COLLATZ CONJECTURE 3n±1 AS A NEWTON BINOMIAL PROBLEM

Abstract

The power transformation of Newton's binomial forms two equal 3n±1 algorithms for transformations of numbers n belongs to N, each of which have one infinite cycle with a unit lower limit of oscillations. It is shown that in the reverse direction, the Kollatz sequence is formed by the lower limits of the corresponding cycles, and the last element goes to a multiple of three odd numbers. It was found that for infinite transformation cycles 3n-1 isolated from the main graph with minimum amplitudes of 5, 7, 17 lower limits of oscillations, additional conditions are fulfilled.