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Analysis of the Collatz Problem

Problem

n is a positive whole number

Formulae:
when n is even, divide by 2
when n is odd, multiply by 3 and add 1

n/2
3n+1

Prove that for any whole number, the continuous application of these simple formulae eventually reduces to 4, then to 2, then to 1. At 1, the problem then cycles from 4 to 2 to 1 continuously.

All starting numbers less than 5 * 2^60 have been checked and they all reduce to the 4,2,1 cycle.

In the 4,2,1 cycle, the only entry point of the cycle is 4, as 2 can only be reached by halving 4, and 1 can only be reached by halving 2. So any one whole number, as a starting point, eventually reaches 4 by continuous application of the formulae.

Analysis to follow…

Ronald L Conte Jr
April 7, 2021

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