The Collatz Conjecture - Introduced By Lothar Collatz

Introduced by Lothar Collatz, the Collatz conjecture explains what happens to sequences defined with the following piecewise defined function:

In other words, start with any positive number. If the number is odd, multiply the number by 3 and add one. If the number is even, divide the number by 2. Then apply the rule to the resulting number and continue over and over again. The resulting sequence is often known as the Collatz sequence.

The Collatz conjecture says that for every n that is a positive integer, a sequence created with the function above will eventually fall to the number one. Thus, these numbers are sometimes called hailstone numbers. This is because hailstones are formed when raindrops get carried upward by updrafts into the colder areas of the atmosphere and freeze. They will continue to do so until they get so heavy that they fall to the ground. Like hailstones, sequences created by the Collatz conjecture problem will often get higher and then lower, but eventually will fall to the ''ground,''.

The reason the sequence is usually not continued after reaching 1 is that a cycle of 3 numbers keeps repeating. After 1, the next number generated is 4 (1*3+1). After 4, the next term is 2 and then 1. After 1, we get 2,4,1. This cycle will repeat infinitely so the sequence is generally not continued after this point.

The interesting part of this conjecture is that all numbers that have been tested till now result in the cycle eventually reaching 1. The conjecture has been tested up to 264 (a 20-digit number) and a number not “falling” to 1 has never been found. Even more interesting is the fact that despite all the calculative proof, the conjecture has never been proven theoretically. Most mathematicians choose to accept the fact that the conjecture is probably true due to the experimental proof.

However, there still remains a possibility of some numbers larger than 2^64 existing which cause their own cycle to be repeated again and again just like 1,2,4. It is possible that the sequence generated by a number causes the original number to be generated again, resulting in an infinite cycle. A number like that has never been found but the conjecture has never been proved either, making the Collatz conjecture one of the most intriguing and famous unsolved problems of all times.