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*Unsolved Simple Arithmetic–– Understanding the Collatz Conjecture*

*Unsolved Simple Arithmetic–– Understanding the Collatz Conjecture*

November 14, 2015

In today’s society, it is a common assumption that higher-level mathematics is so complex and jargon-heavy that the average person cannot comprehend the problem. However, the Collatz Conjecture, one of the most well-known unsolved problems in mathematics today, easily disproves this assumption. The only requirement of the conjecture is a basic understanding of arithmetic. First proposed by German mathematician Lother Collatz in 1937, the conjecture offers the following procedure:

Take any natural number n.

If n is an even number, divide it by 2 to obtain n/2.

If n is an odd number, multiply it by 3 and add 1 to it to obtain 3n+1.

Repeat this process with the numbers obtained from this procedure.

The conjecture states that all natural numbers, when they go through this procedure, will eventually converge to 1; that is, the list of numbers that come from repeatedly doing the procedure will be 1, 4, 2, 1, 4, 2…, which will continue indefinitely.

Let’s take an example:

Start with the number 5.

It is odd, so we multiply by 3 and add 1 to obtain 16.

It is even, so we divide by 2 to obtain 8.

It is even, so we divide by 2 to obtain 4.

It is even, so we have 2.

It is even, so we have 1.

Then, from here, the numbers that come out from this process will continue to be the cycle of numbers 1, 4, 2.

The Collatz Conjecture gets even more interesting when it is modified. For example, consider that instead of dividing by 2 if the number n is even, that one would divide by a natural number b if divisible by b, and instead of multiplying by 3 and adding 1 if it is odd, that one would multiply by a natural number a and add 1 in cases when it is not divisible by b. What would be the behavior of such functions? As it turns out, there are some cases that seem to diverge, or grow without bound, some cases that seem to eventually converge to the value 1, and some cases that seem to cycle indefinitely. The case of the Collatz Conjecture, where b is 2 and a is 3, is not the only case that has an interesting result to it; there are many more like it.

However, despite only involving arithmetic, the Collatz Conjecture remains one of mathematics’s most difficult unsolved puzzles. There is very strong evidence to support that this conjecture is universally true, but there is no definitive proof. Experimentally, the conjecture has been checked with a computer for all natural numbers up to 2^64. No number in this range disproves the conjecture. And yet, there has been no mathematical proof to support the conjecture for all cases.

Paul Erdos, a very prolific 20th century mathematician, once commented on the conjecture that “mathematics may not be ready for such problems,” and even offered $500 for a proof of the conjecture. And yet, the Collatz Conjecture is only one of seemingly simple but exceedingly difficult problems. Sometimes, the simplest problems are the hardest to solve.

Image Sources: https://en.wikipedia.org/wiki/Collatz_conjecture, http://www.mathblog.dk/wp-content/uploads/2011/06/collatz_conjecture-206x300.png

About the Author: Daniel Kim

Daniel is a high school senior from California, USA.

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