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GPT-4 ·
Prove it or provide a counterexample: Consider the following operation on an arbitrary positive integer, If the number is even, divide it by two. If the number is odd, triple it and add one. This process will eventually reach the number 1, regardless of which positive integer is chosen initially.
GPT-4
This proposition is known as the Collatz Conjecture, and as of my knowledge cutoff date in September 2021, it remains an open problem in mathematics. No one has been able to prove it true for all positive integers, nor has anyone been able to provide a counterexample.
The conjecture can be expressed as follows:
Given a function f(n) where
f(n) = n/2, if n is even
f(n) = 3n + 1, if n is odd
For any positive integer n, iterating the function f will eventually reach the number 1.
Despite extensive computational evidence suggesting that the conjecture is true for all positive integers, no general proof or counterexample has been found. If the conjecture is proven true, it would imply that every positive integer eventually reaches a cycle, specifically the cycle (1, 4, 2). If a counterexample is found, it would mean that there exists at least one positive integer that does not follow this pattern.
Due to the current state of mathematical knowledge, it is not possible to provide a proof or counterexample for the Collatz Conjecture. You may find more information on the conjecture and its history here.
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