. Math. Oxford Ser. (2), Vol 27 (1976): pp. 349–353.
Jones, B. W. “A second variation on a problem of Diophantus and Davenport.” The Fibonacci Quarterly, Vol. 16 (1978): pp. 155–165.
Mohanty, S. P. & Ramasamy, M. S. “The simultaneous Diophantine equations 5y 2 — 20 = x 2 and 2x 2 + 1 = z 2.” Journal of Number Theory, Vol. 18 (1984): pp. 356–359.
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Dujella, A. (1998). On the Exceptional Set in the Problem of Diophantus and Davenport. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5020-0_10
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DOI: https://doi.org/10.1007/978-94-011-5020-0_10
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