No. Starting at 6 you get the sequence above in DaveE's post.Dries vanlandschoot said:
And there's the proof we won't hit 2. Once we hit the 2nd 7, we'll be looping indefinitely.Drakkith said:
Maybe we can tweak it into a new result that this version will end in 1,5 or 17? Just a Wil guess. Someone used to call it the " Law of Small Numbers"; wild guesses from limited data.DaveE said:
The Modified Collatz Conjecture is a variation of the original Collatz Conjecture. In this version, the sequence is generated by the following rules: if the number is even, divide it by 2; if the number is odd, multiply it by 3 and add 1, then divide by 2. The conjecture posits that no matter what positive integer you start with, the sequence will eventually reach 1, 5, or 17.
The original Collatz Conjecture involves iterating the sequence by dividing even numbers by 2 and replacing odd numbers with 3n + 1. The Modified Collatz Conjecture adds an additional step for odd numbers: after multiplying by 3 and adding 1, you then divide by 2. This modification changes the dynamics of the sequence and, according to the conjecture, leads to the sequence eventually reaching 1, 5, or 17.
As of now, the Modified Collatz Conjecture, like the original Collatz Conjecture, has not been proven. It remains an open problem in mathematics. While it has been tested for many numbers and no counterexamples have been found, a formal proof that it holds for all positive integers is still lacking.
In the Modified Collatz Conjecture, the numbers 1, 5, and 17 are special because they are the conjectured endpoints of the sequence. When the sequence reaches one of these numbers, it enters a loop or a state where it will repeatedly cycle through the same values. For example, starting from 1, the sequence will stay at 1; starting from 5, it will cycle through 5, 8, 4, 2, 1; and starting from 17, it will eventually cycle through a series of numbers that return to 17.
Studying the Modified Collatz Conjecture, like the original Collatz Conjecture, is significant because it touches on deep questions about number theory, iterative processes, and mathematical patterns. Understanding why certain sequences behave in specific ways can provide insights into more complex mathematical phenomena. Additionally, exploring these conjectures can lead to the development of new mathematical techniques and tools.