Start with any positive integer n. If n is even, divide it by 2; if n is odd, multiply it by 3 and add 1. The conjecture posits that no matter what value of n, this process will eventually reach the value 1.
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The conjecture you are referring to is known as the Collatz Conjecture. Named after mathematician Lothar Collatz, it has been a subject of study in number theory for several decades. The statement of the conjecture, as you described it, suggests that for any positive integer starting value, the iterative process will eventually reach the number 1.
To illustrate the process, let's take some examples:
1) Starting with n = 6:
since 6 is even, divide it by 2: n = 3
since 3 is odd, multiply it by 3 and add 1: n = 10
since 10 is even, divide it by 2: n = 5
since 5 is odd, multiply it by 3 and add 1: n = 16
since 16 is even, divide it by 2: n = 8
since 8 is even, divide it by 2: n = 4
since 4 is even, divide it by 2: n = 2
since 2 is even, divide it by 2: n = 1
The process has reached the value 1.
2) Starting with n = 10:
since 10 is even, divide it by 2: n = 5
since 5 is odd, multiply it by 3 and add 1: n = 16 (same as the previous example)
The process continues from here and eventually reaches 1.
While the conjecture seems deceptively simple, it has proven to be an incredibly challenging problem in mathematics. Despite extensive computational testing and analysis, no counterexamples have been found, meaning that the conjecture holds true for all values tested so far. However, a rigorous proof of the conjecture still eludes mathematicians, and it remains an unsolved problem.
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