Journal of Probability and Statistics

Volume 2019, Article ID 6814378, 11 pages

https://doi.org/10.1155/2019/6814378

## On the Probabilistic Proof of the Convergence of the Collatz Conjecture

Prince Mohammad Bin Fahd University, Al Khobar, Saudi Arabia

Correspondence should be addressed to Kamal Barghout; as.ude.ump@tuohgrabk

Received 20 January 2019; Accepted 27 May 2019; Published 1 August 2019

Academic Editor: Alessandro De Gregorio

Copyright © 2019 Kamal Barghout. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new approach towards probabilistic proof of the convergence of the Collatz conjecture is described via identifying a sequential correlation of even natural numbers by divisions by that follows a recurrent pattern of the form , where represents divisions by 2 more than once. The sequence presents a probability of 50:50 of division by 2 more than once as opposed to division by 2 once over the even natural numbers. The sequence also gives the same 50:50 probability of consecutive Collatz even elements when counted for division by 2 more than once as opposed to division by 2 once and a ratio of 3:1. Considering Collatz function producing random numbers and over sufficient number of iterations, this probability distribution produces numbers in descending order that lead to the convergence of the Collatz function to 1, assuming that the only cycle of the function is 1-4-2-1.

#### 1. Introduction

The Collatz conjecture concerns natural numbers treated as () of positive even integers. It is defined by the functionIt simply asks you to keep dividing any positive even integer repeatedly by 2 until it becomes an odd integer, then convert it to even integer by tripling it and adding 1 to it, and then repeat the process. The conjecture has been widely studied [1, 2]. It predicts that the recurring process will always form a sequence that descends on the natural numbers to cycle around the trivial cycle 1-4-2-1. The conjecture involves the natural numbers and it simply asks, under any complete process of the conjecture, why it is always the case that, over statistically sufficient number of iterations, the decrease made by the divisions by 2 exceeds the increase made by the conversions from oddness to evenness. It has been noticed here that, from a start odd positive integer, one iteration either increases the number when the result-even number is divided by 2* only once* to obtain an odd number or decreases the start number when the result-even number is divided by 2* more than once*. Therefore, we seek here to quantify the decrease and the increase probabilistically of the start number after every iteration and generalize that over a sufficient number of iterations to check convergence of the function. It is claimed here that the function decreases the start number until it reaches a cycle, because statistically the sequence of all of the consecutive even integers of the elements of the Collatz function over the natural numbers (validated by deducting 1 from every even positive integer and then checking divisibility by 3) has a recurrent pattern of of division by 2 more than once compared to division by 2 once of probability 50:50 and ratio of about 3:1, where is division by 2 more than once.

Collatz conjecture function seems to produce random numbers and generate a random walk process locally but globally converges to 1. Therefore, to prove the convergence of the conjecture probabilistically it is sufficient to show that globally the recurrence of divisions of Collatz even elements by 2 more than once to reach an odd number has the same probability as that of their recurrent divisions by 2 once, denoted here as recurrent frequency (RF), and averages by the ratio of about 3:1. Summing over the respective divisions will always lead by a margin that offsets the increase of the recurrent sum made by the recursive conversion process of the odd Collatz number to even number by tripling it and adding 1 to it. This is easily noticeable if we recognize that if the positive even integers were sequenced by increase by 2, e.g., , division by 2 over the positive even integers follows a sequential order that is described as follows: if any of the sequence’s even elements produces an odd number when divided by 2 once, the following element in the sequence must produce an odd number by division by 2 more than once. This hidden regularity produces a 50-50 probabilistic RF of division by 2 over the positive even integers and turns what seems a random distribution of division by 2 to a global process that makes the events of division by 2 recurrence over the whole positive even integers progress according to the sequence , where is the number of divisions of the even number by 2 more than once to produce an odd number. Here we prove that Collatz-even numbers also follow the same 50:50 probability distribution that leads to descent convergence of the sequence made by the function to a cycle. The proposed proof of the Collatz conjecture here is complete if its process only cycles about 1, 4, and 2, since the decrease of the sequence of the global Collatz process is assembled from perfect correlated probabilistic events defined by the sequence ., over the function’s even elements. This probabilistic correlation is not heuristically derived as opposed to the well-known heuristic argument of the function found in many references [3–5] which states that the function averages division by 2 once of the time and division by 2 twice of the time and division by 2 three times of the time, etc., which produces a decrease of of the preceding number each iteration on average. In this paper it is claimed that the function produces an increase of the odd start number of 50%, of the time as opposed to a decrease of the odd start number of 62% the other of the time, averaged over a sample of sufficiently large number of Collatz even integers if we assume that the mixing properties of the function’s even integers are truly picked at random in the process.

