The Collatz Conjecture

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Mathematical inquiry can lead down jagged paths hewed deeply in the landscape of abstract thought. The English cleric Charles Caleb Colton once said, “The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.” This magnificence can be observed directly in the process of making a new discovery. Poised with creativity and a thorough knowledge, mathematicians often discover generalizations that transcend time as much as they do common understanding.

Sometimes, mathematicians can produce inquiries that are more recreational in nature. One such conundrum was raised by the German mathematician Lothar Collatz in 1937. Suppose I create a rule that if I have any even natural number, then I divide it by two, and if I have any odd natural number, then I multiply it by three and add one. More formally, f(n) = n/2 if n is an even natural number, and f(n)= 3n + 1 if n is an odd natural number. Collatz asserted that if one starts with any natural number n, and repeats the rule again and again, with each new appearing natural number, then the process will lead to the number one every time. For example, let’s start with the natural number 10. If I divide by two, I get 5. Taking 5 and multiplying by three and adding one, I get 16. Furthermore, if I notice that 16 = 2^4, then I can divide by two four consecutive times to come to the natural number one as the Collatz Conjecture (as it is originally named) proposes.

There was a power of two in the last step of the example.  It is easily seen that the conjecture is true for any power of two, since all powers of two are even and dividing them by their power number of times by two leads to one. This could spark another question that is equivalent: Starting with any natural number $n$ does repeating the rule always lead to a power of two? The simplicity of such a question can be very deceiving in the world of mathematics.

The prolific mathematician, Paul Erdős, in speaking of the Collatz Conjecture, once said, “Mathematics is not ready for such problems.” He actually offered $500 dollars for its solution (which is written about here). I agree with Erdős. Mathematics is not necessarily equipped at this time to handle such inquiries. However, it could be interesting to consider the existence of a counterexample. Can you show one? What would one look like? Would it lead to any contradicitons? These are the type questions that spark a journey down the paths of mathematical inquiry and lead to unexpected solutions.

About Avery Carr

I am a grad student studying mathematics at Emporia State University. I obtained a B.S. in Mathematics from the University of Memphis in 2010 where I served as President of the math club my senior year. My research interests include Graph Theory, Number Theory, Recreational Mathematics, Mathematical Physics, and Operator Theory.
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4 Responses to The Collatz Conjecture

  1. Tim Roberts says:

    This fascinating problem and others of similar quality can be found at the Unsolved problems web site at unsolvedproblems.org

  2. Steven J Miller says:

    I would like to expand a bit on what Erdos said. I agree that mathematics is not yet ready to SOLVE such problems, but I do feel it is ready to ASK such problems. Frequently asking a hard question can lead to other questions, and to connections between various fields. The 3x+1 Problem has been shown to be connected to a variety of interesting problems. Probabilistic attacks on it have shown remarkable agreement with this deterministic system.

    For people interested in more information about this problem, there are many great resources. There’s the book “The Ultimate Challenge: The 3x+1 Problem” (http://www.amazon.com/The-Ultimate-Challenge-3x-Problem/dp/0821849409). See also the nice collection of papers (including two bibliographies of work in the field) here: http://www.math.lsa.umich.edu/~lagarias/3x+1.html .

  3. Avery Carr says:

    Dr. Miller,

    Thank you for your insightful comments. I did not know that is what Erdős said, as I have only seen it quoted elsewhere the way I stated it in the article. However, it makes better sense that he would have said it that way. Thanks again.

  4. Wyatt Iles says:

    I am fascinated by this conjecture also. The conjecture works for all real numbers 1 to 10. As all real numbers greater than 10 are a multiple of at least one or more of the first 10 real numbers, it stands to reason the conjecture works for all real numbers 1…∞ It is also interesting that for each “odd” number multiplied by three and adding 1, creates an even number. For each even number, once divided by two, it will create another even number or an odd number (that upon the next step will create an even number once manipulated per the conjecture). At some point, one of the even numbers will eventually become a power of two. When this happens, the entire conjecture falls like a house of cards, ending in one. I too believe that manipulation of any real number per the parameters of the conjecture will eventually lead to a power of two, which will ultimately lead to one. Developing a function of X that proves this, however, continues to elude me, and most modern mathematicians.

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