This problem is named after Lothar Collatz, who first proposed it in 1937. It has many other names, in particular, hail numbers, the Syracuse sequence, the 3n + 1 problem, etc.). This problem has a very simple and accessible formulation, but it is still considered unsolved.

To explain the essence of the hypothesis, we take any natural number n. If it is even, then we divide it by 2, and if it is odd, then we multiply by 3 and add 1 (we get 3n + 1). We perform the same actions on the resulting number, and so on.

The resulting sequence of numbers is also called a sequence of hailstones or simply hailstones (since its numbers sharply increase and fall like hailstones during thunderstorms and storms) or else "wandering numbers."

Collatz's hypothesis is that no matter what initial number we take, sooner or later we will get one.

For example, for the number 3 we get:

3 - odd, 3 × 3 + 1 = 10

10 - even, 10: 2 = 5

5 - odd, 5 × 3 + 1 = 16

16 - even, 16: 2 = 8

8 - even, 8: 2 = 4

4 - even, 4: 2 = 2

2 - even, 2: 2 = 1

1 - odd, 1 × 3 + 1 = 4

To test the Collatz hypothesis on large numbers, several distributed computing projects have been launched. As of 2019, all natural numbers less than 1 152 921 504 606 846 976 have been tested and each of them met the conditions of the Collatz Hypothesis in a finite number of steps.