Entertaining mathematics. The most economical number system

We all know from the school course what the number system (SS) is. But not everyone thinks about how costly SS. Those. what set of numbers we need to represent the number in this CC. When we have a limited set of unique elements (multi-colored stones of different sizes) with which we can represent a number, what is the maximum number we can imagine using these elements? (all red pebbles are zero, green - one, blue - two, etc., small - zero discharge, medium - first, large - second, etc.). Where is the line at which the base of the SS plays a greater role than the digit capacity of the number?

Take for example n - the number of elements equal to 60. Breaking down the elements into 2 groups (binary number system) we get 30 digits. 30 units - at most 30-digit number with a base of 2. If we add 1 to it, we get one with 30 zeros, i.e. 2 in the 30th, since every zero is a power of two, and the digits start from 0, and we will not forget to subtract the one that was added.

For other SS similarly $$$y = x ^ {60 / x} -1$, where y is the maximum number, x is the base of the degree.

Construction points:
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$$$2 ^ {30} -1 = 1 \, 073 \, 741 \, 823$.
$$$3 ^ {20} -1 = 3 \, 486 \, 784 \, $ 40$.
$$$4 ^ {15} -1 = 1 \, 073 \, 741 \, $ 82$
$$$5 ^ {12} -1 = 244 \, 140 \, 624$
$$$6 ^ {10} -1 = 60 \, 466 \, $ 17$
$$$10 ^ {6} -1 = 999 \, $ 99$
$$$12 ^ {5} -1 = 248 \, $ 83$

Function graph:



From the graph it is seen that with the increase in the base of the SS, starting from three, its cost increases and the function has an upper extremum. By bringing it to general appearance one can get $$$y = (\ sqrt [x] x) ^ {n} -1$and the maximum function $$$\ sqrt [x] x$achieved at $$$x = e$.



Function graph $$$y = \ sqrt [x] x$

Those. the most economical SS is the system as close as possible to $$$e$or 3.

PS: In addition to that, the SS with base 3 is odd, which means it does not have a rounding problem (bring 0.5 to 0 or 1), and if the numbers are written symmetrically (-1.0.1 instead of 0.1.2), then simplicity appears representations of negative numbers (10-1 is 8, -101 is -8, where minus is not a sign, but part of a digit that can be replaced with Z), but this is a completely different arithmetic)))