Lower Bounds for Lucas Chains
نویسنده
چکیده
Lucas chains are a special type of addition chains satisfying an extra condition: for the representation ak = aj + ai of each element ak in the chain, the difference aj − ai must also be contained in the chain. In analogy to the relation between addition chains and exponentiation, Lucas chains yield computation sequences for Lucas functions, a special kind of linear recurrences. We show that the great majority of natural numbers n does not have Lucas chains shorter than (1− ) logφ n for any > 0, where φ is the golden ratio. Peter L. Montgomery was the first to consider Lucas chains, in the early eighties. He discovered a decomposition theorem for Lucas chains and a lower bound on their length in terms of Fibonacci numbers. His work was not published. Therefore several of Montgomery’s original ideas are represented in this paper.
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عنوان ژورنال:
- SIAM J. Comput.
دوره 31 شماره
صفحات -
تاریخ انتشار 2002