| viXra.org > Number Theory > viXra:2501.0053 |
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| viXra.org > Number Theory > viXra:2501.0053 |
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Number Theory |
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Authors: Theophilus Agama
In this note, we study the distribution of the product of consecutive terms in an addition chain of a given length. If $1,2,ldots,s_{delta(n)-1},s_{delta(n)}=n$ is an addition chain producing $n$ and of length $delta(n)$, with associated sequence of generators begin{align}1+1,s_2=a_2+r_2,ldots,s_{delta(n)-1}=a_{delta(n)-1}+r_{delta(n)-1},s_{delta(n)}=a_{delta(n)}+r_{delta(n)}=nonumberend{align} then $$sum limits_{l=1}^{delta(n)}log s_l=delta(n)log n-O(delta(n)).$$ It follows in particular that $$prod limits_{l=1}^{delta(n)}s_lsim n^{delta(n)}.$$
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[v1] 2025-01-09 21:09:43
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