Collatz conjecture
The Collatz conjecture, also known as the 3n + 1 problem, is one of the most famous and easy-to-state mathematical problems, yet also one of the most challenging to solve. It was proposed by the German mathematician Lothar Collatz in 1937 and remains unproven or disproven.
Statement of the Conjecture
The conjecture is based on the following rule for any positive integer \(n\):
- If \(n\) is even, divide it by \(2\): \(n = \frac{n}{2}\).
- If \(n\) is odd, multiply it by \(3\) and add \(1\): \(n = 3n + 1\).
Repeat these rules iteratively for the new value of \(n\). The conjecture states that:
No matter the initial number \(n\), the sequence will eventually reach \(1\).
Interesting Properties
- Simplicity and Complexity: Although the rules are very simple, proving the conjecture has proven extremely difficult, making it an iconic problem in mathematics.
- Cyclic Sequences: So far, all known sequences end in the cycle \(4\rightarrow2\rightarrow1\), but it has not been proven that this holds for all numbers.
- Computational Exploration: The conjecture has been verified for numbers up to around \(2^{68}\), but this does not constitute a general proof.
- Mathematical Connections: The conjecture links to various areas of mathematics, such as number theory, discrete dynamics, and computational systems.
Current Status
The Collatz conjecture remains an open problem; that is, no mathematical proof has been found to confirm it for all numbers, nor has a counterexample been discovered. The mathematician Paul Erdős commented on the problem:
"Mathematics is not yet ready for such problems."
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