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On the Exceptional Set in the Problem of Diophantus and Davenport

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Applications of Fibonacci Numbers

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

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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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