Lower Bounds for Lucas Chains

13 years 10 months ago

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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 - ) log n for any > 0, where 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. Key words. Lucas chain, addition chain, Lucas function, lower bound, Fibonacci number, golden ratio, smooth number AMS subject classifications. 11Y55, 1...

Martin Kutz

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