# What's The Discrete Math Derivative Equivalent?

• I

## Main Question or Discussion Point

$$ƒ = b^n$$
$$b,n,I ∈ ℤ$$

Condition: Upon choosing a base value $b$..

$$n | b^n ≤ I$$
(n is determined based off the value of $b$ to yield the highest ƒ without going over $I$)
$$1<b<L , L<<I$$
where $I$ is some large number, and $L$ 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 $ƒ$ that is closest to $I$?

## Answers and Replies

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andrewkirk
Science Advisor
Homework Helper
Gold Member
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##.

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