The most efficient numerical base system

An optimization problem with constraints

Credit per StackExchange.

Assume there are $V$ independent states of information. Then we can represent approximately $\frac{V}{N}$ digits in base $N$.

The amount of information we can represent is $I=N^{\frac{V}{N}}$.

The value of $N$that maximizes $I$ (either where the derivative is $0$(if the second derivative is negative) or at infinity (if the second derivative is positive)) is the most "efficient" base.

So we take the natural log: $\ln(I)=\frac{V}{N}\ln(N)$

And take the derivative to $N$: $(\ln(I))^′=V\frac{(1−\ln(N))}{N^2}$.

We then set $(\ln(I))′=0$. Solving, $N=e$.

Take the second derivative: $(\ln(I))′′=V\frac{(2\ln(N)−3)}{N^3}$

When $N=e$, $(\ln(I))^{′′}=−\frac{V}{e^3}$ which is negative (recall that $V$ is positive), so $I$ reaches its maximum when $N=e$.