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I What's The Discrete Math Derivative Equivalent?

  1. Apr 2, 2016 #1
    $$ƒ = b^n$$
    $$ b,n,I ∈ ℤ $$

    Condition: Upon choosing a base value [itex]b[/itex]..

    $$ n | b^n ≤ I $$
    (n is determined based off the value of [itex]b[/itex] to yield the highest ƒ without going over [itex]I[/itex])
    $$1<b<L , L<<I$$
    where [itex]I[/itex] is some large number, and [itex]L[/itex] is also sufficiently large such that we want to avoid going through each base integer via trial and error....

    How might I determine the base value that yields a value [itex]ƒ[/itex] that is closest to [itex]I[/itex]?
     
  2. jcsd
  3. Apr 4, 2016 #2

    andrewkirk

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    I'm not sure exactly what you are asking.
    Are you asking how to, given ##L,I\in\mathbb{Z}## with ##0<<L<<I##, find the combination ##b,n\in\mathbb{Z}## that maximizes ##b^n## subject to the constraints
    1. ##1<b<L##; and
    2. ##b^n<I##

    If so then you could set ##Hi=\lfloor\log_2 I\rfloor## and ##Lo=\lceil \log_L I\rceil##, then find
    $$b^n=max_{Lo\leq m\leq Hi}\bigg( \lfloor I^\tfrac{1}{m}\rfloor\bigg)^m$$

    where ##n## is the value of ##m## that delivers that maximum.
    That will give a much smaller search space than ##1,...,L##.
     
    Last edited: Apr 4, 2016
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