Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Fig. 1. Graph of values for interesting numbers-granules of the algorithm

When the path of one number is so much different even from the neighboring one, how do you even approach the proof of such a hypothesis? Of course, all mathematicians were at a loss and absolutely no one could solve this problem. So Jeffrey Lagarias is a world expert on this problem, and he said that no one should take up this problem if he wants to become a mathematician. A large-scale work was carried out and a huge number of hailstones were studied, trying to find a pattern. Here it can be argued that all values come to one, however, what can be said about the path that all numbers take? The interesting thing is that this path is absolutely random.

For example, we can give a graph of all the values of this algorithm from 1 to 100 (Fig. 2).

Fig. 2. Graph of values for numbers-granules from 1 to 100

As you can see, most often growth begins initially and after a sharp decline, while the value of the number is simply not considered, however, if you make the graph logarithmic, there is a downward trend in its fluctuations. It can also be observed on the stock market on the day of the collapse, which is not accidental, because these are examples of geometric Brownian motion, that is, if you take logarithms and calculate the linear component, the fluctuations seem random, as if a coin was thrown at each step. And if we consider this function analysis as part of mathematical analysis, then there begins to be an obvious connection with probability theory. From where it turns out that when heads are obtained, the line goes up, and when tails go down, from where a special graph is obtained.

If we consider this chart when compared with the same exchange, then it is more likely in a short-term analysis, although in the long term, stocks are still growing, and "3x+1" is falling. You can also pay attention to the highest digit of the gradient numbers – this means a histogram, which is obtained by counting the number of digits from which the numbers begin in a number of granules for a particular number of the algorithm. If you add these values each time, for 1, 2, 3, etc., more and more data is obtained, while the ratio of the height of the columns becomes more and more ordered.

So for the first billion sequences, the most frequent value is one, 29.94% of all cases, 2 – 17,47%, 3 – 12,09%, 4 – 10,63%, 5 – 7,94%, 6 – 6,16%, 7 – 5,76%, 8 – 5,31%, 9 – 4,7% and the larger the figure, the less often it turns out to be ahead.

This arrangement is typical not only for hailstone numbers, there are many examples, these are the population of countries, the value of companies, all physical constants or Fibonacci numbers, and much more. This law is called Benford's law. Surprisingly, if you trace the violation of the Benford law in tax returns, you can even determine the fact of fraud. This law also helps to identify anomalies in the counting of votes in elections or many other things.