Collatz Conjecture Remains Unproven Despite its Easy Arithmetic

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Lothar Collatz Image by Konrad Jacobs

Lothar Collatz proposed the “Collatz problem” in 1937. It is still an unproven conjecture. Although it has not been disproved, neither is there an accepted proof.

Defining the Collatz Function and Series

A neat and tidy mathematical description of the problem is:

Let “n”, “i”, “j” and “m” be any positive integers, and define a function f(n):

f(n)=3*n + 1, if “n” is odd (( and greater than 1 ))
f(n)=n/2, if “n” is even

Then define a series a[j] of positive integers, where

a[1]=m for an arbitrary positive integer “m”, and
a=f(a[i])

What Does The Collatz Conjecture Really Mean?

Here is an English description. First, we define a function on a positive integer, “n”, that has two rules. If “n” is odd, multiply “n” by three and then add one. If “n” is even, divide “n” by two. Let’s call this whole function “f(n)”.

Second, we create a series of positive integers. Choose any positive integer as the starting number. Create the next number in the series by applying that two-rule function “f(n)” to the current number in the series.

It would be decent of us to point out that the series is really boring if you start with “1”. It simply repeats “1, 4, 2, 1, 4, 2, 1, 4, 2, 1…” forever.

Collatz Sample Image by Mike DeHaan

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