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Thus, the main mathematical result, obtained in Ref. [16] at the synthesis of the optimal (n, k, 1)-algorithm, is the fact that these studies led us to the arithmetic square and binomial coefficients. We emphasize that this result is much unexpected because by starting the synthesis of this algorithm, we did not assume its connection with the binomial coefficients. This gives us a reason to call the (n, k, 1)-algorithms the binomial algorithms of measurement.

What is the significance of the binomial measurement algorithms for mathematics? Here, it is important to emphasize that the optimal (n, k, 1)-algorithm is a generalization of the counting algorithm (Fig. 1.5), which, as mentioned above, historically underlies the elementary number theory, the fundamentals of which are outlined in Euclid’s Elements. Therefore, we can assume that the above-considered binomial measurement algorithms can be used for further development of the elementary number theory created by the ancient Greeks.

There is another idea arising from the interpretation of the optimal (n, k, S)-algorithms as new positional representations of natural numbers. With this approach, the binomial algorithms can be of practical importance for modern computer science. And maybe some of our readers, who can be fascinated by such unusual numeral system, will design a new (binomial) computer. Those readers, who are interested in the binomial computers can refer to the book [136] of the Ukrainian scientist Alexey Borisenko (Sumy University).