#### 2. Division by 2 Sequence of Positive Even Integers

For comparison and to easily identify the RF sequence of division by 2 for Collatz function elements, we first generate the RF sequence of positive even integers.

Lemma 1. *Let be any positive even integer that can be divided by 2 only once to yield an odd positive integer; then the next even integer must be divided by 2 by more than once to yield an odd positive integer.*

*Proof. *If by initial definition, then . Adding the LHS expressions yields , an odd number. This necessitates that and the term is divisible by 2 more than once.

Lemma 2. *Let be any positive even integer that can be divided by 2 only once to yield an odd positive integer; then the second to next even integer must be divided by 2 only once to yield an odd positive integer.*

*Proof. *If by initial definition, then . Adding the LHS expressions yields , an even number. This necessitates and the term is divisible by 2 only once to obtain an odd number.

From Lemmas 1 and 2, we generate a table of positive even integers and their corresponding frequencies of division by 2 until reaching an odd parity. Starting with the first row as the even integers made by the term , with elements as the frequencies of division by 2, and spanning the natural numbers we can construct a “RF table” over all positive integers that identifies Collatz elements with the back-bone as the line of integers that collapse to 1 by repeated divisions by 2 made by the even numbers , as Collatz function requires. This row makes a symmetrical line that contains all even numbers made by Collatz function that collapse to the trivial cycle 1-4-2-1, e.g., 4, 8, 16, 32. We then construct columns in ascending order by increase by 2 to produce all even positive integers with each column ending by an even number that is two less than the next integer on the collapsing symmetrical line. We observe that the symmetrical line in the table has symmetrical sequential frequencies for all of the columns to infinity along the rows and makes rows with equal frequencies because of the ordered repeated frequencies for each column, which allows us to estimate relative RFs, a key probability distribution that allows us to conclude that Collatz conjecture converges probabilistically to a cycle. The table is constructed in this order mainly to be able to count frequencies of divisions by 2 and approximate the relative RFs of even positive integers to yield an odd number. It follows that consecutive Collatz function’s even elements (in italic) also follow the same pattern as those of the sequence of the table of . We also construct the table with the variable spanning all positive integers of on the symmetrical line, not just even Collatz function elements that contribute to the collapse process of Collatz function, to produce a line of all powers of twos.

Starting with any natural number, Collatz function produces numbers in seemingly random way locally but globally the numbers decrease and the process proceeds toward the collapsing symmetrical line and to the left on the table and it eventually hits a number on the symmetrical line and then collapses to 1 and cycles around 1-4-2-1 in a deterministic process.

The symmetrical distribution of frequencies of divisions by 2 of even natural numbers as in the table exhibits a classical probability distribution about the collapsing symmetrical line over the natural numbers. Only those numbers on the symmetrical line that satisfy Collatz function can branch out and contribute to the collapse process to 1 (those numbers with* s* an even integer) and the branches that are connected via the function with odd start numbers making new subbranches on the Collatz tree (see Figure 1) that if a branch is reached, the process will collapse to its start odd number; i.e., on the trunk of the tree, the number 2^{8} (256) contributes to the collapse process because you can deduct 1 from it and divide by 3 to get a whole number, but the number 2^{9} (512) does not, and the number 341 leads to 2^{10} (1024) on the symmetrical line that collapses to 1 while the odd number 357913941 ends with 2^{30} (1073741824) on the symmetrical line as well. Those numbers on the symmetrical line that can be traced backward by the function act as points for branching out to trace the Collatz tree where the symmetrical line is the tree’s trunk.