The perplexing Collatz conjecture

-

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 

Comments

Post a Comment

Popular posts from this blog

The Famous Basel Problem: The process that led Euler to the answer!

-
Image
Euler is considered to be one of the greatest mathematician of the 18 th century and many believe that he was among the greatest of all time. The answer to the Basel problem is considered to be among his many gems. It is the sum of the series shown below and the answer is astounding! What is π doing there? Mathematician at the time summed up the series to the first 10, 100, and 1000 terms and the slow rate of convergence gave very little clue to what the answer could be.  Euler realized that he would have to find a function that would converge much faster. After some trial and error he found the function shown below whose power series expansion is shown.   So if we integrate the above expression from 0 to 1 This is what he was looking for! To evaluate the integral in the left, he broke it into two parts. Then he evaluated each of the integrals in turn. Then the next integral Put x=1-t When x=1/2, t=1/2 When x=1, t=0
Read more

Millennium Problem: Riemann Hypothesis

-
Image
A prime number is a number that is divisible only by itself and 1. Some examples being  2, 3, 5, 7, 9, 11, 13.…   The great mathematician Euclid proved centuries ago that there are an infinite number of them continuing without pattern ad infinitum. However, their real importance is given by what is known as the Fundamental Theorem of Arithmetic. It states that every number can be uniquely expressed as a product of prime numbers. Let’s take the example of 364 = 2 × 2 × 7 × 13. This is the only way that 364 can be obtained by prime numbers! Therefore, they are the atoms or building blocks of numbers. But in 1859, the great German mathematician Bernhard Riemann hypothesized that the spacing of the primes logically follows from other numbers, now known as the “nontrivial zeros” of the Riemann zeta function. The Riemann zeta function takes inputs not only real numbers but also  complex numbers — meaning they have both “real” and “imaginary” components — and yields other numbers
Read more

How to make a cool million Dollars by solving a math problem?

-
Image
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of June 2020, six of the problems remain unsolved. A correct solution to any of the problems results in a US $1,000,000 prize sometimes called a Millennium Prize being awarded by the institute.  Solved Problem: -  Poincare conjecture. When the Clay institute first announced these seven problems, it was generally perceived by the mathematics community that these seven problems would remain unsolved for the foreseeable future. To their surprise, Grigori Perelman a recluse Russian mathematician solved the Poincare conjecture and in July 2010 he famously declined the 1 million dollar prize!   Furthermore, in August 2006, he declined the Fields medal which is considered even more prestigious than the Nobel Prize as it is given every four years to a mathematician under the age of 40. The Nobel Prize is given every year with no age limit. In a seri
Read more