價格：免費
更新日期：2012-10-09
檔案大小：726k
目前版本：0.8.5
版本需求：Android 1.6 以上版本
官方網站：mailto:EtymOniroApp@gmail.com
Explore the Collatz Conjecture!
This app approaches a mathematical problem that has been eluding even the brightest of professional mathematicians for 75 years now. Trace the orbit of individual numbers on their way through the Collatz iteration, step by step. If your number of steps to 1 is high enough, enter the Collatz Top 100 Hall of Fame with your name and number.
Or enter batch mode and let the app crunch millions of numbers overnight, relentlessly hunting for new step records and highest intermediate results (peaks). Records and peaks can be saved to a file, too.
If in the mood, make it a game with your friends. Agree on a range of numbers to pick from (e.g. all positive integers from 2 to 100,000,000,000), enter your candidates and see if they yield step numbers high enough to make it into the top 100. Can you beat your friends’ record holders?
If you would like to calculate a sequence of your own:
1. Pick a positive integer greater than 1.
2. This is "your number".
3. If your number is odd, multiply it by 3 and add 1 to this product. The result is your (increased) new number.
4. If your number is even, divide it by two. The result of this division is your new (diminished) number.
5. If your number is now 1, stop calculating. Otherwise, continue with step 3.
Lothar Collatz (1910-1990), a German mathematician, proposed in 1937 that eventually every starting number greater than 1 would end at 1. Despite a lot of theoretical research (and even distributed computing projects on the internet), no one has managed yet to either come up with a number that does not end at 1 (goes into a cycle) or find a mathematically conclusive way of proving that every number will in fact end at 1.
To calculate the orbit (intermediate results) of an individual number, enter a number of your choice in the field labeled "Start number" and leave the field labeled "End number" empty. Then tap on the picture of Mr. Collatz, and your chosen number will be crunched. The intermediate results of the iteration will be displayed in the box below the three check boxes. You can show or leave out the display of single intermediate results.
If a new step record is achieved (one that is elegible to enter the Top 100), you will be presented with a box to enter your name, and then your name, your number and its step number will be immortalized in the Collatz Top 100 Hall of Fame.
If you tick "Output to file" (*before* you tap Mr. Collatz on the shoulder), you will find the results in a newly created file on your Android file system. When the app starts up, it will create a folder on the external storage medium (an SD card, for instance), or, should no external storage be available, in the internal memory. This is why the app needs permission to create folders and files in your file system.
If you want to crunch a wide range of numbers, make sure to uncheck the "Single steps" checkbox. This will greatly speed up the calculation and save megabytes of memory. Even if you uncheck the "Single steps" box, new step records will still be listed, and also new record intermediate results (peaks) will continue to be shown.
Researching a wide range of numbers is much more efficient if you check the "odd only" box. Then even numbers will not be considered, as they quickly become much smaller (already after the first step of the iteration). So if you want to check out the numbers between, say, 3 and 300,000, it is enough to check the odd numbers from 150,001 to 299,999, which will save you quite some time, especially if your range of numbers encompasses numbers that are by magnitudes greater. Still, if, for instance, you are interested in pairs of numbers for which both yield the same step number, uncheck the "odd only" box and check "Single steps", so can inspect these "step number twins" manually.
Enjoy!
P.S. Paul Erdös said about the Collatz conjecture: "Mathematics is not yet ready for such problems." He offered $500 USD for its solution.
