In this document, I present an argument that challenges the widely accepted belief in the validity of the 3x+1 Conjecture. This conjecture, also known as the Collatz Conjecture, posits that starting with any positive integer, repeated application of a particular function will eventually lead to the number 1.
In my first paper, co-authored with Y. Sinai in 2002, I demonstrated that the paths generated by the 3x+1 function can be modeled as a geometric Brownian motion in a precise asymptotic sense, with a drift of log(3/4) < 0. This finding suggests that typical trajectories exhibit a decay pattern, supporting the previously established fact that almost every initial seed eventually reaches a value below itself. However, this process cannot be iterated indefinitely, as the paths may diverge into very sparse trajectories.
Initially, like many others, I believed that the 3x+1 Conjecture was likely true for a multitude of reasons, including heuristic and probabilistic arguments, along with numerical verifications extending up to 2^60. However, during a visit to UCLA a few years ago, a conversation with Igor Pak prompted me to reconsider this stance.
Pak's insight highlighted that the general consensus surrounding a conjecture often leads to a lack of rigorous efforts to disprove it. This raises the question: could the same be true for the 3x+1 Conjecture?
In 1972, John Conway demonstrated that certain generalizations of the 3x+1 problem were undecidable by linking them to the halting problem. This connection likely inspired Conway's creation of the FRACTRAN universal Turing machine.
The 3x+1 map can be conceptualized as hardware that executes programs, where each initial seed represents software that generates specific behaviors when processed by this hardware.
While it is almost certainly true that this hardware is not a universal Turing machine, the question remains: what exactly does it accomplish? Every program input into this system ultimately halts at the value 1; however, one cannot dismiss the possibility of an exceptionally long program—potentially thousands of bits in length—that exhibits different behavior.
From this perspective, the verification of the 3x+1 Conjecture up to 2^60 should not be overly impressive. The conclusion that all programs of length 60 characters eventually halt does not provide significant insight. The heuristic and probabilistic arguments suggest a trend towards average behavior, which may be accurate, yet it is entirely conceivable that nearly all initial seeds decay to 1, while some extraordinarily unique seeds do not. This scenario complicates the problem since it obscures the objective of what one should be attempting to prove.
A close analogy can be drawn with Conway's Game of Life, which is also a universal Turing machine. Within this framework, gliders and other structures were crucial in demonstrating its universality.
What if there exists a particular piece of 3x+1 software (an initial seed) that produces a "glider"—a discernible, recurring, yet expanding pattern—when executed on the 3x+1 hardware?
To further explore this notion, as outlined in K-Sinai, it suffices to consider numbers that are coprime to 6 and to eliminate all powers of 2. Thus, the mapping can be represented as:
[ x \rightarrow \frac{3x+1}{2^k} ]
where ( k \geq 1 ) is maximized while ensuring the result remains an integer (notably, the images generated are always coprime to 3). The values of ( k ) are pivotal in this analysis.
The Structure Theorem in K-Sinai asserts that for any desired sequence of these ( k ) values, there exists an arithmetic progression of initial seeds that will yield the specified sequence. For instance, for the ( k )-path (1, 1, 2, 2), the initial seed ( x \equiv 1 \mod 6 ) is 199, which can be verified to produce this sequence.
The following illustrates a 3x+1 path. Each row represents a number in binary; the black dots signify 1's, while the white dots represent 0's. The solid line on the right indicates that every number is odd. The seed shown is:
887570260646934643447331259693373543698140232874500619768489156911.
In conclusion, while the 3x+1 Conjecture has garnered significant attention and validation, the arguments presented herein suggest that it may not be universally applicable. Further exploration and rigorous investigation are warranted to fully understand the complexities surrounding this conjecture.
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