The Collatz conjecture, proposed by Lothar Collatz in 1937, explores a seemingly pointless process of generating sequences of numbers following a simple rule.
Beginning with some (strictly) positive integer \(n\), we:
- Divide \(n\) by two if \(n \) is even.
- Multiply \(n\) by three and add one if \(n\) is odd.
- Repeat.
That is, we repeatedly apply the following function on \(n\).
\(f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\text{ even}\\[4px]3n+1&{\text{if }}n\text{ odd}.\end{cases}}\)
The Collatz conjecture hypothesizes that this process will eventually yield the integer 1 (before looping \(1 \rightarrow 4 \rightarrow 2 \rightarrow 1 \rightarrow 4 \rightarrow \cdots\)) regardless of the initial choice for \(n\). This is an unsolved problem in mathematics, as there is currently no known proof (or counterproof) of this fact.
Here, we explore a problem inspired by the Collatz conjecture.
For each positive integer \(n\), we define the "endurance of \(n\)" to be the count of the number of applications of the Collatz function on \(n\) before it reaches 1. For example;
- The endurance of 13 is 9, because under repeated application the sequence \(13\rightarrow 40\rightarrow 20\rightarrow 10\rightarrow 5\rightarrow 16\rightarrow 8\rightarrow 4\rightarrow 2\rightarrow 1\) is generated. This required nine applications of the Collatz function. Hence the endurance of 13 is 9.
- The endurance of 3,141,592 is 76.
- The endurance of 1,618,033 is 132.
Consider all positive integers \(< 10^{10}\). Sort the 30 numbers with longest endurance.
Which number is on place 26 on this list? Enter this number as keyword for the checker.
Take the list with the 30 numbers with you to the location. It helps to unlock the cache.
This cache was inspired by https://coord.info/GC984JD
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More Reference https://research.vu.nl/files/123386476/2008.13643v3.pdf