Math Problem Statement
The query submitted by the user concerns a function defined by the Collatz conjecture, which applies the following rules based on whether a number is odd, even, or equal to 1.
Solution
The function shown in the image is a definition of the Collatz function, commonly associated with the Collatz conjecture. It operates as follows:
\begin{cases} 3n + 1 & \text{if } n \text{ is odd} \\ \frac{n}{2} & \text{if } n \text{ is even} \\ \text{stop} & \text{if } n = 1 \end{cases}$$ ### Explanation 1. **If $$n$$ is odd**: The function applies $$3n + 1$$. 2. **If $$n$$ is even**: The function applies $$\frac{n}{2}$$. 3. **If $$n = 1$$**: The function stops. This function is used to illustrate the Collatz sequence. Starting from any positive integer $$n$$, applying $$f_c$$ repeatedly is hypothesized to eventually reach the number 1, regardless of the starting integer. This has been checked for very large values of $$n$$, but a proof or disproof that it holds for all positive integers remains unknown. Would you like a detailed example of how this function works on a specific number, or further exploration of the Collatz conjecture? ### Related Questions 1. How can the Collatz sequence be visualized for a given number? 2. What are the longest known Collatz sequences for small integers? 3. Is there any known proof that the Collatz sequence always reaches 1 for all integers? 4. What are the practical applications of the Collatz conjecture? 5. How does this sequence behave for large values of $$n$$? ### Tip Try applying this function to different starting values to observe how quickly they reach 1. This is a fun way to experiment with the Collatz conjecture!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Collatz Conjecture
Recursion
Number Theory
Formulas
f_c(n) = 3n + 1 if n is odd
f_c(n) = n / 2 if n is even
f_c(n) stops if n = 1
Theorems
Collatz conjecture
Suitable Grade Level
Grades 9-12