The perplexing Collatz conjecture

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Collatz conjecture is perhaps the most elementary open math problem which mathematicians still do not have a clue how to prove or find a counter example despite hundreds of papers written on the subject!
In 1937, German mathematician Lothar Collatz made the following observation

Start with a whole number.
If it’s even, divide by 2.
If it’s odd, multiply by 3 and add 1.
Repeat.
Eventually, you’ll always end up at 1.
In closed form, this can be written as



Here are some examples,

20,10,5,16,8,4,2,1
17,52,26,13,40,20,10,5,16,8,4,2,1
 27,82,41,124,62…………91,274…9232……….16,8,4,2,1

The number of steps it takes to reach 1 is called the final stopping time. For example, when we start with 27 it takes 111 steps to reach 1.
To Date, numbers have been checked out to 87×260 and no counter-example has been found to disprove the conjecture.

However, extreme caution is advised to accept Collatz conjecture to be true. History is replete with many such conjectures which have failed the test of time and effort.

For example, in 1919 Hungarian mathematician George Polya suggested the following known as the Polya’s conjecture.


More than half of the positive whole numbers less than any given number have an odd number of prime factors.

This was assumed to be true for a long time before counter-examples were found. The smallest being 906,150,257.

Many mathematicians have commented about the Collatz conjecture. 


It is to be noted that currently many mathematicians are actively working on Collatz problem either to prove it or find a counter example.

Acknowledgement

I would like to thank Prof Brian Conrad for telling me about the Collatz conjecture about two years back. I remember writing a program to generate these numbers and looking for patterns. Eventually I gave up! Maybe you can do better!

Reference:

Collatz Conjecture: Wikipedia 

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