Abstract
From 1876 to 1880 Lucas developed his theory of the functions v n and u n which now bear his name. Today these functions find use in primality testing and integer factorization, among other computational techniques. The functions v n and u n can be expressed in terms of the nth powers of the zeroes of a quadratic polynomial, and throughout his writings Lucas speculated about the possible extension of these functions to those which could be expressed in terms of the nth powers of the zeroes of a cubic polynomial or of a quartic polynomial. Indeed, at the end of his life he stated that by searching for the addition formulas of the numerical functions which originate from recurrence sequences of the third or fourth degree and by studying in a general way the laws of the residues of these functions for prime moduli, we would arrive at important new properties of prime numbers. We only have scattered hints concerning what functions Lucas had in mind because he provided so little information about them in his published and unpublished work. In this paper we discuss two pairs of functions that are easily expressed as certain combinations of the n th powers of the zeroes of a quartic polynomial and of a sextic polynomial, respectively. We also present several new results, which illustrate the striking similarity between these functions and those of Lucas. The methods that we use to obtain these results are for the most part elementary and would likely have been known to Lucas.
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Notes
- 1.
It seems that G is always 1, but we have not been able to prove this in general.
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H.C. Williams: Research supported in part by NSERC of Canada.
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Dedicated to the memory of Alfred J. van der Poorten (1942–2010)
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Roettger, E.L., Williams, H.C., Guy, R.K. (2013). Some Extensions of the Lucas Functions. In: Borwein, J., Shparlinski, I., Zudilin, W. (eds) Number Theory and Related Fields. Springer Proceedings in Mathematics & Statistics, vol 43. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6642-0_15
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