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Sylvester’s Algorithm and Fibonacci Numbers

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Abstract

We begin by describing an elementary process for expressing a positive proper fraction as a sum of reciprocals of positive integers. Starting with m 1/n 1, where m 1 < n 1, we subtract the largest fraction with numerator unity to leave either zero or another positive fraction, say m 2/n 2. We then subtract the largest fraction with numerator unity from m 2/n 2 and repeat this process until we end up with zero. Eves [1] attributes this process to J. J. Sylvester (1814–1897), whose original account is given in [12]. In this section we will review Sylvester’s algorithm and summarize his findings. In Section 2 we make some further observations on such sums of reciprocals and in Section 3 we extend the Sylvester process from positive proper fractions to all real numbers between 0 and 1. In Section 4 we give some related results concerning sums of reciprocals of Fibonacci numbers.

AMS Classification Numbers

  • 11A67
  • 11B39

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References

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© 1999 Springer Science+Business Media Dordrecht

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Freitag, H.T., Phillips, G.M. (1999). Sylvester’s Algorithm and Fibonacci Numbers. In: Howard, F.T. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4271-7_16

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  • DOI: https://doi.org/10.1007/978-94-011-4271-7_16

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5851-3

  • Online ISBN: 978-94-011-4271-7

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