The 3n+1 Conjecture - Behaviour of the Stopping Time Function - Extended Essay

Sep 22, 2026

The Collatz (3n+1) conjecture asserts that iterating the function on any starting positive integer eventually reaches the cycle 2-1, with all trajectories falling into the repeating 2,1 pattern. The essay studies this process by introducing the total stopping time function, σ(n), defined as the number of iterations until first reaching 1. It outlines trajectories via altitudes, summarizes established background and motivation, and develops computational and explanatory approaches for observed stopping-time patterns, including congruences modulo 20(5)-1.

LanguageEnglish

CategoryExam

FormatPDF

Pages29

Reading Time1 h 13 min

AccessFree to download

Abstract

Statement of the Problem

Total stopping time function and motivation

Collatz function definition and trajectories

Trigonometric approach and function extension

Simplification for positive integers

The 3n+1Conjecture:Behaviour of the Stopping Time Function Alex Lupsasca Alex Lupsasca Atlantic College Extended Essay The 3n+1 Conjecture:Behaviour of the Stopping Time Function Tutor:Irene Stewart Word Count:3250 The 3n+1 Conjecture:Behaviour of the Stopping Time Function # Abstract Pick any natural number n.If it is even,divide it by two;if it is odd,multiply itby three and add one to make it even,and then divide the resulting number bytwo.Take the number you obtained and repeat this process,again and again. The 3n+1 Conjecture states that no matter what positive integer you start outwith,you will eventually reach 1 and then fall into the trivial cycle 2-1. The proof of this statement is a longstanding unsolved problem in numbertheory.Many mathematicians have tried to assail it but none have succeeded,prompting mathematician Paul Erdos to declare that “Mathematics is not yetready for such problems.” Nevertheless,because of the contrast between the problem's apparent simplicityand the underlying difficulty in making any progress towards finding a solution,many people find it captivating and have investigated various aspects of thefunction. Thus,I shall quickly outline this deceptively simple question that has stumpedmathematicians for decades and then raise an intriguing question whichimmediately arises from the study of the 3n+1 iterative map:“how many timesdoes the aforementioned process need to be repeated on a given starting numberfor it to reach 1?” This interrogation leads to the investigation of the so-called total stopping timefunction,denoted by σ(n),which I define and further discuss in the essay.Finally,I report on the findings of my examination of the patterns it exhibits,and provide an explanation for the phenomenon I observed,based on theaccumulation of coalescences congruent modulo 20(5)-1 Word Count:260 Alex Lupsasca Table of Contents Statement of the Problem 1 Origins of the conjecture5 Flight Time 7 A Heuristic Approach8 A Computational Approach9 Observation of the Pattern14 Explanation of the Phenomenon 14 Conclusion 19 Bibliography 23 Acknowledgments23 Appendix 23 The 3n+1 Conjecture:Behaviour of the Stopping Time Function Alex Lupsasca # Statement of the Problem In mathematical notation,the Collatz function T(n)is defined as: For a given starting integer n,which shall henceforth be referred to as the"flight number,"the iteration of this function produces an infinite Collatzsequence The numbers that such a sequence contains are termed “altitudes”of the flightnand form what is called the "forward orbit,"or “trajectory,"of that flight. For instance,flight number 11 has the following trajectory: 11,17,26,13,20,10,5,8,4,2,1,2,1,2,1... As another example,the forward orbit of flight 15 is: 15,23,35,53,80,40,20,10,5,8,4,2,1,2,1,2,1.. It is obvious from these examples that if a flight ever reaches the altitude 8,itwill obligatorily fall to 4 and then into the never-ending cycle 2-1.But do allthe starting values of n generate trajectories which fly through 8 and thus endup alternating between 2 and 1 ad infinitum? The Collatz conjecture answers this question affirmatively.It states that: "The process of iterating the Collatz function yields orbits which eventuallyreach the cycle 2-1,regardless of the positive integer initially chosen." The 3n+1Conjecture:Behaviour of the Stopping Time Function Alex Lupsasca More formally,this is expressed as: Vn∈N>03i∈N:(ao=n→a₁=1) If the conjecture is false,it can only be because there is some natural numberwhich gives rise to a sequence that does not comprise the altitude 1.Such anorbit might enter a repeating cycle that excludes 1 or increase without bound. If the conjecture is true,an interesting aspect of the problem that lends itself toinvestigation is the "total stopping time function,"which I shall introduce andformally develop over the course of this essay in

FAQs

What does the 3n+1 (Collatz) conjecture claim about iterating T(n)?

It claims that starting from any positive integer, repeated application of the Collatz function eventually reaches the trivial cycle 2-1. After finitely many steps, the process falls into alternating between 2 and 1.

What is the total stopping time function σ(n) in this essay?

σ(n) is defined as the number of altitudes (or iterations) in the forward orbit of a starting value until it first reaches altitude 1. It is used to quantify how long a number takes to enter the 2-1 cycle.

What computational or analytical goal does the essay pursue?

The essay aims to compute total stopping times for a wide range of starting values and identify patterns in σ(n). It then seeks an explanation for observed behavior, based on congruences mentioned in the work.

Sep 22, 2026

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