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The Collatz Conjecture Is a Simple Problem That Mathematicians Can’t Solve

Written by Jay Bennett

Part of the beauty of mathematics is that seemingly simple patterns lead to much more complicated questions and theories. The Collatz conjecture, which is the subject of a new video from YouTube channel Numberphile, is the perfect example of a simple problem that even the greatest mathematical minds in the world haven’t been able to solve.

The essence of the question is simple. We have an interesting pattern: If you take a positive integer, and then divide by 2 if it is even, or multiply by 3 and add 1 if it is odd, then repeat this process with the resulting number, eventually you will end up with the number 1. Professor David Eisenbud of the University of California, Berkeley, demonstrates this process, also known as the hailstone sequence, using the number 7 in the video.

So the question is this: Will you eventually end up with the number 1 if you perform the hailstone sequence on any positive integer? It is a subject of much debate, one that has an entire textbook and many studies devoted to it. Prolific Hungarian mathematician Paul Erdős famously said that, “Mathematics may not be ready for such problems.”

Some mathematicians have theorized that there must be starting numbers that either enter a never-ending loop (number x will reach number y, which will return to number x, meaning the number will never reach 1), or that there must be numbers that continually increase rather than eventually decrease to 1—a hypothesis that makes sense when you consider the fact that you multiply odd numbers by 3, but only divide even numbers by 2.

Despite these seemingly logical deductions, no number has been found that will not eventually equal 1 after repeating the “Half Or Triple Plus One” (HOTPO) process. If you want to know which number lower than 100,000,000 requires the most steps to finally get to 1, then you will just have to watch the video.

Source: Numberphile