**Dr. Iago de Moor**

Adjunct Professor, Department of Linear Machine Learning

Diploma Mhill College, Reno, Nevada, USA

**ABSTRACT**

We prove a famous long standing open conjecture in mathematics known as the Collatz conjecture or the Hailstone conjecture or the 3n+1 conjecture or the Syracuse Problem or the Ulam conjecture or Kakutani’s problem or the Thwaites conjecture. The problem is described and more formally known as “a conspiracy to slow down mathematical research in the U.S.” by Kakutani, “an extraordinarily difficult problem, completely out of reach of present day mathematics” according to Jeffrey Lagarias, “The infamous Collatz conjecture” by Terence Tao, and the problem in which “mathematics may not be ready” as stated by Paul Erdős, who offered 500$ for its solution^{1}.

^{1} We accept PayPal, Venmo, direct deposit, or money order

**1. Introduction**

Mathematics has, and will remain, mankind’s most important endeavor-from the caveman who represented the positive integers in tallies on the walls to the shining mathematical stars of the modern period-Fermat, Gauss, Euler, and Terence Tao; from the stardust of creation to the inevitable throes of nuclear winter, and the eventual heat death of the universe. Nothing is more pressing or concerning for the transcendence and evolution of the minds of men than the immaculate beauty and unstained, child-like, virgin innocence of the work of the mathematician and the practice of mathematics. We offer a humble, yet seemingly paltry, contribution to this endeavor by proving the extremely important Collatz Conjecture with many applications (see section 5), which states:

**1.1 Collatz Conjecture **

Given any positive integer n, define

*f(n) = 3n+1 if n is odd and f(n)=n/2 if n is even *

Then the conjecture holds if *inf({f ^{0}(n), f^{1}(n), …}) =1. *

An example of a trajectory under this dynamical system is as follows:

*7 ->22->11->34->17->52->26->13->40->20->10->5->16->8->4->2->1*

For every starting number n, there must exist a k such that *f ^{k}(n)=1* (or every starting number eventually reaches 1 after repeatedly applying the function to n, f(n), and so on). Here we have f

^{16}(7)=1, and thus the conjecture delightfully holds for n=7.

The chaotic randomness is seemingly impenetrable, and the problem has eluded the greatest minds of our time. However, we provide novel insight that will satisfy the departmental machine, and we escape the threat of “publish or peril” yet again by solving a problem who’s ramifications will have absolutely no impact on the lives of anybody, but will provide us with fame and bragging rights among all 12 mathematicians left in academia after budget cuts due to COVID-19, and the increasing unwillingness of taxpayers to funnel funds into legitimate critical theory.

Figure 1. The trajectory of n=7

**1.2 Barriers to progress **

Given that the problem has such a simple statement, one must wonder, “why hasn’t this problem been solved sooner?” We propose a few hypotheses:

Firstly, the standards of mathematical rigor are so rigorous that no problem solution may be formally accepted anymore. This would explain why no paper of ours has been accepted previously by any journal.

Secondly, it might be that the snobbery of academic mathematicians is so off-putting that the greatest minds have chosen an alternative course of study.

Thirdly, the mind of the mathematician is vast, and many math PhD holders decide to work in an applications-oriented setting, the most notable applications appearing in retail and food services.

Fourthly, it is possible that people care so little about the problem that nobody seriously considers it as a legitimate project for research. We reject this hypothesis in favor of the other three on the grounds that a solution to this problem is indeed important.

**1.3 The Importance of a solution **

Rather than asking why the outcome is important, sometimes it is more important to recognize the smaller achievements we made along the way such as the addition of friends, colleagues, and cash prizes.

**2. Statement of Main Result**

The Collatz conjecture is true because every possible trajectory is bounded and nonperiodic (outside of the usual period). In layman’s terms, for every periodic orbit *P≠{4,2,1},*

*{n∈ N: lim_{ t-> ∞} f^{t}(n) exists and is finite and {f^{0}(n), f^{1}(n),…}∩P = ∅}=N.*

**3.** **Proof of Main Result**

Suppose the above result fails. A trajectory can fail the first condition or the second condition. To show that no trajectory fails the first condition, suppose for the sake of contradiction some trajectory does indeed fail. Then there must be an initial value, say n, with minimal value that does so. Necessarily, n must be odd, or else f(n)=n/2 < n, which contradicts the minimality of n. Since n is odd, 3n+1>n. Repeated iteration proves that eventually f^{k}(n) <n.

To show that no trajectory fails the second condition, suppose that one does. Then not every integer is connected to the collatz tree, and that’s not possible. Though we have omitted some minor details, we are confident that the astute reader may fill them in as an exercise. □

**4. Corollary**

An immediate corollary is the solution to the Goldbach conjecture. Though the application is trivial, we expect the details to be a fun, family-friendly exercise for the average fourth grader to become acquainted with the rich, exciting field of arithmetic topology with applications to Riemannian manifolds.

Additionally, physical systems such as the dynamics of hailstones or anything bouncy that can be modeled as a discrete dynamical system that exhibits similar properties will plausibly have a well-defined end behavior.

**5. Applications Research**

As an additional note, we looked very hard for more real-world applications of the Collatz conjecture. We searched far and wide in numerous industries-from truly inspiring applications in string theory and particle physics to more primitive and mundane applications in engineering. We took the initiative and sent a lot of emails to experts, most of which received no response, or were met with questions, such as “What is a Collatz? Is that like a type of Machine Learning?” In the end, our greatest asset became Google Scholar, for numerous applications were found in the textbook and publishing industries. It turns out, a solution to such a problem would probably receive positive comments on Terence Tao’s blog and warrant a new Numberphile video-the greatest rewards of all. It might even generate enough traffic from eager mathematicians to produce enough ad revenue to offset one-sixteenth of the time-cost complexity for the 4 years we spent on this proof. We also expect the Field’s Medal we will receive will come with a nice financial reward and hopefully a job offer at a Massachusetts university.

**6. Conclusion**

In conclusion, we have proven the greatest minds wrong, and the Collatz conjecture is indeed not only plausible with current mathematical techniques and technology, but the conjecture is rather accessible to the slightly clever challenger. We question the value and necessity for peer-review in favor of an openness to accept new ideas, especially from non-technical audiences.

**7. References**

Brady, H. [Numberphile]. (2016, August 8). UNCRACKABLE? The Collatz Conjecture – Numberphile [Video].YouTube. https://www.youtube.com/watch?v=5mFpVDpKX70&ab_channel=Numberphile

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Congratulations! This is really big news, I’ll have to share your work with my department once I can barge into people’s offices. Nobody answers my emails anymore!

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