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Diophantine Representation of Fibonacci Numbers Over Natural Numbers

In: Applications of Fibonacci Numbers

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  • James P. Jones

Abstract

The sequence of Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, …, defined by F0 = 0, F1 = 1, F n+2 = F n + F n+1, played an important role in the solution of one of the Hilbert Problems. The Fibonacci sequence was used in 1970 by the Russian mathematician Y.V. Matijasevič to solve the Tenth Problem of Hilbert. The Tenth Problem of Hilbert was the problem of existence of an algorithm for deciding solvability of Diophantine equations. Matijasevič [8] [9] made use of divisibility properties of the Fibonacci sequence to prove that every recursively enumerable set is Diophantine. This solved Hilbert’s Tenth Problem in the negative.

Suggested Citation

  • James P. Jones, 1990. "Diophantine Representation of Fibonacci Numbers Over Natural Numbers," Springer Books, in: G. E. Bergum & A. N. Philippou & A. F. Horadam (ed.), Applications of Fibonacci Numbers, pages 197-201, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-1910-5_22
    DOI: 10.1007/978-94-009-1910-5_22
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